| 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 |
| 34.1-a1 |
34.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
34.1 |
\( 2 \cdot 17 \) |
\( 2^{12} \cdot 17^{2} \) |
$3.08193$ |
$(a-7), (17,a)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \cdot 3 \) |
$1.578233770$ |
$20.21098874$ |
2.977710999 |
\( \frac{3048625}{1088} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -3\) , \( 1\bigr] \) |
${y}^2+{x}{y}={x}^3-3{x}+1$ |
| 34.1-a2 |
34.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
34.1 |
\( 2 \cdot 17 \) |
\( 2^{2} \cdot 17^{12} \) |
$3.08193$ |
$(a-7), (17,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$9.469402625$ |
$2.245665415$ |
2.977710999 |
\( \frac{159661140625}{48275138} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -113\) , \( -329\bigr] \) |
${y}^2+{x}{y}={x}^3-113{x}-329$ |
| 34.1-a3 |
34.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
34.1 |
\( 2 \cdot 17 \) |
\( 2^{6} \cdot 17^{4} \) |
$3.08193$ |
$(a-7), (17,a)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$3.156467541$ |
$20.21098874$ |
2.977710999 |
\( \frac{8805624625}{2312} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -43\) , \( 105\bigr] \) |
${y}^2+{x}{y}={x}^3-43{x}+105$ |
| 34.1-a4 |
34.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
34.1 |
\( 2 \cdot 17 \) |
\( 2^{4} \cdot 17^{6} \) |
$3.08193$ |
$(a-7), (17,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{3} \) |
$4.734701312$ |
$2.245665415$ |
2.977710999 |
\( \frac{120920208625}{19652} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -103\) , \( -411\bigr] \) |
${y}^2+{x}{y}={x}^3-103{x}-411$ |
| 34.1-b1 |
34.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
34.1 |
\( 2 \cdot 17 \) |
\( 2^{12} \cdot 17^{2} \) |
$3.08193$ |
$(a-7), (17,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2^{2} \) |
$1$ |
$13.90059457$ |
3.892945149 |
\( \frac{3048625}{1088} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( 43\) , \( 47\bigr] \) |
${y}^2+w{x}{y}={x}^3-{x}^2+43{x}+47$ |
| 34.1-b2 |
34.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
34.1 |
\( 2 \cdot 17 \) |
\( 2^{2} \cdot 17^{12} \) |
$3.08193$ |
$(a-7), (17,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$16$ |
\( 2^{2} \) |
$1$ |
$3.475148644$ |
3.892945149 |
\( \frac{159661140625}{48275138} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -67\) , \( -63\bigr] \) |
${y}^2+w{x}{y}={x}^3-{x}^2-67{x}-63$ |
| 34.1-b3 |
34.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
34.1 |
\( 2 \cdot 17 \) |
\( 2^{6} \cdot 17^{4} \) |
$3.08193$ |
$(a-7), (17,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$16$ |
\( 2^{2} \) |
$1$ |
$3.475148644$ |
3.892945149 |
\( \frac{8805624625}{2312} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( 3\) , \( -217\bigr] \) |
${y}^2+w{x}{y}={x}^3-{x}^2+3{x}-217$ |
| 34.1-b4 |
34.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
34.1 |
\( 2 \cdot 17 \) |
\( 2^{4} \cdot 17^{6} \) |
$3.08193$ |
$(a-7), (17,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2^{2} \) |
$1$ |
$13.90059457$ |
3.892945149 |
\( \frac{120920208625}{19652} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -57\) , \( 59\bigr] \) |
${y}^2+w{x}{y}={x}^3-{x}^2-57{x}+59$ |
| 34.1-c1 |
34.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
34.1 |
\( 2 \cdot 17 \) |
\( 2^{12} \cdot 3^{12} \cdot 17^{2} \) |
$3.08193$ |
$(a-7), (17,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$0.562629009$ |
$13.90059457$ |
1.095141937 |
\( \frac{3048625}{1088} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -27\) , \( -27\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2-27{x}-27$ |
| 34.1-c2 |
34.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
34.1 |
\( 2 \cdot 17 \) |
\( 2^{2} \cdot 3^{12} \cdot 17^{12} \) |
$3.08193$ |
$(a-7), (17,a)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \cdot 3 \) |
$3.375774058$ |
$3.475148644$ |
1.095141937 |
\( \frac{159661140625}{48275138} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -1017\) , \( 8883\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2-1017{x}+8883$ |
| 34.1-c3 |
34.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
34.1 |
\( 2 \cdot 17 \) |
\( 2^{6} \cdot 3^{12} \cdot 17^{4} \) |
$3.08193$ |
$(a-7), (17,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{3} \) |
$1.125258019$ |
$3.475148644$ |
1.095141937 |
\( \frac{8805624625}{2312} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -387\) , \( -2835\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2-387{x}-2835$ |
| 34.1-c4 |
34.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
34.1 |
\( 2 \cdot 17 \) |
\( 2^{4} \cdot 3^{12} \cdot 17^{6} \) |
$3.08193$ |
$(a-7), (17,a)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1.687887029$ |
$13.90059457$ |
1.095141937 |
\( \frac{120920208625}{19652} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -927\) , \( 11097\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2-927{x}+11097$ |
| 34.1-d1 |
34.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
34.1 |
\( 2 \cdot 17 \) |
\( 2^{12} \cdot 3^{12} \cdot 17^{2} \) |
$3.08193$ |
$(a-7), (17,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$20.21098874$ |
8.490313504 |
\( \frac{3048625}{1088} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 27\) , \( -5\bigr] \) |
${y}^2+w{x}{y}={x}^3+27{x}-5$ |
| 34.1-d2 |
34.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
34.1 |
\( 2 \cdot 17 \) |
\( 2^{2} \cdot 3^{12} \cdot 17^{12} \) |
$3.08193$ |
$(a-7), (17,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$9$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$2.245665415$ |
8.490313504 |
\( \frac{159661140625}{48275138} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( -963\) , \( -12875\bigr] \) |
${y}^2+w{x}{y}={x}^3-963{x}-12875$ |
| 34.1-d3 |
34.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
34.1 |
\( 2 \cdot 17 \) |
\( 2^{6} \cdot 3^{12} \cdot 17^{4} \) |
$3.08193$ |
$(a-7), (17,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$20.21098874$ |
8.490313504 |
\( \frac{8805624625}{2312} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( -333\) , \( 1363\bigr] \) |
${y}^2+w{x}{y}={x}^3-333{x}+1363$ |
| 34.1-d4 |
34.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
34.1 |
\( 2 \cdot 17 \) |
\( 2^{4} \cdot 3^{12} \cdot 17^{6} \) |
$3.08193$ |
$(a-7), (17,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$9$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$2.245665415$ |
8.490313504 |
\( \frac{120920208625}{19652} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( -873\) , \( -14729\bigr] \) |
${y}^2+w{x}{y}={x}^3-873{x}-14729$ |
*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.
