| 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 |
| 16.5-a1 |
16.5-a |
$2$ |
$2$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
16.5 |
\( 2^{4} \) |
\( - 2^{8} \) |
$2.52156$ |
$(-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$152.8488957$ |
2.149605541 |
\( 510768 a^{2} - 1452320 a + 597920 \) |
\( \bigl[a^{2} + a - 2\) , \( a^{2} - a - 3\) , \( a + 1\) , \( 29 a^{2} - 12 a - 116\) , \( -136 a^{2} + 77 a + 587\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(29a^{2}-12a-116\right){x}-136a^{2}+77a+587$ |
| 16.5-a2 |
16.5-a |
$2$ |
$2$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
16.5 |
\( 2^{4} \) |
\( 2^{4} \) |
$2.52156$ |
$(-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$152.8488957$ |
2.149605541 |
\( 342336 a^{2} + 460416 a - 292544 \) |
\( \bigl[a^{2} + a - 2\) , \( -a + 1\) , \( a + 1\) , \( -2775 a^{2} + 1470 a + 11796\) , \( -1678490 a^{2} + 888455 a + 7132145\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-2775a^{2}+1470a+11796\right){x}-1678490a^{2}+888455a+7132145$ |
| 16.5-b1 |
16.5-b |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
16.5 |
\( 2^{4} \) |
\( 2^{16} \) |
$2.52156$ |
$(-a+1)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$200.5904149$ |
0.705255777 |
\( \frac{200481}{16} a^{2} - \frac{325027}{8} a + \frac{233973}{8} \) |
\( \bigl[a^{2} - 3\) , \( a^{2} - 3\) , \( a^{2} - 3\) , \( 100434688208 a^{2} - 53161745688 a - 426761105098\) , \( -23577610985047683 a^{2} + 12480020414213596 a + 100184582628814686\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(100434688208a^{2}-53161745688a-426761105098\right){x}-23577610985047683a^{2}+12480020414213596a+100184582628814686$ |
| 16.5-b2 |
16.5-b |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
16.5 |
\( 2^{4} \) |
\( 2^{13} \) |
$2.52156$ |
$(-a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$50.14760373$ |
0.705255777 |
\( \frac{89069256294443128323}{2} a^{2} - 125302919348047971845 a + 49111676741552886754 \) |
\( \bigl[a^{2} - 3\) , \( -a - 1\) , \( a + 1\) , \( 213 a^{2} - 108 a - 917\) , \( -2667 a^{2} + 1411 a + 11362\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(213a^{2}-108a-917\right){x}-2667a^{2}+1411a+11362$ |
| 16.5-b3 |
16.5-b |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
16.5 |
\( 2^{4} \) |
\( 2^{28} \) |
$2.52156$ |
$(-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$12.53690093$ |
0.705255777 |
\( \frac{4108077233}{65536} a^{2} - \frac{5810733523}{32768} a + \frac{2294990397}{32768} \) |
\( \bigl[a^{2} - 3\) , \( -a^{2} + a + 2\) , \( a^{2} + a - 2\) , \( -109074459285818 a^{2} + 57734919768569 a + 463472706625929\) , \( 1508268935406496095109 a^{2} - 798351754803931876384 a - 6408846675837241310282\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(-109074459285818a^{2}+57734919768569a+463472706625929\right){x}+1508268935406496095109a^{2}-798351754803931876384a-6408846675837241310282$ |
| 16.5-b4 |
16.5-b |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
16.5 |
\( 2^{4} \) |
\( - 2^{13} \) |
$2.52156$ |
$(-a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$100.2952074$ |
0.705255777 |
\( -\frac{10154621765056003}{2} a^{2} + 2687505089603077 a + 21574207961612782 \) |
\( \bigl[a^{2} - 3\) , \( -a^{2} + a + 2\) , \( 0\) , \( 297980666 a^{2} - 157726108 a - 1266161726\) , \( -4093950113647 a^{2} + 2166995673374 a + 17395769388977\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(297980666a^{2}-157726108a-1266161726\right){x}-4093950113647a^{2}+2166995673374a+17395769388977$ |
| 16.5-b5 |
16.5-b |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
16.5 |
\( 2^{4} \) |
\( 2^{14} \) |
$2.52156$ |
$(-a+1)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$200.5904149$ |
0.705255777 |
\( \frac{8525185037}{4} a^{2} - \frac{12150636535}{2} a + \frac{5069473275}{2} \) |
\( \bigl[a^{2} - 3\) , \( -a^{2} + a + 2\) , \( 0\) , \( 18623821 a^{2} - 9857898 a - 79135231\) , \( -63948294020 a^{2} + 33848892296 a + 271725288464\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(18623821a^{2}-9857898a-79135231\right){x}-63948294020a^{2}+33848892296a+271725288464$ |
| 16.5-b6 |
16.5-b |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
16.5 |
\( 2^{4} \) |
\( - 2^{14} \) |
$2.52156$ |
$(-a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$100.2952074$ |
0.705255777 |
\( \frac{427}{4} a^{2} + \frac{383}{2} a - \frac{47}{2} \) |
\( \bigl[a + 1\) , \( a^{2} - 3\) , \( a^{2} - 3\) , \( -41098595441852 a^{2} + 21754167987381 a + 174633707952162\) , \( -893212511727360662557 a^{2} + 472792192035817395818 a + 3795385492745243685734\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(-41098595441852a^{2}+21754167987381a+174633707952162\right){x}-893212511727360662557a^{2}+472792192035817395818a+3795385492745243685734$ |
| 16.5-b7 |
16.5-b |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
16.5 |
\( 2^{4} \) |
\( - 2^{16} \) |
$2.52156$ |
$(-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$6.268450466$ |
0.705255777 |
\( \frac{32251567931279}{16} a^{2} + \frac{21655756210331}{8} a - \frac{13765401110581}{8} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a^{2} - 3\) , \( 64001862107159 a^{2} - 33877246772437 a - 271952906795303\) , \( 407543787751541638838 a^{2} - 215719683985399604073 a - 1731710829597940478002\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(64001862107159a^{2}-33877246772437a-271952906795303\right){x}+407543787751541638838a^{2}-215719683985399604073a-1731710829597940478002$ |
| 16.5-b8 |
16.5-b |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
16.5 |
\( 2^{4} \) |
\( 2^{20} \) |
$2.52156$ |
$(-a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$50.14760373$ |
0.705255777 |
\( \frac{130050481}{256} a^{2} + \frac{87514925}{128} a - \frac{54929731}{128} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a^{2} - 3\) , \( 4217266330899 a^{2} - 2232268991762 a - 17919757326708\) , \( 5637971616536361013 a^{2} - 2984271854928490894 a - 23956533748648438226\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(4217266330899a^{2}-2232268991762a-17919757326708\right){x}+5637971616536361013a^{2}-2984271854928490894a-23956533748648438226$ |
*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.
