| 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 |
| 17.1-a1 |
17.1-a |
$6$ |
$8$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17 \) |
$5.76248$ |
$(a^2+a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 1 \) |
$1$ |
$76.35812246$ |
1.687292068 |
\( \frac{3474454835624}{17} a^{3} + \frac{6438155264624}{17} a^{2} - \frac{2035288673232}{17} a - \frac{3771384141720}{17} \) |
\( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( -a^{3} - a^{2} + 3 a + 1\) , \( a + 1\) , \( -12 a^{3} - 23 a^{2} + 11 a + 20\) , \( -38 a^{3} - 68 a^{2} + 26 a + 39\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}-a^{2}+3a+1\right){x}^{2}+\left(-12a^{3}-23a^{2}+11a+20\right){x}-38a^{3}-68a^{2}+26a+39$ |
| 17.1-a2 |
17.1-a |
$6$ |
$8$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17^{4} \) |
$5.76248$ |
$(a^2+a-3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$610.8649797$ |
1.687292068 |
\( \frac{165365068800}{83521} a^{3} - \frac{126928507008}{83521} a^{2} - \frac{564595078656}{83521} a + \frac{433545361088}{83521} \) |
\( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( a^{3} - a^{2} - 4 a + 3\) , \( a^{3} - 2 a + 1\) , \( -a^{3} - 7 a^{2} - 4 a + 8\) , \( -4 a^{3} - 8 a^{2} + 2 a + 5\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-2a-2\right){x}{y}+\left(a^{3}-2a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+3\right){x}^{2}+\left(-a^{3}-7a^{2}-4a+8\right){x}-4a^{3}-8a^{2}+2a+5$ |
| 17.1-a3 |
17.1-a |
$6$ |
$8$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17^{2} \) |
$5.76248$ |
$(a^2+a-3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$610.8649797$ |
1.687292068 |
\( -\frac{12542757888}{289} a^{3} - \frac{9552369536}{289} a^{2} + \frac{42933732864}{289} a + \frac{32833347776}{289} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} - a^{2} + 3 a + 2\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -8 a^{3} - 16 a^{2} + 6 a + 11\) , \( -23 a^{3} - 43 a^{2} + 14 a + 25\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(-a^{3}-a^{2}+3a+2\right){x}^{2}+\left(-8a^{3}-16a^{2}+6a+11\right){x}-23a^{3}-43a^{2}+14a+25$ |
| 17.1-a4 |
17.1-a |
$6$ |
$8$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17 \) |
$5.76248$ |
$(a^2+a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$305.4324898$ |
1.687292068 |
\( -\frac{139504668627740072}{17} a^{3} - \frac{106772250842960368}{17} a^{2} + \frac{476298731643191760}{17} a + \frac{364543266913132200}{17} \) |
\( \bigl[a^{2} + a - 2\) , \( a^{3} - 3 a - 1\) , \( a^{3} + a^{2} - 3 a - 1\) , \( 3 a^{3} + 3 a^{2} - 13 a - 14\) , \( -15 a^{3} - 14 a^{2} + 47 a + 39\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{3}-3a-1\right){x}^{2}+\left(3a^{3}+3a^{2}-13a-14\right){x}-15a^{3}-14a^{2}+47a+39$ |
| 17.1-a5 |
17.1-a |
$6$ |
$8$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17^{8} \) |
$5.76248$ |
$(a^2+a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$152.7162449$ |
1.687292068 |
\( \frac{33560279785800}{6975757441} a^{3} + \frac{82992167336432}{6975757441} a^{2} + \frac{12329668327776}{6975757441} a - \frac{39834319232952}{6975757441} \) |
\( \bigl[a^{3} - 3 a\) , \( a + 1\) , \( 1\) , \( -3 a^{3} - 2 a^{2} + 6 a\) , \( 3 a^{2} + 3 a - 5\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-3a^{3}-2a^{2}+6a\right){x}+3a^{2}+3a-5$ |
| 17.1-a6 |
17.1-a |
$6$ |
$8$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17^{2} \) |
$5.76248$ |
$(a^2+a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$152.7162449$ |
1.687292068 |
\( \frac{4920808663186872}{289} a^{3} - \frac{3766223840964720}{289} a^{2} - \frac{16800691825638240}{289} a + \frac{12858692794214600}{289} \) |
\( \bigl[a\) , \( a^{2} + a - 1\) , \( a + 1\) , \( -14 a^{3} + 14 a^{2} + 52 a - 42\) , \( -43 a^{3} + 36 a^{2} + 150 a - 117\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}+a-1\right){x}^{2}+\left(-14a^{3}+14a^{2}+52a-42\right){x}-43a^{3}+36a^{2}+150a-117$ |
| 17.1-b1 |
17.1-b |
$8$ |
$12$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17^{4} \) |
$5.76248$ |
$(a^2+a-3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$594.0223287$ |
0.729231279 |
\( -\frac{592518613480}{83521} a^{3} - \frac{453481639872}{83521} a^{2} + \frac{2022986926112}{83521} a + \frac{1548430713016}{83521} \) |
\( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( -a^{2} + 3\) , \( a^{3} + a^{2} - 2 a - 1\) , \( -5 a^{3} - 9 a^{2} + 5 a + 9\) , \( -4 a^{3} - 7 a^{2} + 4 a + 6\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(-5a^{3}-9a^{2}+5a+9\right){x}-4a^{3}-7a^{2}+4a+6$ |
| 17.1-b2 |
17.1-b |
$8$ |
$12$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17 \) |
$5.76248$ |
