| 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 |
| 1.1-a1 |
1.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
1.1 |
\( 1 \) |
\( 7^{12} \) |
$0.94146$ |
$\textsf{none}$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$3, 5$ |
3Cn, 5B |
$1$ |
\( 1 \) |
$0.205572591$ |
$13.17843113$ |
1.028554772 |
\( 4096 \) |
\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( -3 a - 16\) , \( 7 a - 31\bigr] \) |
${y}^2+{y}={x}^3+\left(-a+1\right){x}^2+\left(-3a-16\right){x}+7a-31$ |
| 1.1-a2 |
1.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
1.1 |
\( 1 \) |
\( 7^{12} \) |
$0.94146$ |
$\textsf{none}$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$3, 5$ |
3Cn, 5B |
$1$ |
\( 1 \) |
$1.027862956$ |
$2.635686227$ |
1.028554772 |
\( 38477541376 \) |
\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( -563 a - 1486\) , \( -14287 a + 263\bigr] \) |
${y}^2+{y}={x}^3+\left(-a+1\right){x}^2+\left(-563a-1486\right){x}-14287a+263$ |
| 1.1-b1 |
1.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
1.1 |
\( 1 \) |
\( 7^{12} \) |
$0.94146$ |
$\textsf{none}$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$3, 5$ |
3Cn, 5B |
$1$ |
\( 1 \) |
$0.205572591$ |
$13.17843113$ |
1.028554772 |
\( 4096 \) |
\( \bigl[0\) , \( a\) , \( 1\) , \( 3 a - 19\) , \( -7 a - 24\bigr] \) |
${y}^2+{y}={x}^3+a{x}^2+\left(3a-19\right){x}-7a-24$ |
| 1.1-b2 |
1.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
1.1 |
\( 1 \) |
\( 7^{12} \) |
$0.94146$ |
$\textsf{none}$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$3, 5$ |
3Cn, 5B |
$1$ |
\( 1 \) |
$1.027862956$ |
$2.635686227$ |
1.028554772 |
\( 38477541376 \) |
\( \bigl[0\) , \( a\) , \( 1\) , \( 563 a - 2049\) , \( 14287 a - 14024\bigr] \) |
${y}^2+{y}={x}^3+a{x}^2+\left(563a-2049\right){x}+14287a-14024$ |
| 3.1-a1 |
3.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
3.1 |
\( 3 \) |
\( 3^{6} \cdot 7^{12} \) |
$1.23903$ |
$(3,a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2Cs, 3Cn |
$1$ |
\( 2 \) |
$3.388142012$ |
$8.084873186$ |
1.299999940 |
\( \frac{1331}{27} \) |
\( \bigl[1\) , \( -a\) , \( 1\) , \( 2 a - 5\) , \( 15 a + 11\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-a{x}^2+\left(2a-5\right){x}+15a+11$ |
| 3.1-a2 |
3.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
3.1 |
\( 3 \) |
\( 3^{12} \cdot 7^{12} \) |
$1.23903$ |
$(3,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2Cs, 3Cn |
$1$ |
\( 2 \) |
$1.694071006$ |
$4.042436593$ |
1.299999940 |
\( \frac{12008989}{729} \) |
\( \bigl[1\) , \( -a\) , \( 1\) , \( -38 a - 110\) , \( 291 a - 465\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-a{x}^2+\left(-38a-110\right){x}+291a-465$ |
| 3.1-a3 |
3.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
3.1 |
\( 3 \) |
\( 2^{12} \cdot 3^{3} \) |
$1.23903$ |
$(3,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2Cs, 3Cn |
$1$ |
\( 1 \) |
$1.694071006$ |
$8.084873186$ |
1.299999940 |
\( \frac{154154}{9} a + \frac{258641}{9} \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( -3 a + 20\) , \( 3 a - 17\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3-{x}^2+\left(-3a+20\right){x}+3a-17$ |
| 3.1-a4 |
3.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
3.1 |
\( 3 \) |
\( 3^{3} \cdot 5^{12} \) |
$1.23903$ |
$(3,a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2Cs, 3Cn |
$1$ |
\( 1 \) |
$6.776284025$ |
$8.084873186$ |
1.299999940 |
