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 |
392.1-a1 |
392.1-a |
$6$ |
$18$ |
\(\Q(\zeta_{7})^+\) |
$3$ |
$[3, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{27} \cdot 7^{12} \) |
$1.69220$ |
$(-a^2-a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$7.041603260$ |
1.005943322 |
\( \frac{2251439055699625}{25088} \) |
\( \bigl[a^{2} - 2\) , \( -1\) , \( a^{2} + a - 2\) , \( -10922 a^{2} + 13652 a - 5462\) , \( 772040 a^{2} - 1158061 a + 386020\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}-{x}^{2}+\left(-10922a^{2}+13652a-5462\right){x}+772040a^{2}-1158061a+386020$ |
392.1-a2 |
392.1-a |
$6$ |
$18$ |
\(\Q(\zeta_{7})^+\) |
$3$ |
$[3, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( - 2^{54} \cdot 7^{9} \) |
$1.69220$ |
$(-a^2-a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$7.041603260$ |
1.005943322 |
\( -\frac{548347731625}{1835008} \) |
\( \bigl[a^{2} - 2\) , \( -1\) , \( a^{2} + a - 2\) , \( -682 a^{2} + 852 a - 342\) , \( 12232 a^{2} - 18349 a + 6116\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}-{x}^{2}+\left(-682a^{2}+852a-342\right){x}+12232a^{2}-18349a+6116$ |
392.1-a3 |
392.1-a |
$6$ |
$18$ |
\(\Q(\zeta_{7})^+\) |
$3$ |
$[3, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{9} \cdot 7^{24} \) |
$1.69220$ |
$(-a^2-a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$7.041603260$ |
1.005943322 |
\( \frac{4956477625}{941192} \) |
\( \bigl[a^{2} - 2\) , \( -1\) , \( a^{2} + a - 2\) , \( -142 a^{2} + 177 a - 72\) , \( 976 a^{2} - 1465 a + 488\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}-{x}^{2}+\left(-142a^{2}+177a-72\right){x}+976a^{2}-1465a+488$ |
392.1-a4 |
392.1-a |
$6$ |
$18$ |
\(\Q(\zeta_{7})^+\) |
$3$ |
$[3, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{3} \cdot 7^{12} \) |
$1.69220$ |
$(-a^2-a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$7.041603260$ |
1.005943322 |
\( \frac{128787625}{98} \) |
\( \bigl[a^{2} - 2\) , \( -1\) , \( a^{2} + a - 2\) , \( -42 a^{2} + 52 a - 22\) , \( -172 a^{2} + 257 a - 86\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}-{x}^{2}+\left(-42a^{2}+52a-22\right){x}-172a^{2}+257a-86$ |
392.1-a5 |
392.1-a |
$6$ |
$18$ |
\(\Q(\zeta_{7})^+\) |
$3$ |
$[3, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( - 2^{6} \cdot 7^{9} \) |
$1.69220$ |
$(-a^2-a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$7.041603260$ |
1.005943322 |
\( -\frac{15625}{28} \) |
\( \bigl[a^{2} - 2\) , \( -1\) , \( a^{2} + a - 2\) , \( -2 a^{2} + 2 a - 2\) , \( -4 a^{2} + 5 a - 2\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}-{x}^{2}+\left(-2a^{2}+2a-2\right){x}-4a^{2}+5a-2$ |
392.1-a6 |
392.1-a |
$6$ |
$18$ |
\(\Q(\zeta_{7})^+\) |
$3$ |
$[3, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( - 2^{18} \cdot 7^{15} \) |
$1.69220$ |
$(-a^2-a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$7.041603260$ |
1.005943322 |
\( \frac{9938375}{21952} \) |
\( \bigl[a^{2} - 2\) , \( -1\) , \( a^{2} + a - 2\) , \( 18 a^{2} - 23 a + 8\) , \( 80 a^{2} - 121 a + 40\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}-{x}^{2}+\left(18a^{2}-23a+8\right){x}+80a^{2}-121a+40$ |
*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.