| 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 |
| 81.1-a1 |
81.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{22} \) |
$1.10531$ |
$(3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 2 \) |
$1$ |
$8.687738250$ |
4.214172053 |
\( -\frac{13549359104}{243} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 7152 a - 18327\) , \( -472886 a + 1211329\bigr] \) |
${y}^2+{y}={x}^{3}+\left(7152a-18327\right){x}-472886a+1211329$ |
| 81.1-a2 |
81.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{14} \) |
$1.10531$ |
$(3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B.4.1 |
$1$ |
\( 2 \) |
$1$ |
$8.687738250$ |
4.214172053 |
\( \frac{4096}{3} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -48 a + 123\) , \( -134 a + 343\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-48a+123\right){x}-134a+343$ |
| 81.1-b1 |
81.1-b |
$4$ |
$51$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{6} \) |
$1.10531$ |
$(3)$ |
0 |
$\Z/3\Z$ |
$\textsf{potential}$ |
$-51$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$13.47264434$ |
0.726132492 |
\( -1343913984 a - 2098593792 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -6 a - 12\) , \( 14 a + 19\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-6a-12\right){x}+14a+19$ |
| 81.1-b2 |
81.1-b |
$4$ |
$51$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{18} \) |
$1.10531$ |
$(3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-51$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$1.496960482$ |
0.726132492 |
\( -1343913984 a - 2098593792 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -54 a - 108\) , \( -378 a - 520\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-54a-108\right){x}-378a-520$ |
| 81.1-b3 |
81.1-b |
$4$ |
$51$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{6} \) |
$1.10531$ |
$(3)$ |
0 |
$\Z/3\Z$ |
$\textsf{potential}$ |
$-51$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$13.47264434$ |
0.726132492 |
\( 1343913984 a - 3442507776 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 6 a - 18\) , \( -14 a + 33\bigr] \) |
${y}^2+{y}={x}^{3}+\left(6a-18\right){x}-14a+33$ |
| 81.1-b4 |
81.1-b |
$4$ |
$51$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{18} \) |
$1.10531$ |
$(3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-51$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$1.496960482$ |
0.726132492 |
\( 1343913984 a - 3442507776 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 54 a - 162\) , \( 378 a - 898\bigr] \) |
${y}^2+{y}={x}^{3}+\left(54a-162\right){x}+378a-898$ |
| 81.1-c1 |
81.1-c |
$8$ |
$16$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( - 3^{14} \) |
$1.10531$ |
$(3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$6.828410025$ |
0.414033173 |
\( -\frac{396321250}{3} a + 338397375 \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 435 a - 1115\) , \( -7202 a + 18448\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(435a-1115\right){x}-7202a+18448$ |
| 81.1-c2 |
81.1-c |
$8$ |
$16$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( - 3^{14} \) |
$1.10531$ |
$(3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.707102506$ |
0.414033173 |
\( -\frac{31564213125250}{3} a + \frac{80853398915125}{3} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 285 a - 725\) , \( 3676 a - 9366\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(285a-725\right){x}+3676a-9366$ |
| 81.1-c3 |
81.1-c |
$8$ |
$16$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{28} \) |
$1.10531$ |
$(3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.707102506$ |
0.414033173 |
\( \frac{5359375}{6561} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 525 a + 820\) , \( -10064 a - 15716\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(525a+820\right){x}-10064a-15716$ |
| 81.1-c4 |
81.1-c |
$8$ |
$16$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{20} \) |
$1.10531$ |
$(3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$6.828410025$ |
0.414033173 |
\( \frac{274625}{81} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 195 a - 500\) , \( 1442 a - 3694\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(195a-500\right){x}+1442a-3694$ |
| 81.1-c5 |
81.1-c |
$8$ |
$16$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{16} \) |
$1.10531$ |
$(3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$6.828410025$ |
0.414033173 |
\( -\frac{14326000}{9} a + \frac{36913625}{9} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 15 a - 50\) , \( 58 a - 132\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(15a-50\right){x}+58a-132$ |
| 81.1-c6 |
81.1-c |
$8$ |
$16$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{16} \) |
$1.10531$ |
$(3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$6.828410025$ |
0.414033173 |
\( \frac{14326000}{9} a + \frac{22587625}{9} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -15 a - 35\) , \( -58 a - 74\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-15a-35\right){x}-58a-74$ |
| 81.1-c7 |
81.1-c |
$8$ |
$16$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( - 3^{14} \) |
$1.10531$ |
$(3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$6.828410025$ |
0.414033173 |
\( \frac{396321250}{3} a + \frac{618870875}{3} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -435 a - 680\) , \( 7202 a + 11246\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-435a-680\right){x}+7202a+11246$ |
| 81.1-c8 |
81.1-c |
$8$ |
$16$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( - 3^{14} \) |
$1.10531$ |
$(3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.707102506$ |
0.414033173 |
\( \frac{31564213125250}{3} a + 16429728596625 \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -285 a - 440\) , \( -3676 a - 5690\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-285a-440\right){x}-3676a-5690$ |
*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.
