| 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 |
| 17.2-a1 |
17.2-a |
$6$ |
$8$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
17.2 |
\( 17 \) |
\( 17^{4} \) |
$5.76248$ |
$(-a^3-a^2+3a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$610.8649797$ |
1.687292068 |
\( \frac{68499872256}{83521} a^{3} + \frac{126928507008}{83521} a^{2} - \frac{40134547968}{83521} a - \frac{74168666944}{83521} \) |
\( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( 1\) , \( a^{3} - 2 a + 1\) , \( 5 a^{3} + 4 a^{2} - 17 a - 14\) , \( 9 a^{3} + 7 a^{2} - 31 a - 25\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-2a-2\right){x}{y}+\left(a^{3}-2a+1\right){y}={x}^{3}+{x}^{2}+\left(5a^{3}+4a^{2}-17a-14\right){x}+9a^{3}+7a^{2}-31a-25$ |
| 17.2-a2 |
17.2-a |
$6$ |
$8$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
17.2 |
\( 17 \) |
\( 17^{2} \) |
$5.76248$ |
$(-a^3-a^2+3a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$610.8649797$ |
1.687292068 |
\( -\frac{5305459200}{289} a^{3} + \frac{9552369536}{289} a^{2} + \frac{3373619712}{289} a - \frac{5376130368}{289} \) |
\( \bigl[a^{2} - 2\) , \( a^{2} - a - 2\) , \( a^{2} + a - 1\) , \( 17 a^{3} + 15 a^{2} - 60 a - 51\) , \( 54 a^{3} + 42 a^{2} - 186 a - 145\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}+a-1\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(17a^{3}+15a^{2}-60a-51\right){x}+54a^{3}+42a^{2}-186a-145$ |
| 17.2-a3 |
17.2-a |
$6$ |
$8$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
17.2 |
\( 17 \) |
\( 17 \) |
$5.76248$ |
$(-a^3-a^2+3a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 1 \) |
$1$ |
$76.35812246$ |
1.687292068 |
\( -\frac{8388075833640}{17} a^{3} - \frac{6438155264624}{17} a^{2} + \frac{28638682336544}{17} a + \frac{21981236916776}{17} \) |
\( \bigl[a^{2} + a - 2\) , \( -a^{3} + a^{2} + 4 a - 3\) , \( a^{2} + a - 1\) , \( 26 a^{3} + 20 a^{2} - 88 a - 68\) , \( 108 a^{3} + 83 a^{2} - 369 a - 283\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{2}+a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-3\right){x}^{2}+\left(26a^{3}+20a^{2}-88a-68\right){x}+108a^{3}+83a^{2}-369a-283$ |
| 17.2-a4 |
17.2-a |
$6$ |
$8$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
17.2 |
\( 17 \) |
\( 17 \) |
$5.76248$ |
$(-a^3-a^2+3a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$305.4324898$ |
1.687292068 |
\( -\frac{57784725759971544}{17} a^{3} + \frac{106772250842960368}{17} a^{2} + \frac{33849508652174560}{17} a - \frac{62545736458709272}{17} \) |
\( \bigl[a^{2} + a - 2\) , \( -a^{2} - a + 2\) , \( a^{2} + a - 1\) , \( 44 a^{3} + 32 a^{2} - 152 a - 113\) , \( 463 a^{3} + 364 a^{2} - 1582 a - 1245\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{2}+a-1\right){y}={x}^{3}+\left(-a^{2}-a+2\right){x}^{2}+\left(44a^{3}+32a^{2}-152a-113\right){x}+463a^{3}+364a^{2}-1582a-1245$ |
| 17.2-a5 |
17.2-a |
$6$ |
$8$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
17.2 |
\( 17 \) |
\( 17^{2} \) |
$5.76248$ |
$(-a^3-a^2+3a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$152.7162449$ |
1.687292068 |
\( \frac{2038265836077624}{289} a^{3} + \frac{3766223840964720}{289} a^{2} - \frac{1193988845046000}{289} a - \frac{2206202569644280}{289} \) |