\( -\frac{154154}{9} a + \frac{412795}{9} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( -8 a - 10\) , \( -18 a + 141\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(a-1\right){x}^2+\left(-8a-10\right){x}-18a+141$ |
| 3.1-b1 |
3.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
3.1 |
\( 3 \) |
\( 3^{6} \cdot 7^{12} \) |
$1.23903$ |
$(3,a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2Cs, 3Cn |
$1$ |
\( 2 \) |
$3.388142012$ |
$8.084873186$ |
1.299999940 |
\( \frac{1331}{27} \) |
\( \bigl[1\) , \( a - 1\) , \( 1\) , \( -2 a - 3\) , \( -15 a + 26\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+\left(a-1\right){x}^2+\left(-2a-3\right){x}-15a+26$ |
| 3.1-b2 |
3.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
3.1 |
\( 3 \) |
\( 3^{12} \cdot 7^{12} \) |
$1.23903$ |
$(3,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2Cs, 3Cn |
$1$ |
\( 2 \) |
$1.694071006$ |
$4.042436593$ |
1.299999940 |
\( \frac{12008989}{729} \) |
\( \bigl[1\) , \( a - 1\) , \( 1\) , \( 38 a - 148\) , \( -291 a - 174\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+\left(a-1\right){x}^2+\left(38a-148\right){x}-291a-174$ |
| 3.1-b3 |
3.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
3.1 |
\( 3 \) |
\( 3^{3} \cdot 5^{12} \) |
$1.23903$ |
$(3,a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2Cs, 3Cn |
$1$ |
\( 1 \) |
$6.776284025$ |
$8.084873186$ |
1.299999940 |
\( \frac{154154}{9} a + \frac{258641}{9} \) |
\( \bigl[1\) , \( -a\) , \( 0\) , \( 8 a - 18\) , \( 18 a + 123\bigr] \) |
${y}^2+{x}{y}={x}^3-a{x}^2+\left(8a-18\right){x}+18a+123$ |
| 3.1-b4 |
3.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
3.1 |
\( 3 \) |
\( 2^{12} \cdot 3^{3} \) |
$1.23903$ |
$(3,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2Cs, 3Cn |
$1$ |
\( 1 \) |
$1.694071006$ |
$8.084873186$ |
1.299999940 |
\( -\frac{154154}{9} a + \frac{412795}{9} \) |
\( \bigl[a\) , \( -a\) , \( 0\) , \( 3 a + 17\) , \( -3 a - 14\bigr] \) |
${y}^2+a{x}{y}={x}^3-a{x}^2+\left(3a+17\right){x}-3a-14$ |
| 12.2-a1 |
12.2-a |
$2$ |
$5$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{50} \cdot 3^{2} \cdot 7^{12} \) |
$1.75225$ |
$(2,a), (2,a+1), (3,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$5$ |
5B |
$1$ |
\( 2 \cdot 5^{2} \) |
$0.261617015$ |
$0.624987031$ |
3.103884118 |
\( -\frac{136511322949}{100663296} \) |
\( \bigl[1\) , \( a + 1\) , \( 1\) , \( -857 a - 2262\) , \( -43379 a + 24135\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+\left(a+1\right){x}^2+\left(-857a-2262\right){x}-43379a+24135$ |
| 12.2-a2 |
12.2-a |
$2$ |
$5$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{10} \cdot 3^{10} \cdot 7^{12} \) |
$1.75225$ |
$(2,a), (2,a+1), (3,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$5$ |
5B |
$1$ |
\( 2 \cdot 5^{2} \) |
$0.052323403$ |
$3.124935156$ |
3.103884118 |
\( -\frac{1295029}{7776} \) |
\( \bigl[1\) , \( a + 1\) , \( 1\) , \( -17 a - 57\) , \( 259 a + 27\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+\left(a+1\right){x}^2+\left(-17a-57\right){x}+259a+27$ |
| 12.2-b1 |
12.2-b |
$2$ |
$5$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{50} \cdot 3^{2} \cdot 7^{12} \) |
$1.75225$ |
$(2,a), (2,a+1), (3,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$5$ |
5B |
$1$ |
\( 2 \cdot 5^{2} \) |
$0.261617015$ |
$0.624987031$ |
3.103884118 |
\( -\frac{136511322949}{100663296} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( 859 a - 3120\) , \( 42521 a - 16124\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(-a-1\right){x}^2+\left(859a-3120\right){x}+42521a-16124$ |