$(a^2+a-3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$1188.044657$ |
0.729231279 |
\( \frac{6336256}{17} a^{3} - \frac{4283264}{17} a^{2} - \frac{23118848}{17} a + \frac{17368512}{17} \) |
\( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( -a^{3} + a^{2} + 4 a - 1\) , \( a^{2} - 1\) , \( 2 a^{2} + 3 a + 1\) , \( a^{3} + 2 a^{2} + a\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-2a-2\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-1\right){x}^{2}+\left(2a^{2}+3a+1\right){x}+a^{3}+2a^{2}+a$ |
| 17.1-b3 |
17.1-b |
$8$ |
$12$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17^{2} \) |
$5.76248$ |
$(a^2+a-3)$ |
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 |
\( \frac{3698944}{289} a^{3} + \frac{7079808}{289} a^{2} - \frac{1393664}{289} a - \frac{2911808}{289} \) |
\( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( -a^{3} - a^{2} + 3 a + 1\) , \( a^{3} - 2 a + 1\) , \( 8 a^{3} + 5 a^{2} - 29 a - 21\) , \( -9 a^{3} - 7 a^{2} + 30 a + 22\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-2a-2\right){x}{y}+\left(a^{3}-2a+1\right){y}={x}^{3}+\left(-a^{3}-a^{2}+3a+1\right){x}^{2}+\left(8a^{3}+5a^{2}-29a-21\right){x}-9a^{3}-7a^{2}+30a+22$ |
| 17.1-b4 |
17.1-b |
$8$ |
$12$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17^{6} \) |
$5.76248$ |
$(a^2+a-3)$ |
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 |
\( \frac{12570783870096128}{24137569} a^{3} - \frac{9306221858122368}{24137569} a^{2} - \frac{43762768311598592}{24137569} a + \frac{33332412617053120}{24137569} \) |
\( \bigl[a^{2} - 2\) , \( a^{3} - 2 a\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -15 a^{3} + 4 a^{2} + 38 a - 40\) , \( -65 a^{3} + 8 a^{2} + 170 a - 124\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{3}-2a\right){x}^{2}+\left(-15a^{3}+4a^{2}+38a-40\right){x}-65a^{3}+8a^{2}+170a-124$ |
| 17.1-b5 |
17.1-b |
$8$ |
$12$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17^{3} \) |
$5.76248$ |
$(a^2+a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$14.66721799$ |
0.729231279 |
\( \frac{6051196672537254728960}{4913} a^{3} - \frac{4631385425125526708608}{4913} a^{2} - \frac{20660077747963655657984}{4913} a + \frac{15812538931040685648832}{4913} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} - a^{2} + 3 a + 2\) , \( a^{3} - 3 a + 1\) , \( 18 a^{3} - 33 a^{2} - 10 a + 18\) , \( 90 a^{3} - 160 a^{2} - 59 a + 84\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-3a+1\right){y}={x}^{3}+\left(-a^{3}-a^{2}+3a+2\right){x}^{2}+\left(18a^{3}-33a^{2}-10a+18\right){x}+90a^{3}-160a^{2}-59a+84$ |
| 17.1-b6 |
17.1-b |
$8$ |
$12$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17^{3} \) |
$5.76248$ |
$(a^2+a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$14.66721799$ |
0.729231279 |
\( -\frac{7841209391842596488}{4913} a^{3} + \frac{14486119363825311696}{4913} a^{2} + \frac{4596030288025128320}{4913} a - \frac{8483662774447771368}{4913} \) |
\( \bigl[a^{2} + a - 2\) , \( -a^{3} + 3 a + 1\) , \( a^{3} + a^{2} - 2 a - 1\) , \( -27 a^{3} - 88 a^{2} + 8 a + 41\) , \( 237 a^{3} + 260 a^{2} - 168 a - 180\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(-a^{3}+3a+1\right){x}^{2}+\left(-27a^{3}-88a^{2}+8a+41\right){x}+237a^{3}+260a^{2}-168a-180$ |
| 17.1-b7 |
17.1-b |
$8$ |
$12$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17 \) |
$5.76248$ |
$(a^2+a-3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$1188.044657$ |
0.729231279 |
\( \frac{34031804264}{17} a^{3} + \frac{62883544896}{17} a^{2} - \frac{19935919648}{17} a - \frac{36835732008}{17} \) |
\( \bigl[a^{3} - 3 a\) , \( a + 1\) , \( a^{3} - 3 a + 1\) , \( 4 a^{3} + a^{2} - 10 a - 9\) , \( 9 a^{3} + 4 a^{2} - 27 a - 21\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{3}-3a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(4a^{3}+a^{2}-10a-9\right){x}+9a^{3}+4a^{2}-27a-21$ |
| 17.1-b8 |
17.1-b |
$8$ |
$12$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17^{12} \) |
$5.76248$ |
$(a^2+a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \) |
$1$ |
$7.333608997$ |
0.729231279 |
\( \frac{25337090011897879957000}{582622237229761} a^{3} + \frac{46831566314755966139568}{582622237229761} a^{2} - \frac{14823378335892769704832}{582622237229761} a - \frac{27446515898763049270664}{582622237229761} \) |
\( \bigl[a\) , \( a^{2} + a - 1\) , \( a^{3} - 2 a + 1\) , \( 28 a^{3} + 6 a^{2} - 79 a - 53\) , \( 52 a^{3} - 20 a^{2} - 119 a - 58\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-2a+1\right){y}={x}^{3}+\left(a^{2}+a-1\right){x}^{2}+\left(28a^{3}+6a^{2}-79a-53\right){x}+52a^{3}-20a^{2}-119a-58$ |
*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.
