| Label |
Class |
Class size |
Class degree |
Base field |
Field degree |
Field signature |
Conductor |
Conductor norm |
Discriminant norm |
Root analytic conductor |
Bad primes |
Rank |
Torsion |
CM |
CM |
Sato-Tate |
$\Q$-curve |
Base change |
Semistable |
Potentially good |
Nonmax $\ell$ |
mod-$\ell$ images |
$Ш_{\textrm{an}}$ |
Tamagawa |
Regulator |
Period |
Leading coeff |
j-invariant |
Weierstrass coefficients |
Weierstrass equation |
| 27.2-a6 |
27.2-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.2 |
\( 3^{3} \) |
\( 2^{12} \cdot 3^{23} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.589586966$ |
0.662903589 |
\( \frac{67902559538}{43046721} a + \frac{66394340881}{43046721} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -17 a - 78\) , \( -130 a - 80\bigr] \) |
${y}^2+a{x}{y}={x}^3+a{x}^2+\left(-17a-78\right){x}-130a-80$ |
| 27.3-a6 |
27.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 3^{23} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.589586966$ |
0.662903589 |
\( \frac{67902559538}{43046721} a + \frac{66394340881}{43046721} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( -8 a - 4\) , \( -22 a + 43\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(a+1\right){x}^2+\left(-8a-4\right){x}-22a+43$ |
| 144.14-a6 |
144.14-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
144.14 |
\( 2^{4} \cdot 3^{2} \) |
\( 2^{12} \cdot 3^{17} \) |
$1.48454$ |
$(2,a+1), (3,a), (3,a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.335501588$ |
$1.376622694$ |
1.533399817 |
\( \frac{67902559538}{43046721} a + \frac{66394340881}{43046721} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( 2 a - 29\) , \( -4 a + 68\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-a+1\right){x}^2+\left(2a-29\right){x}-4a+68$ |
| 144.2-b6 |
144.2-b |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
144.2 |
\( 2^{4} \cdot 3^{2} \) |
\( 2^{24} \cdot 3^{17} \) |
$1.48454$ |
$(2,a), (3,a), (3,a+2)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$0.333875397$ |
$1.376622694$ |
1.533399817 |
\( \frac{67902559538}{43046721} a + \frac{66394340881}{43046721} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -48 a + 74\) , \( 9 a - 475\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+{x}^2+\left(-48a+74\right){x}+9a-475$ |
| 576.2-d6 |
576.2-d |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
576.2 |
\( 2^{6} \cdot 3^{2} \) |
\( 2^{30} \cdot 3^{17} \) |
$2.09946$ |
$(2,a), (3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$0.973419242$ |
3.247551087 |
\( \frac{67902559538}{43046721} a + \frac{66394340881}{43046721} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 77 a + 125\) , \( 85 a - 1363\bigr] \) |
${y}^2={x}^3+{x}^2+\left(77a+125\right){x}+85a-1363$ |
| 576.20-d6 |
576.20-d |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
576.20 |
\( 2^{6} \cdot 3^{2} \) |
\( 2^{30} \cdot 3^{17} \) |
$2.09946$ |
$(2,a+1), (3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$0.973419242$ |
3.247551087 |
\( \frac{67902559538}{43046721} a + \frac{66394340881}{43046721} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -103 a + 97\) , \( -139 a + 1517\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-103a+97\right){x}-139a+1517$ |
| 1296.23-b6 |
1296.23-b |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
1296.23 |
\( 2^{4} \cdot 3^{4} \) |
\( 2^{12} \cdot 3^{29} \) |
$2.57131$ |
$(2,a+1), (3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$0.458874231$ |
1.530910264 |
\( \frac{67902559538}{43046721} a + \frac{66394340881}{43046721} \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 30 a - 282\) , \( 57 a - 1519\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3-{x}^2+\left(30a-282\right){x}+57a-1519$ |
| 1296.3-b6 |
1296.3-b |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
1296.3 |
\( 2^{4} \cdot 3^{4} \) |
\( 2^{24} \cdot 3^{29} \) |
$2.57131$ |
$(2,a), (3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$0.458874231$ |
1.530910264 |
\( \frac{67902559538}{43046721} a + \frac{66394340881}{43046721} \) |
\( \bigl[a\) , \( -a\) , \( 0\) , \( -426 a + 639\) , \( 177 a + 12227\bigr] \) |
${y}^2+a{x}{y}={x}^3-a{x}^2+\left(-426a+639\right){x}+177a+12227$ |
| 1728.2-c6 |
1728.2-c |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
1728.2 |
\( 2^{6} \cdot 3^{3} \) |
\( 2^{30} \cdot 3^{23} \) |
$2.76305$ |
$(2,a), (3,a), (3,a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$3.372475190$ |
$0.562003861$ |
3.161652482 |
\( \frac{67902559538}{43046721} a + \frac{66394340881}{43046721} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 308 a - 141\) , \( -2589 a - 2995\bigr] \) |
${y}^2+a{x}{y}={x}^3+\left(308a-141\right){x}-2589a-2995$ |
| 1728.26-e6 |
1728.26-e |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
1728.26 |
\( 2^{6} \cdot 3^{3} \) |
\( 2^{30} \cdot 3^{23} \) |
$2.76305$ |
$(2,a+1), (3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.562003861$ |
0.937487247 |
\( \frac{67902559538}{43046721} a + \frac{66394340881}{43046721} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -194 a + 669\) , \( -2189 a - 4654\bigr] \) |
${y}^2={x}^3-a{x}^2+\left(-194a+669\right){x}-2189a-4654$ |
| 1728.27-c6 |
1728.27-c |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
1728.27 |
\( 2^{6} \cdot 3^{3} \) |
\( 2^{18} \cdot 3^{23} \) |
$2.76305$ |
$(2,a+1), (3,a), (3,a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{7} \) |
$0.210779699$ |
$0.562003861$ |
3.161652482 |
\( \frac{67902559538}{43046721} a + \frac{66394340881}{43046721} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( 11 a + 178\) , \( -364 a + 459\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3-a{x}^2+\left(11a+178\right){x}-364a+459$ |
| 1728.3-e6 |
1728.3-e |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
1728.3 |
\( 2^{6} \cdot 3^{3} \) |
\( 2^{18} \cdot 3^{23} \) |
$2.76305$ |
$(2,a), (3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.562003861$ |
0.937487247 |
\( \frac{67902559538}{43046721} a + \frac{66394340881}{43046721} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( 53 a - 158\) , \( -358 a + 780\bigr] \) |
${y}^2+a{x}{y}={x}^3-{x}^2+\left(53a-158\right){x}-358a+780$ |
*The rank, regulator and analytic order of Ш are
not known for all curves in the database; curves for which these are
unknown will not appear in searches specifying one of these
quantities.
