| 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-b4 |
16.1-b |
$16$ |
$48$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$6.06589$ |
$(a^3-4a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-192$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$626.2283084$ |
0.815401443 |
\( 2242365893060213250 a^{3} - 1160733998424384000 a^{2} - 8368623442060794000 a + 4331918256193377000 \) |
\( \bigl[a^{2} + a - 2\) , \( -a^{2} + 3\) , \( a + 1\) , \( 19 a^{3} + 25 a^{2} - 28 a - 17\) , \( 38 a^{3} + 86 a^{2} + 15 a - 10\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(19a^{3}+25a^{2}-28a-17\right){x}+38a^{3}+86a^{2}+15a-10$ |
| 16.1-b13 |
16.1-b |
$16$ |
$48$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$6.06589$ |
$(a^3-4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-192$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$16$ |
\( 1 \) |
$1$ |
$9.784817320$ |
0.815401443 |
\( 2242365893060213250 a^{3} - 1160733998424384000 a^{2} - 8368623442060794000 a + 4331918256193377000 \) |
\( \bigl[a^{3} - 4 a + 1\) , \( -a^{3} + a^{2} + 3 a - 2\) , \( a^{2} - 1\) , \( 19 a^{3} + 27 a^{2} - 28 a - 22\) , \( -19 a^{3} - 60 a^{2} - 43 a - 11\bigr] \) |
${y}^2+\left(a^{3}-4a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-2\right){x}^{2}+\left(19a^{3}+27a^{2}-28a-22\right){x}-19a^{3}-60a^{2}-43a-11$ |
| 256.1-a13 |
256.1-a |
$16$ |
$48$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{20} \) |
$8.57847$ |
$(a^3-4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-192$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$78.27853856$ |
1.630802886 |
\( 2242365893060213250 a^{3} - 1160733998424384000 a^{2} - 8368623442060794000 a + 4331918256193377000 \) |
\( \bigl[a^{3} - 3 a\) , \( a^{3} - a^{2} - 5 a + 3\) , \( a^{3} + a^{2} - 3 a - 1\) , \( 323 a^{3} + 164 a^{2} - 1197 a - 614\) , \( 4298 a^{3} + 2225 a^{2} - 16054 a - 8309\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a+3\right){x}^{2}+\left(323a^{3}+164a^{2}-1197a-614\right){x}+4298a^{3}+2225a^{2}-16054a-8309$ |
| 256.1-a14 |
256.1-a |
$16$ |
$48$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{20} \) |
$8.57847$ |
$(a^3-4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-192$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$4$ |
\( 2^{2} \) |
$1$ |
$19.56963464$ |
1.630802886 |
\( 2242365893060213250 a^{3} - 1160733998424384000 a^{2} - 8368623442060794000 a + 4331918256193377000 \) |
\( \bigl[a^{3} - 3 a\) , \( -a^{3} + a^{2} + 5 a - 1\) , \( a^{2} - 1\) , \( 321 a^{3} + 166 a^{2} - 1187 a - 617\) , \( -3976 a^{3} - 2061 a^{2} + 14862 a + 7693\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a-1\right){x}^{2}+\left(321a^{3}+166a^{2}-1187a-617\right){x}-3976a^{3}-2061a^{2}+14862a+7693$ |
| 324.1-a13 |
324.1-a |
$16$ |
$48$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{8} \cdot 3^{12} \) |
$8.83482$ |
$(a^3-4a+1), (a^2-2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{potential}$ |
$-192$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1.306469355$ |
$78.27853856$ |
2.840791994 |
\( 2242365893060213250 a^{3} - 1160733998424384000 a^{2} - 8368623442060794000 a + 4331918256193377000 \) |
\( \bigl[a^{3} + a^{2} - 4 a - 2\) , \( a^{3} + a^{2} - 5 a - 1\) , \( a^{3} - 3 a\) , \( 56 a^{3} + 83 a^{2} - 82 a - 69\) , \( -243 a^{3} - 390 a^{2} + 315 a + 244\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-4a-2\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(a^{3}+a^{2}-5a-1\right){x}^{2}+\left(56a^{3}+83a^{2}-82a-69\right){x}-243a^{3}-390a^{2}+315a+244$ |
| 324.1-a14 |
324.1-a |
$16$ |
$48$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{8} \cdot 3^{12} \) |
$8.83482$ |
$(a^3-4a+1), (a^2-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-192$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 2^{2} \) |
$3.919408065$ |
$8.697615395$ |
2.840791994 |
\( 2242365893060213250 a^{3} - 1160733998424384000 a^{2} - 8368623442060794000 a + 4331918256193377000 \) |
\( \bigl[a + 1\) , \( -a^{2} + a + 2\) , \( a^{3} - 4 a + 1\) , \( 55 a^{3} + 81 a^{2} - 75 a - 66\) , \( 298 a^{3} + 472 a^{2} - 391 a - 313\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-4a+1\right){y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(55a^{3}+81a^{2}-75a-66\right){x}+298a^{3}+472a^{2}-391a-313$ |
*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.