| 12.2-b2 |
12.2-b |
$2$ |
$5$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{10} \cdot 3^{10} \cdot 7^{12} \) |
$1.75225$ |
$(2,a), (2,a+1), (3,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$5$ |
5B |
$1$ |
\( 2 \cdot 5^{2} \) |
$0.052323403$ |
$3.124935156$ |
3.103884118 |
\( -\frac{1295029}{7776} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( 19 a - 75\) , \( -277 a + 361\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(-a-1\right){x}^2+\left(19a-75\right){x}-277a+361$ |
| 14.2-a1 |
14.2-a |
$2$ |
$5$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
14.2 |
\( 2 \cdot 7 \) |
\( 2^{10} \cdot 5^{12} \cdot 7^{5} \) |
$1.82109$ |
$(2,a), (7,a+6)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B |
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$3.306981059$ |
6.277695028 |
\( -\frac{6874449269}{17210368} a - \frac{2990176993}{2458624} \) |
\( \bigl[a + 1\) , \( -1\) , \( 1\) , \( -24 a - 22\) , \( 150 a + 193\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^3-{x}^2+\left(-24a-22\right){x}+150a+193$ |
| 14.2-a2 |
14.2-a |
$2$ |
$5$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
14.2 |
\( 2 \cdot 7 \) |
\( 2^{50} \cdot 5^{12} \cdot 7 \) |
$1.82109$ |
$(2,a), (7,a+6)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B |
$1$ |
\( 2 \cdot 5^{2} \) |
$1$ |
$0.661396211$ |
6.277695028 |
\( \frac{141298995282504251}{7881299347898368} a + \frac{50655190979215887}{1125899906842624} \) |
\( \bigl[a + 1\) , \( -1\) , \( 1\) , \( -49 a - 1047\) , \( -10468 a - 4645\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^3-{x}^2+\left(-49a-1047\right){x}-10468a-4645$ |
| 14.2-b1 |
14.2-b |
$2$ |
$5$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
14.2 |
\( 2 \cdot 7 \) |
\( 2^{22} \cdot 7^{5} \) |
$1.82109$ |
$(2,a), (7,a+6)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B |
$1$ |
\( 2 \) |
$0.209784422$ |
$3.306981059$ |
0.526785050 |
\( -\frac{6874449269}{17210368} a - \frac{2990176993}{2458624} \) |
\( \bigl[0\) , \( -a + 1\) , \( a\) , \( -3 a + 4\) , \( -4 a - 52\bigr] \) |
${y}^2+a{y}={x}^3+\left(-a+1\right){x}^2+\left(-3a+4\right){x}-4a-52$ |
| 14.2-b2 |
14.2-b |
$2$ |
$5$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
14.2 |
\( 2 \cdot 7 \) |
\( 2^{62} \cdot 7 \) |
$1.82109$ |
$(2,a), (7,a+6)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B |
$1$ |
\( 2 \) |
$1.048922112$ |
$0.661396211$ |
0.526785050 |
\( \frac{141298995282504251}{7881299347898368} a + \frac{50655190979215887}{1125899906842624} \) |
\( \bigl[0\) , \( -a + 1\) , \( a\) , \( -33 a - 36\) , \( -84 a + 3276\bigr] \) |
${y}^2+a{y}={x}^3+\left(-a+1\right){x}^2+\left(-33a-36\right){x}-84a+3276$ |
| 14.3-a1 |
14.3-a |
$2$ |
$5$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
14.3 |
\( 2 \cdot 7 \) |
\( 2^{10} \cdot 5^{12} \cdot 7^{5} \) |
$1.82109$ |
$(2,a+1), (7,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B |
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$3.306981059$ |
6.277695028 |
\( \frac{6874449269}{17210368} a - \frac{6951422055}{4302592} \) |
\( \bigl[a\) , \( -a\) , \( 1\) , \( 23 a - 45\) , \( -150 a + 343\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^3-a{x}^2+\left(23a-45\right){x}-150a+343$ |
| 14.3-a2 |
14.3-a |
$2$ |
$5$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
14.3 |
\( 2 \cdot 7 \) |
\( 2^{50} \cdot 5^{12} \cdot 7 \) |
$1.82109$ |
$(2,a+1), (7,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B |
