| 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 |
| 22.2-a1 |
22.2-a |
$4$ |
$15$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
22.2 |
\( 2 \cdot 11 \) |
\( 2 \cdot 11^{5} \) |
$0.67040$ |
$(a+1), (2a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B.1.1, 5B.4.2 |
$1$ |
\( 1 \) |
$1$ |
$19.48623858$ |
0.625021394 |
\( \frac{3800943658260597}{322102} a - \frac{6583595299744607}{322102} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 1813 a - 3142\) , \( -55168 a + 95554\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(1813a-3142\right){x}-55168a+95554$ |
| 22.2-a2 |
22.2-a |
$4$ |
$15$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
22.2 |
\( 2 \cdot 11 \) |
\( 2^{15} \cdot 11^{3} \) |
$0.67040$ |
$(a+1), (2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B.1.2, 5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$2.165137620$ |
0.625021394 |
\( -\frac{7452136447}{340736} a - \frac{12920117437}{340736} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( -18 a - 33\) , \( -83 a - 145\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-18a-33\right){x}-83a-145$ |
| 22.2-a3 |
22.2-a |
$4$ |
$15$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
22.2 |
\( 2 \cdot 11 \) |
\( 2^{3} \cdot 11^{15} \) |
$0.67040$ |
$(a+1), (2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B.1.2, 5B.4.2 |
$1$ |
\( 1 \) |
$1$ |
$2.165137620$ |
0.625021394 |
\( \frac{14621235235888115443}{16708992677662604} a - \frac{20728089694692551503}{16708992677662604} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 1688 a - 2922\) , \( -63194 a + 109458\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(1688a-2922\right){x}-63194a+109458$ |
| 22.2-a4 |
22.2-a |
$4$ |
$15$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
22.2 |
\( 2 \cdot 11 \) |
\( 2^{5} \cdot 11 \) |
$0.67040$ |
$(a+1), (2a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B.1.1, 5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$19.48623858$ |
0.625021394 |
\( \frac{26727}{88} a + \frac{49507}{88} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( 2 a + 2\) , \( a + 1\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(2a+2\right){x}+a+1$ |
| 22.2-b1 |
22.2-b |
$4$ |
$15$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
22.2 |
\( 2 \cdot 11 \) |
\( 2 \cdot 11^{5} \) |
$0.67040$ |
$(a+1), (2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B.1.2, 5B.1.2 |
$1$ |
\( 5 \) |
$1$ |
$0.597897022$ |
0.862990016 |
\( \frac{3800943658260597}{322102} a - \frac{6583595299744607}{322102} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 1813 a - 3143\) , \( 55168 a - 95555\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(1813a-3143\right){x}+55168a-95555$ |
| 22.2-b2 |
22.2-b |
$4$ |
$15$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
22.2 |
\( 2 \cdot 11 \) |
\( 2^{15} \cdot 11^{3} \) |
$0.67040$ |
$(a+1), (2a+1)$ |
0 |
$\Z/15\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B.1.1, 5B.1.1 |
$1$ |
\( 3^{2} \cdot 5 \) |
$1$ |
$14.94742555$ |
0.862990016 |
\( -\frac{7452136447}{340736} a - \frac{12920117437}{340736} \) |
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( -20 a - 32\) , \( 64 a + 112\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-20a-32\right){x}+64a+112$ |
| 22.2-b3 |
22.2-b |
$4$ |
$15$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
22.2 |
\( 2 \cdot 11 \) |
\( 2^{3} \cdot 11^{15} \) |
$0.67040$ |
$(a+1), (2a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B.1.1, 5B.1.2 |
$1$ |
\( 3^{2} \cdot 5 \) |
$1$ |
$0.597897022$ |
0.862990016 |
\( \frac{14621235235888115443}{16708992677662604} a - \frac{20728089694692551503}{16708992677662604} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 1688 a - 2923\) , \( 63194 a - 109459\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(1688a-2923\right){x}+63194a-109459$ |
| 22.2-b4 |
22.2-b |
$4$ |
$15$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
22.2 |
\( 2 \cdot 11 \) |
\( 2^{5} \cdot 11 \) |
$0.67040$ |
$(a+1), (2a+1)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B.1.2, 5B.1.1 |
$1$ |
\( 5 \) |
$1$ |
$14.94742555$ |
0.862990016 |
\( \frac{26727}{88} a + \frac{49507}{88} \) |
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( 3\) , \( 1\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+3{x}+1$ |
*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.
