| 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 |
| 208.1-a1 |
208.1-a |
$6$ |
$8$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
208.1 |
\( 2^{4} \cdot 13 \) |
\( 2^{4} \cdot 13 \) |
$2.64614$ |
$(a^2-a-2), (a^2-2a-2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$214.5471665$ |
1.102229278 |
\( \frac{1896640}{13} a^{2} + \frac{2231040}{13} a - \frac{856512}{13} \) |
\( \bigl[a^{2} - 1\) , \( a^{2} - 2 a - 2\) , \( a + 1\) , \( -42901 a^{2} - 50199 a + 19772\) , \( 8876792 a^{2} + 10386613 a - 4090525\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-2a-2\right){x}^{2}+\left(-42901a^{2}-50199a+19772\right){x}+8876792a^{2}+10386613a-4090525$ |
| 208.1-a2 |
208.1-a |
$6$ |
$8$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
208.1 |
\( 2^{4} \cdot 13 \) |
\( 2^{8} \cdot 13^{2} \) |
$2.64614$ |
$(a^2-a-2), (a^2-2a-2)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$214.5471665$ |
1.102229278 |
\( \frac{1834400}{169} a^{2} - \frac{4540000}{169} a + \frac{1544832}{169} \) |
\( \bigl[a^{2} - 1\) , \( a^{2} - 2\) , \( 0\) , \( 3 a^{2} + 2 a - 6\) , \( 2 a + 4\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(3a^{2}+2a-6\right){x}+2a+4$ |
| 208.1-a3 |
208.1-a |
$6$ |
$8$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
208.1 |
\( 2^{4} \cdot 13 \) |
\( - 2^{4} \cdot 13 \) |
$2.64614$ |
$(a^2-a-2), (a^2-2a-2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$214.5471665$ |
1.102229278 |
\( \frac{10295074668}{13} a^{2} - \frac{25548710024}{13} a + \frac{6960588668}{13} \) |
\( \bigl[a + 1\) , \( 1\) , \( a^{2} - a - 2\) , \( a^{2} + 3 a - 2\) , \( 2 a^{2} + a - 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+{x}^{2}+\left(a^{2}+3a-2\right){x}+2a^{2}+a-1$ |
| 208.1-a4 |
208.1-a |
$6$ |
$8$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
208.1 |
\( 2^{4} \cdot 13 \) |
\( - 2^{8} \cdot 13^{2} \) |
$2.64614$ |
$(a^2-a-2), (a^2-2a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$6.704598954$ |
1.102229278 |
\( -\frac{653401310948}{169} a^{2} + \frac{847055130906}{169} a + \frac{2688174503518}{169} \) |
\( \bigl[a + 1\) , \( 1\) , \( a^{2} - 1\) , \( -22 a^{2} - 48 a - 20\) , \( -199 a^{2} - 294 a + 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+{x}^{2}+\left(-22a^{2}-48a-20\right){x}-199a^{2}-294a+1$ |
| 208.1-a5 |
208.1-a |
$6$ |
$8$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
208.1 |
\( 2^{4} \cdot 13 \) |
\( 2^{4} \cdot 13^{4} \) |
$2.64614$ |
$(a^2-a-2), (a^2-2a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$53.63679163$ |
1.102229278 |
\( -\frac{725118876}{28561} a^{2} + \frac{1213367848}{28561} a + \frac{3447890788}{28561} \) |
\( \bigl[a + 1\) , \( 1\) , \( a^{2} - 1\) , \( -2 a^{2} - 3 a\) , \( -5 a^{2} - 7 a\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+{x}^{2}+\left(-2a^{2}-3a\right){x}-5a^{2}-7a$ |
| 208.1-a6 |
208.1-a |
$6$ |
$8$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
208.1 |
\( 2^{4} \cdot 13 \) |
\( 2^{8} \cdot 13^{8} \) |
$2.64614$ |
$(a^2-a-2), (a^2-2a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$6.704598954$ |
1.102229278 |
\( \frac{168959003658436}{815730721} a^{2} - \frac{419215196828106}{815730721} a + \frac{114055804392930}{815730721} \) |
\( \bigl[a + 1\) , \( 1\) , \( a^{2} - 1\) , \( -2 a^{2} + 2 a\) , \( -9 a^{2} - 6 a - 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+{x}^{2}+\left(-2a^{2}+2a\right){x}-9a^{2}-6a-1$ |