$1$ |
\( 2 \cdot 5^{2} \) |
$1$ |
$0.661396211$ |
6.277695028 |
\( -\frac{141298995282504251}{7881299347898368} a + \frac{123971333034253865}{1970324836974592} \) |
\( \bigl[a\) , \( -a\) , \( 1\) , \( 48 a - 1095\) , \( 10468 a - 15113\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^3-a{x}^2+\left(48a-1095\right){x}+10468a-15113$ |
| 14.3-b1 |
14.3-b |
$2$ |
$5$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
14.3 |
\( 2 \cdot 7 \) |
\( 2^{22} \cdot 7^{5} \) |
$1.82109$ |
$(2,a+1), (7,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B |
$1$ |
\( 2 \) |
$0.209784422$ |
$3.306981059$ |
0.526785050 |
\( \frac{6874449269}{17210368} a - \frac{6951422055}{4302592} \) |
\( \bigl[0\) , \( a\) , \( a + 1\) , \( 3 a + 1\) , \( 3 a - 56\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^3+a{x}^2+\left(3a+1\right){x}+3a-56$ |
| 14.3-b2 |
14.3-b |
$2$ |
$5$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
14.3 |
\( 2 \cdot 7 \) |
\( 2^{62} \cdot 7 \) |
$1.82109$ |
$(2,a+1), (7,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B |
$1$ |
\( 2 \) |
$1.048922112$ |
$0.661396211$ |
0.526785050 |
\( -\frac{141298995282504251}{7881299347898368} a + \frac{123971333034253865}{1970324836974592} \) |
\( \bigl[0\) , \( a\) , \( a + 1\) , \( 33 a - 69\) , \( 83 a + 3192\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^3+a{x}^2+\left(33a-69\right){x}+83a+3192$ |
| 16.1-a1 |
16.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$1.88291$ |
$(2,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$8.847515954$ |
3.359076204 |
\( 0 \) |
\( \bigl[0\) , \( a + 1\) , \( a\) , \( a - 9\) , \( -a + 3\bigr] \) |
${y}^2+a{y}={x}^3+\left(a+1\right){x}^2+\left(a-9\right){x}-a+3$ |
| 16.1-a2 |
16.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \cdot 3^{12} \) |
$1.88291$ |
$(2,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$8.847515954$ |
3.359076204 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( a\) , \( 0\) , \( -a + 4\bigr] \) |
${y}^2+a{y}={x}^3-a+4$ |
| 16.5-a1 |
16.5-a |
$2$ |
$3$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
16.5 |
\( 2^{4} \) |
\( 2^{8} \cdot 3^{12} \) |
$1.88291$ |
$(2,a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$8.847515954$ |
3.359076204 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( a + 1\) , \( 0\) , \( 3\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^3+3$ |
| 16.5-a2 |
16.5-a |
$2$ |
$3$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
16.5 |
\( 2^{4} \) |
\( 2^{8} \) |
$1.88291$ |
$(2,a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$8.847515954$ |
3.359076204 |
\( 0 \) |
\( \bigl[0\) , \( -a - 1\) , \( a + 1\) , \( a - 9\) , \( 11\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^3+\left(-a-1\right){x}^2+\left(a-9\right){x}+11$ |
| 20.3-a1 |
20.3-a |
$3$ |
$9$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
20.3 |
\( 2^{2} \cdot 5 \) |
\( 2^{12} \cdot 5^{6} \cdot 7^{12} \) |
$1.99094$ |
$(2,a), (2,a+1), (5,a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$2.928844251$ |
0.370658125 |
\( -\frac{2813494887}{8000000} a - \frac{5213014499}{2000000} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( 38 a + 206\) , \( 372 a - 2412\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^3+\left(a+1\right){x}^2+\left(38a+206\right){x}+372a-2412$ |
| 20.3-a2 |
20.3-a |
$3$ |
$9$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
20.3 |
\( 2^{2} \cdot 5 \) |
\( 2^{4} \cdot 5^{18} \cdot 7^{12} \) |
