| 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 |
| 57.1-a1 |
57.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( 3^{4} \cdot 19^{10} \) |
$1.85372$ |
$(4a+13), (10a-43)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.4.2 |
$4$ |
\( 2^{2} \) |
$1$ |
$3.435258684$ |
7.280178051 |
\( -\frac{9358714467168256}{22284891} \) |
\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( -53035227 a - 173685972\) , \( 408531845304 a + 1337907974141\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-53035227a-173685972\right){x}+408531845304a+1337907974141$ |
| 57.1-a2 |
57.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( 3^{20} \cdot 19^{2} \) |
$1.85372$ |
$(4a+13), (10a-43)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.4.1 |
$4$ |
\( 2^{2} \) |
$1$ |
$3.435258684$ |
7.280178051 |
\( \frac{841232384}{1121931} \) |
\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( 237573 a + 778038\) , \( 139619034 a + 457240781\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(237573a+778038\right){x}+139619034a+457240781$ |
| 57.1-b1 |
57.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( 3^{2} \cdot 19^{2} \) |
$1.85372$ |
$(4a+13), (10a-43)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$5.765563733$ |
3.054670288 |
\( \frac{1413120}{19} a - \frac{18030592}{57} \) |
\( \bigl[0\) , \( a - 1\) , \( 1\) , \( 264 a - 1125\) , \( 4176 a - 17851\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(264a-1125\right){x}+4176a-17851$ |
| 57.1-c1 |
57.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( 3^{2} \cdot 19^{2} \) |
$1.85372$ |
$(4a+13), (10a-43)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$5.765563733$ |
3.054670288 |
\( -\frac{1413120}{19} a - \frac{13791232}{57} \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( -264 a - 861\) , \( -4176 a - 13675\bigr] \) |
${y}^2+{y}={x}^{3}-a{x}^{2}+\left(-264a-861\right){x}-4176a-13675$ |
| 57.1-d1 |
57.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( 3^{2} \cdot 19^{8} \) |
$1.85372$ |
$(4a+13), (10a-43)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.262154252$ |
0.299629650 |
\( \frac{67419143}{390963} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( 102428 a + 335449\) , \( -104265070 a - 341459474\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(102428a+335449\right){x}-104265070a-341459474$ |
| 57.1-d2 |
57.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( - 3 \cdot 19 \) |
$1.85372$ |
$(4a+13), (10a-43)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$36.19446804$ |
0.299629650 |
\( -\frac{276137246}{57} a + \frac{1180481203}{57} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 11 a + 36\) , \( 88 a + 288\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+\left(11a+36\right){x}+88a+288$ |
| 57.1-d3 |
57.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( 3^{2} \cdot 19^{2} \) |
$1.85372$ |
$(4a+13), (10a-43)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$36.19446804$ |
0.299629650 |
\( \frac{389017}{57} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( -18372 a - 60161\) , \( 2300500 a + 7533946\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-18372a-60161\right){x}+2300500a+7533946$ |
| 57.1-d4 |
57.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( 3^{4} \cdot 19^{4} \) |
$1.85372$ |
$(4a+13), (10a-43)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$9.048617011$ |
0.299629650 |
\( \frac{30664297}{3249} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( -78772 a - 257966\) , \( -21027235 a - 68862455\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-78772a-257966\right){x}-21027235a-68862455$ |
| 57.1-d5 |
57.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( - 3 \cdot 19 \) |
$1.85372$ |
$(4a+13), (10a-43)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$36.19446804$ |
0.299629650 |
\( \frac{276137246}{57} a + \frac{904343957}{57} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -11 a + 47\) , \( -88 a + 376\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+\left(-11a+47\right){x}-88a+376$ |
| 57.1-d6 |
57.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( 3^{8} \cdot 19^{2} \) |
$1.85372$ |
$(4a+13), (10a-43)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.262154252$ |
0.299629650 |
\( \frac{115714886617}{1539} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( -1226372 a - 4016261\) , \( -1434654120 a - 4698373480\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-1226372a-4016261\right){x}-1434654120a-4698373480$ |
| 57.1-e1 |
57.1-e |
$1$ |
$1$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( 3^{4} \cdot 19^{2} \) |
$1.85372$ |
$(4a+13), (10a-43)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
|
|
$1$ |
\( 2^{3} \) |
$1.194156810$ |
$3.679988085$ |
9.313015529 |
\( -\frac{1404928}{171} \) |