\( \bigl[a^{3} - 3 a\) , \( -a^{3} - a^{2} + 3 a + 3\) , \( a^{3} - 3 a + 1\) , \( -11 a^{3} - 14 a^{2} + 19 a + 14\) , \( -22 a^{3} - 36 a^{2} + 23 a + 27\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{3}-3a+1\right){y}={x}^{3}+\left(-a^{3}-a^{2}+3a+3\right){x}^{2}+\left(-11a^{3}-14a^{2}+19a+14\right){x}-22a^{3}-36a^{2}+23a+27$ |
| 17.2-a6 |
17.2-a |
$6$ |
$8$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
17.2 |
\( 17 \) |
\( 17^{8} \) |
$5.76248$ |
$(-a^3-a^2+3a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$152.7162449$ |
1.687292068 |
\( -\frac{113010507685176}{6975757441} a^{3} - \frac{82992167336432}{6975757441} a^{2} + \frac{372591802841328}{6975757441} a + \frac{292134350112776}{6975757441} \) |
\( \bigl[a\) , \( -a^{3} + 3 a + 1\) , \( 1\) , \( 3 a^{3} + 2 a^{2} - 12 a - 8\) , \( -3 a^{3} - 3 a^{2} + 9 a + 7\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a^{3}+3a+1\right){x}^{2}+\left(3a^{3}+2a^{2}-12a-8\right){x}-3a^{3}-3a^{2}+9a+7$ |
| 17.2-b1 |
17.2-b |
$8$ |
$12$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
17.2 |
\( 17 \) |
\( 17^{3} \) |
$5.76248$ |
$(-a^3-a^2+3a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$14.66721799$ |
0.729231279 |
\( \frac{18927597887502661144}{4913} a^{3} - \frac{14486119363825311696}{4913} a^{2} - \frac{64624003054350579920}{4913} a + \frac{49460814680853475416}{4913} \) |
\( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( -a + 1\) , \( a^{3} + a^{2} - 2 a - 1\) , \( 71 a^{3} + 86 a^{2} - 241 a - 307\) , \( -545 a^{3} - 262 a^{2} + 1871 a + 864\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(71a^{3}+86a^{2}-241a-307\right){x}-545a^{3}-262a^{2}+1871a+864$ |
| 17.2-b2 |
17.2-b |
$8$ |
$12$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
17.2 |
\( 17 \) |
\( 17 \) |
$5.76248$ |
$(-a^3-a^2+3a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$1188.044657$ |
0.729231279 |
\( \frac{4110080}{17} a^{3} + \frac{4283264}{17} a^{2} - \frac{5993984}{17} a + \frac{235456}{17} \) |
\( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( a^{3} + a^{2} - 2 a - 1\) , \( a^{2} - 1\) , \( 2 a^{3} + 4 a^{2} - 2 a - 3\) , \( a^{3} + 2 a^{2} + a\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-2a-2\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}+a^{2}-2a-1\right){x}^{2}+\left(2a^{3}+4a^{2}-2a-3\right){x}+a^{3}+2a^{2}+a$ |
| 17.2-b3 |
17.2-b |
$8$ |
$12$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
17.2 |
\( 17 \) |
\( 17^{2} \) |
$5.76248$ |
$(-a^3-a^2+3a+1)$ |
0 |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$2376.089315$ |
0.729231279 |
\( -\frac{9703168}{289} a^{3} - \frac{7079808}{289} a^{2} + \frac{32808448}{289} a + \frac{25407424}{289} \) |
\( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( 1\) , \( a^{3} - 2 a + 1\) , \( -3 a^{3} + 2 a^{2} + 11 a - 7\) , \( -a^{3} + a^{2} + 4 a - 3\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-2a-2\right){x}{y}+\left(a^{3}-2a+1\right){y}={x}^{3}+{x}^{2}+\left(-3a^{3}+2a^{2}+11a-7\right){x}-a^{3}+a^{2}+4a-3$ |
| 17.2-b4 |
17.2-b |
$8$ |
$12$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
17.2 |
\( 17 \) |
\( 17^{3} \) |
$5.76248$ |
$(-a^3-a^2+3a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$14.66721799$ |
0.729231279 |