| 208.1-b1 |
208.1-b |
$2$ |
$3$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
208.1 |
\( 2^{4} \cdot 13 \) |
\( - 2^{15} \cdot 13^{3} \) |
$2.64614$ |
$(a^2-a-2), (a^2-2a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$0.027494208$ |
$63.63376441$ |
1.725755341 |
\( -\frac{936828579535}{4394} a^{2} + \frac{1669497916531}{4394} a - \frac{428712816945}{4394} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{2} - a - 1\) , \( a^{2} - a - 2\) , \( -443 a^{2} + 1079 a - 295\) , \( 8550 a^{2} - 21172 a + 5758\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{2}-a-1\right){x}^{2}+\left(-443a^{2}+1079a-295\right){x}+8550a^{2}-21172a+5758$ |
| 208.1-b2 |
208.1-b |
$2$ |
$3$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
208.1 |
\( 2^{4} \cdot 13 \) |
\( - 2^{13} \cdot 13 \) |
$2.64614$ |
$(a^2-a-2), (a^2-2a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$0.009164736$ |
$190.9012932$ |
1.725755341 |
\( \frac{47519}{26} a^{2} - \frac{15283}{13} a - \frac{161671}{26} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{2} - a - 1\) , \( a^{2} - a - 2\) , \( -3 a^{2} + 9 a - 5\) , \( 14 a^{2} - 34 a + 8\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{2}-a-1\right){x}^{2}+\left(-3a^{2}+9a-5\right){x}+14a^{2}-34a+8$ |
| 208.1-c1 |
208.1-c |
$1$ |
$1$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
208.1 |
\( 2^{4} \cdot 13 \) |
\( - 2^{11} \cdot 13^{5} \) |
$2.64614$ |
$(a^2-a-2), (a^2-2a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \cdot 5 \) |
$0.005630301$ |
$62.82464199$ |
1.744544407 |
\( -\frac{650158650}{371293} a^{2} - \frac{1006681203}{371293} a + \frac{376033455}{371293} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{2} - 2\) , \( 0\) , \( 3 a^{2} - 7 a + 2\) , \( 4 a^{2} - 10 a + 3\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(3a^{2}-7a+2\right){x}+4a^{2}-10a+3$ |
| 208.1-d1 |
208.1-d |
$2$ |
$7$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
208.1 |
\( 2^{4} \cdot 13 \) |
\( - 2^{13} \cdot 13^{7} \) |
$2.64614$ |
$(a^2-a-2), (a^2-2a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$7$ |
7B.6.3 |
$1$ |
\( 2 \cdot 7 \) |
$1$ |
$1.223126103$ |
1.407564848 |
\( \frac{2930762217049733062408935}{125497034} a^{2} - \frac{1009489594983775464711535}{62748517} a - \frac{9420406871887862637780499}{125497034} \) |
\( \bigl[a^{2} - a - 2\) , \( -a^{2} + 1\) , \( a^{2} - a - 2\) , \( -5673342 a^{2} - 6638321 a + 2614308\) , \( 13557805294 a^{2} + 15863804701 a - 6247587632\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a^{2}+1\right){x}^{2}+\left(-5673342a^{2}-6638321a+2614308\right){x}+13557805294a^{2}+15863804701a-6247587632$ |
| 208.1-d2 |
208.1-d |
$2$ |
$7$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
208.1 |
\( 2^{4} \cdot 13 \) |
\( - 2^{19} \cdot 13 \) |
$2.64614$ |
$(a^2-a-2), (a^2-2a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$7$ |
7B.6.1 |
$1$ |
\( 2 \) |
$1$ |
$8.561882721$ |
1.407564848 |
\( -\frac{192290305}{104} a^{2} + \frac{119296865}{26} a - \frac{129845369}{104} \) |
\( \bigl[a^{2} - a - 2\) , \( -a^{2} + 2 a + 2\) , \( 0\) , \( a + 3\) , \( 15 a^{2} - 10 a - 48\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}={x}^{3}+\left(-a^{2}+2a+2\right){x}^{2}+\left(a+3\right){x}+15a^{2}-10a-48$ |
*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.