\( \bigl[0\) , \( a - 1\) , \( 1\) , \( -28187 a - 92304\) , \( -5536728 a - 18132329\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-28187a-92304\right){x}-5536728a-18132329$ |
| 57.1-f1 |
57.1-f |
$1$ |
$1$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( 3^{4} \cdot 19^{2} \) |
$1.85372$ |
$(4a+13), (10a-43)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.011598383$ |
$30.86363048$ |
0.758624779 |
\( -\frac{1404928}{171} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -2\) , \( 2\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}-2{x}+2$ |
| 57.1-g1 |
57.1-g |
$6$ |
$8$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( 3^{2} \cdot 19^{8} \) |
$1.85372$ |
$(4a+13), (10a-43)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$3.892773922$ |
$4.711524088$ |
4.858622600 |
\( \frac{67419143}{390963} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 8\) , \( 29\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+8{x}+29$ |
| 57.1-g2 |
57.1-g |
$6$ |
$8$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( - 3 \cdot 19 \) |
$1.85372$ |
$(4a+13), (10a-43)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$7.785547844$ |
$9.423048177$ |
4.858622600 |
\( -\frac{276137246}{57} a + \frac{1180481203}{57} \) |
\( \bigl[1\) , \( -a - 1\) , \( a\) , \( 543 a - 2315\) , \( 13825 a - 59110\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(543a-2315\right){x}+13825a-59110$ |
| 57.1-g3 |
57.1-g |
$6$ |
$8$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( 3^{2} \cdot 19^{2} \) |
$1.85372$ |
$(4a+13), (10a-43)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$3.892773922$ |
$18.84609635$ |
4.858622600 |
\( \frac{389017}{57} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -2\) , \( -1\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-2{x}-1$ |
| 57.1-g4 |
57.1-g |
$6$ |
$8$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( 3^{4} \cdot 19^{4} \) |
$1.85372$ |
$(4a+13), (10a-43)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1.946386961$ |
$18.84609635$ |
4.858622600 |
\( \frac{30664297}{3249} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -7\) , \( 5\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-7{x}+5$ |
| 57.1-g5 |
57.1-g |
$6$ |
$8$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( - 3 \cdot 19 \) |
$1.85372$ |
$(4a+13), (10a-43)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$7.785547844$ |
$9.423048177$ |
4.858622600 |
\( \frac{276137246}{57} a + \frac{904343957}{57} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( -542 a - 1772\) , \( -14368 a - 47057\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-542a-1772\right){x}-14368a-47057$ |
| 57.1-g6 |
57.1-g |
$6$ |
$8$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( 3^{8} \cdot 19^{2} \) |
$1.85372$ |
$(4a+13), (10a-43)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.973193480$ |
$18.84609635$ |
4.858622600 |
\( \frac{115714886617}{1539} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -102\) , \( 385\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-102{x}+385$ |
| 57.1-h1 |
57.1-h |
$1$ |
$1$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( 3^{2} \cdot 19^{2} \) |
$1.85372$ |
$(4a+13), (10a-43)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.039109271$ |
$22.71862741$ |
0.941487090 |
\( -\frac{1413120}{19} a - \frac{13791232}{57} \) |
\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( -38 a + 166\) , \( 9 a - 40\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-38a+166\right){x}+9a-40$ |
| 57.1-i1 |
57.1-i |
$1$ |
$1$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( 3^{2} \cdot 19^{2} \) |
$1.85372$ |
$(4a+13), (10a-43)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.039109271$ |
$22.71862741$ |
0.941487090 |
\( \frac{1413120}{19} a - \frac{18030592}{57} \) |
\( \bigl[0\) , \( a\) , \( 1\) , \( 38 a + 128\) , \( -9 a - 31\bigr] \) |
${y}^2+{y}={x}^{3}+a{x}^{2}+\left(38a+128\right){x}-9a-31$ |
| 57.1-j1 |
57.1-j |
$2$ |
$5$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( 3^{4} \cdot 19^{10} \) |
$1.85372$ |
$(4a+13), (10a-43)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.1.2 |
$1$ |
\( 2^{3} \) |
$6.359928028$ |
$0.085998375$ |
1.159110933 |
\( -\frac{9358714467168256}{22284891} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -4390\) , \( -113432\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}-4390{x}-113432$ |
| 57.1-j2 |
57.1-j |
$2$ |
$5$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( 3^{20} \cdot 19^{2} \) |
$1.85372$ |
$(4a+13), (10a-43)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 2^{3} \cdot 5 \) |
$1.271985605$ |
$2.149959381$ |
1.159110933 |
\( \frac{841232384}{1121931} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( 20\) , \( -32\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}+20{x}-32$ |
*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.
