| 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 |
| 16.1-a1 |
16.1-a |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{11} \) |
$5.47911$ |
$(-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$9$ |
\( 2^{2} \) |
$1$ |
$5.875276403$ |
1.368946263 |
\( -643928257795 a^{2} + 1049079504293 a + 5142910167338 \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 66 a^{2} - 70 a - 632\) , \( 694 a^{2} - 972 a - 6090\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(66a^{2}-70a-632\right){x}+694a^{2}-972a-6090$ |
| 16.1-a2 |
16.1-a |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{10} \) |
$5.47911$ |
$(-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$9$ |
\( 2^{2} \) |
$1$ |
$5.875276403$ |
1.368946263 |
\( 70234953 a^{2} - 219146556 a - 166914917 \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( -14 a^{2} + 60 a + 8\) , \( -88 a^{2} + 304 a + 140\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-14a^{2}+60a+8\right){x}-88a^{2}+304a+140$ |
| 16.1-b1 |
16.1-b |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
16.1 |
\( 2^{4} \) |
\( - 2^{8} \) |
$5.47911$ |
$(-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$212.8151395$ |
2.754787558 |
\( 22765256684 a^{2} + 62673649730 a + 30329193250 \) |
\( \bigl[a^{2} - a - 6\) , \( 1\) , \( 0\) , \( 53 a^{2} - 85 a - 420\) , \( -269 a^{2} + 438 a + 2148\bigr] \) |
${y}^2+\left(a^{2}-a-6\right){x}{y}={x}^{3}+{x}^{2}+\left(53a^{2}-85a-420\right){x}-269a^{2}+438a+2148$ |
| 16.1-b2 |
16.1-b |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$5.47911$ |
$(-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$425.6302790$ |
2.754787558 |
\( -26300 a^{2} - 92424 a - 47580 \) |
\( \bigl[a^{2} - a - 6\) , \( 1\) , \( 0\) , \( -2 a^{2} + 5 a + 20\) , \( -19 a^{2} + 32 a + 154\bigr] \) |
${y}^2+\left(a^{2}-a-6\right){x}{y}={x}^{3}+{x}^{2}+\left(-2a^{2}+5a+20\right){x}-19a^{2}+32a+154$ |
| 16.1-c1 |
16.1-c |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
16.1 |
\( 2^{4} \) |
\( - 2^{8} \) |
$5.47911$ |
$(-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$28.19422359$ |
0.364960390 |
\( 22765256684 a^{2} + 62673649730 a + 30329193250 \) |
\( \bigl[a + 1\) , \( -a^{2} + 2 a + 7\) , \( a^{2} - 2 a - 5\) , \( 5 a^{2} - 18 a - 3\) , \( 315 a^{2} - 990 a - 741\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2a-5\right){y}={x}^{3}+\left(-a^{2}+2a+7\right){x}^{2}+\left(5a^{2}-18a-3\right){x}+315a^{2}-990a-741$ |
| 16.1-c2 |
16.1-c |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$5.47911$ |
$(-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$56.38844718$ |
0.364960390 |
\( -26300 a^{2} - 92424 a - 47580 \) |
\( \bigl[a + 1\) , \( -a^{2} + 2 a + 7\) , \( a^{2} - 2 a - 5\) , \( -20 a^{2} + 62 a + 57\) , \( 101 a^{2} - 315 a - 234\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2a-5\right){y}={x}^{3}+\left(-a^{2}+2a+7\right){x}^{2}+\left(-20a^{2}+62a+57\right){x}+101a^{2}-315a-234$ |
| 16.1-d1 |
16.1-d |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{11} \) |
$5.47911$ |
$(-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$99.72686981$ |
2.581830792 |
\( -643928257795 a^{2} + 1049079504293 a + 5142910167338 \) |
\( \bigl[a + 1\) , \( -a^{2} + 3 a + 7\) , \( a^{2} - 2 a - 5\) , \( -138 a^{2} - 316 a - 136\) , \( 2382 a^{2} + 6714 a + 3272\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2a-5\right){y}={x}^{3}+\left(-a^{2}+3a+7\right){x}^{2}+\left(-138a^{2}-316a-136\right){x}+2382a^{2}+6714a+3272$ |
| 16.1-d2 |
16.1-d |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{10} \) |
$5.47911$ |
$(-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$99.72686981$ |
2.581830792 |
\( 70234953 a^{2} - 219146556 a - 166914917 \) |
\( \bigl[a + 1\) , \( -a^{2} + 3 a + 7\) , \( a^{2} - 2 a - 5\) , \( -18 a^{2} + 14 a + 24\) , \( -8 a^{2} + 140 a + 92\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2a-5\right){y}={x}^{3}+\left(-a^{2}+3a+7\right){x}^{2}+\left(-18a^{2}+14a+24\right){x}-8a^{2}+140a+92$ |
*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.
