| 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 |
| 75.1-a1 |
75.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
75.1 |
\( 3 \cdot 5^{2} \) |
\( 3^{32} \cdot 5^{2} \) |
$1.98537$ |
$(4a+13), (5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$2.547989231$ |
2.699915346 |
\( -\frac{147281603041}{215233605} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( -1329051 a - 4352529\) , \( 3040012667 a + 9955789822\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-1329051a-4352529\right){x}+3040012667a+9955789822$ |
| 75.1-a2 |
75.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
75.1 |
\( 3 \cdot 5^{2} \) |
\( 3^{2} \cdot 5^{2} \) |
$1.98537$ |
$(4a+13), (5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$10.19195692$ |
2.699915346 |
\( -\frac{1}{15} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( -251 a - 819\) , \( -680263 a - 2227808\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-251a-819\right){x}-680263a-2227808$ |
| 75.1-a3 |
75.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
75.1 |
\( 3 \cdot 5^{2} \) |
\( 3^{4} \cdot 5^{16} \) |
$1.98537$ |
$(4a+13), (5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$2.547989231$ |
2.699915346 |
\( \frac{4733169839}{3515625} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( 422549 a + 1383816\) , \( 155447492 a + 509077665\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(422549a+1383816\right){x}+155447492a+509077665$ |
| 75.1-a4 |
75.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
75.1 |
\( 3 \cdot 5^{2} \) |
\( 3^{8} \cdot 5^{8} \) |
$1.98537$ |
$(4a+13), (5)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$10.19195692$ |
2.699915346 |
\( \frac{111284641}{50625} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( -121051 a - 396429\) , \( 20377967 a + 66736152\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-121051a-396429\right){x}+20377967a+66736152$ |
| 75.1-a5 |
75.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
75.1 |
\( 3 \cdot 5^{2} \) |
\( 3^{4} \cdot 5^{4} \) |
$1.98537$ |
$(4a+13), (5)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{3} \) |
$1$ |
$10.19195692$ |
2.699915346 |
\( \frac{13997521}{225} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( -60651 a - 198624\) , \( -15687988 a - 51376865\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-60651a-198624\right){x}-15687988a-51376865$ |
| 75.1-a6 |
75.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
75.1 |
\( 3 \cdot 5^{2} \) |
\( 3^{16} \cdot 5^{4} \) |
$1.98537$ |
$(4a+13), (5)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$10.19195692$ |
2.699915346 |
\( \frac{272223782641}{164025} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( -1631051 a - 5341554\) , \( 2198868842 a + 7201113427\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-1631051a-5341554\right){x}+2198868842a+7201113427$ |
| 75.1-a7 |
75.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
75.1 |
\( 3 \cdot 5^{2} \) |
\( 3^{2} \cdot 5^{2} \) |
$1.98537$ |
$(4a+13), (5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \) |
$1$ |
$2.547989231$ |
2.699915346 |
\( \frac{56667352321}{15} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( -966651 a - 3165699\) , \( -1006909063 a - 3297543830\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-966651a-3165699\right){x}-1006909063a-3297543830$ |
| 75.1-a8 |
75.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
75.1 |
\( 3 \cdot 5^{2} \) |
\( 3^{8} \cdot 5^{2} \) |
$1.98537$ |
$(4a+13), (5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$10.19195692$ |
2.699915346 |
\( \frac{1114544804970241}{405} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( -26093051 a - 85452579\) , \( 140914623017 a + 461483725132\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-26093051a-85452579\right){x}+140914623017a+461483725132$ |
| 75.1-b1 |
75.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
75.1 |
\( 3 \cdot 5^{2} \) |
\( 3^{32} \cdot 5^{2} \) |
$1.98537$ |
$(4a+13), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$16.44430019$ |
$0.490422220$ |
1.068189016 |
\( -\frac{147281603041}{215233605} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -110\) , \( -880\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-110{x}-880$ |
| 75.1-b2 |
75.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
75.1 |
\( 3 \cdot 5^{2} \) |
\( 3^{2} \cdot 5^{2} \) |
$1.98537$ |
$(4a+13), (5)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.027768762$ |
$31.38702211$ |
1.068189016 |
\( -\frac{1}{15} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 0\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}$ |
| 75.1-b3 |
75.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
75.1 |
\( 3 \cdot 5^{2} \) |
\( 3^{4} \cdot 5^{16} \) |
$1.98537$ |
$(4a+13), (5)$ |
$1$ |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$8.222150098$ |
$1.961688882$ |
1.068189016 |
\( \frac{4733169839}{3515625} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 35\) , \( -28\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+35{x}-28$ |
| 75.1-b4 |
75.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
75.1 |
\( 3 \cdot 5^{2} \) |
\( 3^{8} \cdot 5^{8} \) |
$1.98537$ |
$(4a+13), (5)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$4.111075049$ |
$7.846755528$ |
1.068189016 |
\( \frac{111284641}{50625} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -10\) , \( -10\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-10{x}-10$ |
| 75.1-b5 |
75.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
75.1 |
\( 3 \cdot 5^{2} \) |
\( 3^{4} \cdot 5^{4} \) |
$1.98537$ |
$(4a+13), (5)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$2.055537524$ |
$31.38702211$ |
1.068189016 |
\( \frac{13997521}{225} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -5\) , \( 2\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-5{x}+2$ |
| 75.1-b6 |
75.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
75.1 |
\( 3 \cdot 5^{2} \) |
\( 3^{16} \cdot 5^{4} \) |
$1.98537$ |
$(4a+13), (5)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$8.222150098$ |
$1.961688882$ |
1.068189016 |
\( \frac{272223782641}{164025} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -135\) , \( -660\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-135{x}-660$ |
| 75.1-b7 |
75.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
75.1 |
\( 3 \cdot 5^{2} \) |
\( 3^{2} \cdot 5^{2} \) |
$1.98537$ |
$(4a+13), (5)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.027768762$ |
$31.38702211$ |
1.068189016 |
\( \frac{56667352321}{15} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -80\) , \( 242\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-80{x}+242$ |
| 75.1-b8 |
75.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
75.1 |
\( 3 \cdot 5^{2} \) |
\( 3^{8} \cdot 5^{2} \) |
$1.98537$ |
$(4a+13), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$16.44430019$ |
$0.490422220$ |
1.068189016 |
\( \frac{1114544804970241}{405} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -2160\) , \( -39540\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-2160{x}-39540$ |
*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.
