| 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 |
| 104.2-a1 |
104.2-a |
$2$ |
$7$ |
\(\Q(\zeta_{7})^+\) |
$3$ |
$[3, 0]$ |
104.2 |
\( 2^{3} \cdot 13 \) |
\( - 2^{21} \cdot 13^{7} \) |
$1.35647$ |
$(a^2-2a-2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.1.6[3] |
$1$ |
\( 7 \) |
$1$ |
$0.693240326$ |
0.693240326 |
\( \frac{274090208021491107}{2007952544} a^{2} - \frac{304013417326180219}{4015905088} a - \frac{2464236390737668359}{8031810176} \) |
\( \bigl[a^{2} + a - 1\) , \( a^{2} - 2\) , \( a + 1\) , \( -10 a^{2} + 51 a - 69\) , \( -105 a^{2} + 256 a - 199\bigr] \) |
${y}^2+\left(a^{2}+a-1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(-10a^{2}+51a-69\right){x}-105a^{2}+256a-199$ |
| 104.2-a2 |
104.2-a |
$2$ |
$7$ |
\(\Q(\zeta_{7})^+\) |
$3$ |
$[3, 0]$ |
104.2 |
\( 2^{3} \cdot 13 \) |
\( - 2^{3} \cdot 13 \) |
$1.35647$ |
$(a^2-2a-2), (2)$ |
0 |
$\Z/7\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.1.1[3] |
$1$ |
\( 1 \) |
$1$ |
$237.7814320$ |
0.693240326 |
\( -\frac{245189}{26} a^{2} - \frac{174769}{26} a + \frac{163157}{26} \) |
\( \bigl[a^{2} + a - 1\) , \( a^{2} - 2\) , \( a + 1\) , \( a + 1\) , \( a^{2} - 1\bigr] \) |
${y}^2+\left(a^{2}+a-1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(a+1\right){x}+a^{2}-1$ |
| 104.2-b1 |
104.2-b |
$3$ |
$9$ |
\(\Q(\zeta_{7})^+\) |
$3$ |
$[3, 0]$ |
104.2 |
\( 2^{3} \cdot 13 \) |
\( - 2^{27} \cdot 13^{3} \) |
$1.35647$ |
$(a^2-2a-2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 3 \) |
$1$ |
$0.173437023$ |
0.668971376 |
\( -\frac{179962171805350369053}{35152} a^{2} - \frac{4618190615991267991205}{1124864} a + \frac{3195887056317157693225}{1124864} \) |
\( \bigl[a^{2} - 2\) , \( a^{2} - 1\) , \( a^{2} + a - 2\) , \( 571 a^{2} - 1130 a - 2302\) , \( 11352 a^{2} - 22648 a - 45902\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{2}-1\right){x}^{2}+\left(571a^{2}-1130a-2302\right){x}+11352a^{2}-22648a-45902$ |
| 104.2-b2 |
104.2-b |
$3$ |
$9$ |
\(\Q(\zeta_{7})^+\) |
$3$ |
$[3, 0]$ |
104.2 |
\( 2^{3} \cdot 13 \) |
\( - 2^{3} \cdot 13^{3} \) |
$1.35647$ |
$(a^2-2a-2), (2)$ |
0 |
$\Z/9\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$1$ |
$126.4355900$ |
0.668971376 |
\( -\frac{153935551}{4394} a^{2} + \frac{347455161}{4394} a - \frac{62073129}{2197} \) |
\( \bigl[a^{2} - 2\) , \( a^{2} - 1\) , \( a^{2} + a - 2\) , \( a^{2} - 2\) , \( -4 a^{2} + 2 a + 8\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{2}-1\right){x}^{2}+\left(a^{2}-2\right){x}-4a^{2}+2a+8$ |
| 104.2-b3 |
104.2-b |
$3$ |
$9$ |
\(\Q(\zeta_{7})^+\) |
$3$ |
$[3, 0]$ |
104.2 |
\( 2^{3} \cdot 13 \) |
\( - 2^{9} \cdot 13^{9} \) |
$1.35647$ |
$(a^2-2a-2), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$4.682799632$ |
0.668971376 |
\( -\frac{5647619780915441}{84835994984} a^{2} - \frac{4561969890781345}{84835994984} a + \frac{3148905634529193}{84835994984} \) |
\( \bigl[a^{2} - 2\) , \( a^{2} - 1\) , \( a^{2} + a - 2\) , \( -9 a^{2} - 5 a + 8\) , \( 78 a^{2} - 74 a - 214\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{2}-1\right){x}^{2}+\left(-9a^{2}-5a+8\right){x}+78a^{2}-74a-214$ |
*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.