$1.99094$ |
$(2,a), (2,a+1), (5,a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$0.976281417$ |
0.370658125 |
\( \frac{1847775539549013}{30517578125000} a + \frac{5248838253585713}{3814697265625} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( -252 a - 1744\) , \( -4846 a + 13502\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^3+\left(a+1\right){x}^2+\left(-252a-1744\right){x}-4846a+13502$ |
| 20.3-a3 |
20.3-a |
$3$ |
$9$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
20.3 |
\( 2^{2} \cdot 5 \) |
\( 2^{36} \cdot 5^{2} \cdot 7^{12} \) |
$1.99094$ |
$(2,a), (2,a+1), (5,a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$0.976281417$ |
0.370658125 |
\( \frac{87441895536793}{3355443200} a + \frac{60246101822861}{838860800} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( -962 a + 3831\) , \( -2777 a - 168971\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^3+\left(a+1\right){x}^2+\left(-962a+3831\right){x}-2777a-168971$ |
| 20.3-b1 |
20.3-b |
$3$ |
$9$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
20.3 |
\( 2^{2} \cdot 5 \) |
\( 2^{12} \cdot 5^{6} \cdot 7^{12} \) |
$1.99094$ |
$(2,a), (2,a+1), (5,a+1)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3^{3} \) |
$0.658483436$ |
$2.928844251$ |
4.393300252 |
\( -\frac{2813494887}{8000000} a - \frac{5213014499}{2000000} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( 21 a - 295\) , \( -285 a + 2085\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(21a-295\right){x}-285a+2085$ |
| 20.3-b2 |
20.3-b |
$3$ |
$9$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
20.3 |
\( 2^{2} \cdot 5 \) |
\( 2^{4} \cdot 5^{18} \cdot 7^{12} \) |
$1.99094$ |
$(2,a), (2,a+1), (5,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1.975450310$ |
$0.976281417$ |
4.393300252 |
\( \frac{1847775539549013}{30517578125000} a + \frac{5248838253585713}{3814697265625} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -189 a + 2155\) , \( 4125 a - 10949\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(-189a+2155\right){x}+4125a-10949$ |
| 20.3-b3 |
20.3-b |
$3$ |
$9$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
20.3 |
\( 2^{2} \cdot 5 \) |
\( 2^{36} \cdot 5^{2} \cdot 7^{12} \) |
$1.99094$ |
$(2,a), (2,a+1), (5,a+1)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3^{5} \) |
$0.219494478$ |
$0.976281417$ |
4.393300252 |
\( \frac{87441895536793}{3355443200} a + \frac{60246101822861}{838860800} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( 1021 a + 2330\) , \( 2197 a + 180872\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(1021a+2330\right){x}+2197a+180872$ |
| 20.4-a1 |
20.4-a |
$3$ |
$9$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
20.4 |
\( 2^{2} \cdot 5 \) |
\( 2^{12} \cdot 5^{6} \cdot 7^{12} \) |
$1.99094$ |
$(2,a), (2,a+1), (5,a+3)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3^{3} \) |
$0.658483436$ |
$2.928844251$ |
4.393300252 |
\( \frac{2813494887}{8000000} a - \frac{23665552883}{8000000} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -21 a - 274\) , \( 285 a + 1800\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(-21a-274\right){x}+285a+1800$ |
| 20.4-a2 |
20.4-a |
$3$ |
$9$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
20.4 |
\( 2^{2} \cdot 5 \) |
\( 2^{4} \cdot 5^{18} \cdot 7^{12} \) |
$1.99094$ |
$(2,a), (2,a+1), (5,a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1.975450310$ |
$0.976281417$ |
4.393300252 |