\( \frac{2506487730351891471104}{4913} a^{3} + \frac{4631385425125526708608}{4913} a^{2} - \frac{1468266518518419684352}{4913} a - \frac{2713002769461421185600}{4913} \) |
\( \bigl[a^{2} - 2\) , \( a^{3} + a^{2} - 2 a - 3\) , \( a^{2} + a - 1\) , \( 2 a^{3} - 52 a^{2} + 5 a + 29\) , \( 112 a^{3} - 458 a^{2} - 64 a + 266\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}+a-1\right){y}={x}^{3}+\left(a^{3}+a^{2}-2a-3\right){x}^{2}+\left(2a^{3}-52a^{2}+5a+29\right){x}+112a^{3}-458a^{2}-64a+266$ |
| 17.2-b5 |
17.2-b |
$8$ |
$12$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
17.2 |
\( 17 \) |
\( 17^{6} \) |
$5.76248$ |
$(-a^3-a^2+3a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.2 |
$9$ |
\( 2 \) |
$1$ |
$29.33443598$ |
0.729231279 |
\( \frac{6050416701310208}{24137569} a^{3} + \frac{9306221858122368}{24137569} a^{2} - \frac{5580466233834496}{24137569} a - \frac{3892474815436352}{24137569} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + 4 a\) , \( a^{2} + a - 1\) , \( 6 a^{3} - 5 a^{2} - 34 a - 22\) , \( 24 a^{3} - 9 a^{2} - 138 a - 90\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}+a-1\right){y}={x}^{3}+\left(-a^{3}+4a\right){x}^{2}+\left(6a^{3}-5a^{2}-34a-22\right){x}+24a^{3}-9a^{2}-138a-90$ |
| 17.2-b6 |
17.2-b |
$8$ |
$12$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
17.2 |
\( 17 \) |
\( 17^{4} \) |
$5.76248$ |
$(-a^3-a^2+3a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$594.0223287$ |
0.729231279 |
\( -\frac{245431085672}{83521} a^{3} + \frac{453481639872}{83521} a^{2} + \frac{143774643536}{83521} a - \frac{265495846472}{83521} \) |
\( \bigl[a^{2} + a - 2\) , \( -a^{3} + a^{2} + 2 a - 1\) , \( a^{3} + a^{2} - 2 a - 1\) , \( 8 a^{3} + 7 a^{2} - 31 a - 23\) , \( 6 a^{3} + 5 a^{2} - 23 a - 18\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+2a-1\right){x}^{2}+\left(8a^{3}+7a^{2}-31a-23\right){x}+6a^{3}+5a^{2}-23a-18$ |
| 17.2-b7 |
17.2-b |
$8$ |
$12$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
17.2 |
\( 17 \) |
\( 17^{12} \) |
$5.76248$ |
$(-a^3-a^2+3a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \) |
$1$ |
$7.333608997$ |
0.729231279 |
\( -\frac{61187891699800870166168}{582622237229761} a^{3} - \frac{46831566314755966139568}{582622237229761} a^{2} + \frac{208900765111300490455504}{582622237229761} a + \frac{159879749360260815287608}{582622237229761} \) |
\( \bigl[a^{3} - 3 a\) , \( -a^{3} - a^{2} + 3 a + 3\) , \( a^{3} - 2 a + 1\) , \( -6 a^{3} - 7 a^{2} + 46 a - 27\) , \( -38 a^{3} + 19 a^{2} + 166 a - 136\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{3}-2a+1\right){y}={x}^{3}+\left(-a^{3}-a^{2}+3a+3\right){x}^{2}+\left(-6a^{3}-7a^{2}+46a-27\right){x}-38a^{3}+19a^{2}+166a-136$ |
| 17.2-b8 |
17.2-b |
$8$ |
$12$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
17.2 |
\( 17 \) |
\( 17 \) |
$5.76248$ |
$(-a^3-a^2+3a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$1188.044657$ |
0.729231279 |
\( -\frac{82159493144}{17} a^{3} - \frac{62883544896}{17} a^{2} + \frac{280510283696}{17} a + \frac{214698447576}{17} \) |
\( \bigl[a\) , \( -a^{3} + 3 a + 1\) , \( a + 1\) , \( -2 a^{3} - a^{2} + 10 a - 5\) , \( -4 a^{2} + 9 a - 5\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+3a+1\right){x}^{2}+\left(-2a^{3}-a^{2}+10a-5\right){x}-4a^{2}+9a-5$ |
*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.
