| 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 |
| 22.1-a1 |
22.1-a |
$4$ |
$15$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2 \cdot 11^{5} \) |
$0.67040$ |
$(a+1), (-2a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B.1.1, 5B.4.2 |
$1$ |
\( 1 \) |
$1$ |
$19.48623858$ |
0.625021394 |
\( -\frac{3800943658260597}{322102} a - \frac{6583595299744607}{322102} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( 14261 a - 24701\) , \( -1229141 a + 2128933\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(14261a-24701\right){x}-1229141a+2128933$ |
| 22.1-a2 |
22.1-a |
$4$ |
$15$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{3} \cdot 11^{15} \) |
$0.67040$ |
$(a+1), (-2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B.1.2, 5B.4.2 |
$1$ |
\( 1 \) |
$1$ |
$2.165137620$ |
0.625021394 |
\( -\frac{14621235235888115443}{16708992677662604} a - \frac{20728089694692551503}{16708992677662604} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( -23694 a + 41039\) , \( -5992615 a + 10379512\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-23694a+41039\right){x}-5992615a+10379512$ |
| 22.1-a3 |
22.1-a |
$4$ |
$15$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{15} \cdot 11^{3} \) |
$0.67040$ |
$(a+1), (-2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B.1.2, 5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$2.165137620$ |
0.625021394 |
\( \frac{7452136447}{340736} a - \frac{12920117437}{340736} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 18 a - 33\) , \( 83 a - 145\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(18a-33\right){x}+83a-145$ |
| 22.1-a4 |
22.1-a |
$4$ |
$15$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{5} \cdot 11 \) |
$0.67040$ |
$(a+1), (-2a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B.1.1, 5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$19.48623858$ |
0.625021394 |
\( -\frac{26727}{88} a + \frac{49507}{88} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -2 a + 2\) , \( -a + 1\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-2a+2\right){x}-a+1$ |
| 22.1-b1 |
22.1-b |
$4$ |
$15$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2 \cdot 11^{5} \) |
$0.67040$ |
$(a+1), (-2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B.1.2, 5B.1.2 |
$1$ |
\( 5 \) |
$1$ |
$0.597897022$ |
0.862990016 |
\( -\frac{3800943658260597}{322102} a - \frac{6583595299744607}{322102} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( 14261 a - 24702\) , \( 1229140 a - 2128935\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(14261a-24702\right){x}+1229140a-2128935$ |
| 22.1-b2 |
22.1-b |
$4$ |
$15$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{3} \cdot 11^{15} \) |
$0.67040$ |
$(a+1), (-2a+1)$ |
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 |
\( -\frac{14621235235888115443}{16708992677662604} a - \frac{20728089694692551503}{16708992677662604} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( -23694 a + 41038\) , \( 5992614 a - 10379514\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-23694a+41038\right){x}+5992614a-10379514$ |
| 22.1-b3 |
22.1-b |
$4$ |
$15$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{15} \cdot 11^{3} \) |
$0.67040$ |
$(a+1), (-2a+1)$ |
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 |
\( \frac{7452136447}{340736} a - \frac{12920117437}{340736} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( 20 a - 32\) , \( -64 a + 112\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(20a-32\right){x}-64a+112$ |
| 22.1-b4 |
22.1-b |
$4$ |
$15$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{5} \cdot 11 \) |
$0.67040$ |
$(a+1), (-2a+1)$ |
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.