\( -\frac{1847775539549013}{30517578125000} a + \frac{43838481568234717}{30517578125000} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( 189 a + 1966\) , \( -4125 a - 6824\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(189a+1966\right){x}-4125a-6824$ |
| 20.4-a3 |
20.4-a |
$3$ |
$9$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
20.4 |
\( 2^{2} \cdot 5 \) |
\( 2^{36} \cdot 5^{2} \cdot 7^{12} \) |
$1.99094$ |
$(2,a), (2,a+1), (5,a+3)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3^{5} \) |
$0.219494478$ |
$0.976281417$ |
4.393300252 |
\( -\frac{87441895536793}{3355443200} a + \frac{328426302828237}{3355443200} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -1021 a + 3351\) , \( -2197 a + 183069\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(-1021a+3351\right){x}-2197a+183069$ |
| 20.4-b1 |
20.4-b |
$3$ |
$9$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
20.4 |
\( 2^{2} \cdot 5 \) |
\( 2^{12} \cdot 5^{6} \cdot 7^{12} \) |
$1.99094$ |
$(2,a), (2,a+1), (5,a+3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$2.928844251$ |
0.370658125 |
\( \frac{2813494887}{8000000} a - \frac{23665552883}{8000000} \) |
\( \bigl[1\) , \( -a - 1\) , \( a\) , \( -37 a + 243\) , \( -335 a - 2283\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^3+\left(-a-1\right){x}^2+\left(-37a+243\right){x}-335a-2283$ |
| 20.4-b2 |
20.4-b |
$3$ |
$9$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
20.4 |
\( 2^{2} \cdot 5 \) |
\( 2^{4} \cdot 5^{18} \cdot 7^{12} \) |
$1.99094$ |
$(2,a), (2,a+1), (5,a+3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$0.976281417$ |
0.370658125 |
\( -\frac{1847775539549013}{30517578125000} a + \frac{43838481568234717}{30517578125000} \) |
\( \bigl[1\) , \( -a - 1\) , \( a\) , \( 253 a - 1997\) , \( 4593 a + 10653\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^3+\left(-a-1\right){x}^2+\left(253a-1997\right){x}+4593a+10653$ |
| 20.4-b3 |
20.4-b |
$3$ |
$9$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
20.4 |
\( 2^{2} \cdot 5 \) |
\( 2^{36} \cdot 5^{2} \cdot 7^{12} \) |
$1.99094$ |
$(2,a), (2,a+1), (5,a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$0.976281417$ |
0.370658125 |
\( -\frac{87441895536793}{3355443200} a + \frac{328426302828237}{3355443200} \) |
\( \bigl[1\) , \( -a - 1\) , \( a\) , \( 963 a + 2868\) , \( 1814 a - 174616\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^3+\left(-a-1\right){x}^2+\left(963a+2868\right){x}+1814a-174616$ |
| 23.1-a1 |
23.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
23.1 |
\( 23 \) |
\( 2^{12} \cdot 23^{4} \) |
$2.06173$ |
$(23,a+10)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.575582415$ |
$4.727680842$ |
2.066256180 |
\( \frac{102662144}{279841} a + \frac{38141952}{279841} \) |
\( \bigl[0\) , \( -a + 1\) , \( a\) , \( -a - 1\) , \( 3\bigr] \) |
${y}^2+a{y}={x}^3+\left(-a+1\right){x}^2+\left(-a-1\right){x}+3$ |
| 23.1-b1 |
23.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
23.1 |
\( 23 \) |
\( 5^{12} \cdot 23^{4} \) |
$2.06173$ |
$(23,a+10)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.143946843$ |
$4.727680842$ |
2.066991953 |
\( \frac{102662144}{279841} a + \frac{38141952}{279841} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -10 a + 7\) , \( -7 a + 156\bigr] \) |
${y}^2+{y}={x}^3-{x}^2+\left(-10a+7\right){x}-7a+156$ |
| 23.2-a1 |
23.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
23.2 |
\( 23 \) |
\( 2^{12} \cdot 23^{4} \) |
$2.06173$ |
$(23,a+12)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.575582415$ |
