I will show images first (explanation - after):
![ellipse with 4 lattice points](https://cdn.statically.io/img/i.sstatic.net/MPLq4.png)
![ellipse with 6 lattice points](https://cdn.statically.io/img/i.sstatic.net/D0vjt.png)
![ellipse with 7 lattice points (by square)](https://cdn.statically.io/img/i.sstatic.net/pQTG9.png)
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/gJc4S.png)
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/878gB.png)
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/aIfZl.png)
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/KPEvs.png)
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/H1t9l.png)
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/mRSjf.png)
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/GbBgA.png)
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/naM7A.png)
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/jippR.png)
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/pNgiP.png)
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/Wr5nW.png)
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/GEE7s.png)
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/pytOV.png)
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/HpbQL.png)
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/i66zb.png)
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/Lh93u.png)
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/aARDC.png)
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/Cq60S.png)
And a few ellipses with 36, 48 lattice points:
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/vWw9V.png)
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/Jur6G.png)
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/NNnpM.png)
We will consider ellipses with format
$a_{11} x^2 + a_{12} xy + a_{22} y^2 + a_{13}x +a_{23} y + a_{33} = 0$,
where
1) $0 < a_{11} \le a_{22}$ (otherwise we can make exchange $x' = y$, $y'=x$);
2) $a_{12} \le 0$ (otherwise we can replace $y' = -y$).
We'll denote:
$a$ $-$ semi-major axis;
$b$ $-$ semi-minor axis;
$S$ $-$ square (area);
$M = \max \{ |a_{11}|, |a_{12}|,|a_{22}|,|a_{13}|,|a_{23}|,|a_{33}| \}$;
$\Sigma = |a_{11}|+|a_{12}| + |a_{22}| + |a_{13}| + |a_{23}| + |a_{33}|$.
I see 2 (or more) kinds of "smallness":
A. by semi-major axis ($a$);
B. by square ($S$);
(C). (by max coefficient);
(D). (by sum of coefficients).
A. Priorities to minimize:
$a \rightarrow S \rightarrow M \rightarrow \Sigma \rightarrow |a_{13}|+|a_{23}| \rightarrow \max\{|a_{13}|,|a_{23}|\} \rightarrow (a_{13} \le 0)$ ;
B. Priorities to minimize:
$S \rightarrow a \rightarrow M \rightarrow \Sigma \rightarrow |a_{13}|+|a_{23}| \rightarrow \max\{|a_{13}|,|a_{23}|\} \rightarrow (a_{13} \le 0)$;
For example, if we will have big amount of ellipses with the same $a$ and $S$, then we will choose ellipse(s) with smallest $M$.
If there are a few ellipses with the same $a$, $S$, $M$, $\Sigma$, then we'll choose ellipse(s) with smallest sum $|a_{13}|+|a_{23}|$.
... while we will get 1 ellipse.
How to calculate $a, S$, if we know ellipse formula?
We will denote
$$
I_1 = 2a_{11}+ 2a_{22} > 0,
$$
$$
I_2 = \left| \begin{array}{cc} 2a_{11} & a_{12} \\ a_{12} & 2a_{22} \end{array} \right| = 4a_{11}a_{22}-a_{12}^2 > 0,
$$
$$
I_3 = - \left| \begin{array}{ccc} 2a_{11} & a_{12} & a_{13} \\ a_{12} & 2a_{22} & a_{23} \\ a_{13} & a_{23} & 2a_{33} \end{array} \right| > 0;
$$
then we can rewrite ellipse equation:
$$
a_{11} x^2 + a_{12} xy + a_{22} y^2 = \frac{I_3}{I_2};
$$
let $\lambda_a, \lambda_b$ are roots of equation
$$
\left| \begin{array}{cc} 2a_{11}-\lambda & a_{12} \\ a_{12} & 2a_{22}-\lambda \end{array} \right| = \lambda^2 - I_1 \lambda + I_2 = 0,
$$
so,
$\lambda_a = (a_{11}+a_{22})+\sqrt{(a_{22}-a_{11})^2+a_{12}^2}$,
$\lambda_b = (a_{11}+a_{22})-\sqrt{(a_{22}-a_{11})^2+a_{12}^2}$,
then
$\displaystyle a = \frac{\sqrt{I_3 \lambda_a}}{I_2}$,
$\displaystyle b = \frac{\sqrt{I_3 \lambda_b}}{I_2}$,
$\displaystyle S = \pi a b = \pi \frac{I_3}{I_2 \sqrt{I_2}}$.