$4.727680842$ |
2.066256180 |
\( -\frac{102662144}{279841} a + \frac{140804096}{279841} \) |
\( \bigl[0\) , \( a\) , \( a + 1\) , \( a - 2\) , \( -a + 3\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^3+a{x}^2+\left(a-2\right){x}-a+3$ |
| 23.2-b1 |
23.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
23.2 |
\( 23 \) |
\( 5^{12} \cdot 23^{4} \) |
$2.06173$ |
$(23,a+12)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.143946843$ |
$4.727680842$ |
2.066991953 |
\( -\frac{102662144}{279841} a + \frac{140804096}{279841} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( 10 a - 3\) , \( 7 a + 149\bigr] \) |
${y}^2+{y}={x}^3-{x}^2+\left(10a-3\right){x}+7a+149$ |
| 24.2-a1 |
24.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
24.2 |
\( 2^{3} \cdot 3 \) |
\( 2^{16} \cdot 3 \cdot 5^{12} \) |
$2.08378$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$4.982543650$ |
5.675065374 |
\( \frac{13475}{768} a + \frac{301151}{192} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -2 a + 52\) , \( -4 a - 136\bigr] \) |
${y}^2+a{x}{y}={x}^3+\left(-a-1\right){x}^2+\left(-2a+52\right){x}-4a-136$ |
| 24.2-a2 |
24.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
24.2 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{14} \) |
$2.08378$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$4.982543650$ |
5.675065374 |
\( \frac{991221}{16} a + \frac{1829963}{12} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -6 a + 3\) , \( -3 a + 24\bigr] \) |
${y}^2+a{x}{y}={x}^3+\left(-a+1\right){x}^2+\left(-6a+3\right){x}-3a+24$ |
| 24.2-b1 |
24.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
24.2 |
\( 2^{3} \cdot 3 \) |
\( 2^{28} \cdot 3 \) |
$2.08378$ |
$(2,a), (2,a+1), (3,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$2.052332333$ |
$4.982543650$ |
1.941186693 |
\( \frac{13475}{768} a + \frac{301151}{192} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -a - 11\) , \( 2 a - 14\bigr] \) |
${y}^2={x}^3-a{x}^2+\left(-a-11\right){x}+2a-14$ |
| 24.2-b2 |
24.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
24.2 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \) |
$2.08378$ |
$(2,a), (2,a+1), (3,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.026166166$ |
$4.982543650$ |
1.941186693 |
\( \frac{991221}{16} a + \frac{1829963}{12} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( -6 a + 11\) , \( 4 a + 36\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(a+1\right){x}^2+\left(-6a+11\right){x}+4a+36$ |
| 24.3-a1 |
24.3-a |
$2$ |
$2$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
24.3 |
\( 2^{3} \cdot 3 \) |
\( 2^{16} \cdot 3 \cdot 5^{12} \) |
$2.08378$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$4.982543650$ |
5.675065374 |
\( -\frac{13475}{768} a + \frac{1218079}{768} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 2 a + 49\) , \( 6 a - 91\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+{x}^2+\left(2a+49\right){x}+6a-91$ |
| 24.3-a2 |
24.3-a |
$2$ |
$2$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
24.3 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{14} \) |
$2.08378$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$4.982543650$ |
5.675065374 |
\( -\frac{991221}{16} a + \frac{10293515}{48} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 6 a - 3\) , \( 3 a + 21\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(6a-3\right){x}+3a+21$ |
*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.