Here is the table of smallest (by $a$ or by $S$) ellipses with $n$
lattice points (up to $n=25$).
\begin{array}{|l|l|l|r|r|}
\hline
\mathrm{Lattice} & \mathrm{equation} & \min a / & a & S \\
\mathrm{points} & & \min S & & \\
\hline 3 & x^2 +xy + y^2 -x-y=0 & S & 0.81649658 & 1.20919958 \\
\hline 4 & x^2 + y^2 -x-y=0 & a, S & 0.70710678 & 1.57079633 \\
\hline 5 & 2x^2 -xy + 2y^2 -x+y-3=0 & a, S & 1.46059349 & 5.19139671 \\
\hline 6 & x^2 -xy + y^2 -1=0 & a, S & 1.41421356 & 3.62759873 \\
\hline 7 & 2x^2 -xy + 2y^2 -5x-4y-6=0 & a & 2.92118697 & 20.76558682 \\
& 2x^2 -xy + 4y^2 -7x-7y-7=0 & S & 3.09818451 & 20.38568713 \\
\hline 8 & x^2+y^2-x-y-2=0 & a, S & 1.58113883 & 7.85398163 \\
\hline 9 & x^2-xy+3y^2-7x-6y=0 & a, S & 4.81580610 & 38.75014739 \\
\hline 10 & x^2-xy+4y^2-5x-5y-6=0 & a, S & 4.17287179 & 25.95698353 \\
\hline 11 & 3x^2-3xy+4y^2-21x-21y-10=0 & a, S & 8.00878321 & 123.82952163 \\
\hline 12 & x^2+y^2-5x-5y=0 & a & 3.53553391 & 39.26990817 \\
& x^2-xy+y^2-2x-2y-3=0 & S & 3.74165739 & 25.39319110 \\
\hline 13 & 2x^2-xy+3y^2-31x-26y-25=0 & a, S & 11.67004201 & 319.90071698 \\
\hline 14 & x^2-xy+4y^2-12x-9y-13=0 & a, S & 8.34574359 & 103.82793410 \\
\hline 15 & 2x^2-xy+5y^2-39x-36y-41=0 & a, S & 13.28106447 & 340.53118449 \\
\hline 16 & x^2+y^2-7x-7y-8=0 & a, S & 5.70087713 & 102.10176124 \\
\hline 17 & 2x^2-xy+3y^2-51x-51y-54=0 & a, S & 20.21310568 & 959.70215095 \\
\hline 18 & x^2-xy+y^2-7x-7y=0 & a, S & 9.89949494 & 177.75233769 \\
\hline 19 & 2x^2-xy+3y^2-61x-61y-6=0 & a, S & 23.34008401 & 1279.60286793 \\
\hline 20 & 2x^2-xy+2y^2-27x-27y-29=0 & a & 13.46600658 & 441.26871995 \\
& x^2+5y^2-31x-25y-12=0 & S & 16.83745824 & 398.30699525 \\
\hline 21 & 2x^2-xy+5y^2-81x-81y-8=0 & a, S & 26.56212894 & 1362.12473794 \\
\hline 22 & 4x^2-2xy+9y^2-194x-179y-30=0 & a & 31.84451541 & 2050.29169319 \\
& x^2-xy+10y^2-73x-61y-24=0 & S & 40.56562661 & 1609.78378121 \\
\hline 23 & 2x^2-xy+3y^2-105x-105y-54=0 & a & 40.42621136 & 3838.80860378 \\
& 3x^2-3xy+4y^2-120x-121y-3=0 & S & 44.04830766 & 3745.84302935 \\
\hline 24 & x^2+y^2-17x-17y-18=0 & a & 12.74754878 & 510.50880621 \\
& x^2-xy+y^2-9x-9y-10=0 & S & 13.49073756 & 330.11148429 \\
\hline 25 & 3x^2-xy+8y^2-279x-271y-84=0 & a, S & 57.49719096 & 6287.89822644 \\
\hline
\cdots & \cdots & \cdots & \cdots & \cdots \\
\end{array}