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 |
2.1-a1 |
2.1-a |
$2$ |
$3$ |
3.3.733.1 |
$3$ |
$[3, 0]$ |
2.1 |
\( 2 \) |
\( - 2^{9} \) |
$2.71558$ |
$(a^2-6)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$43.69625498$ |
1.613958003 |
\( -\frac{153560745}{512} a^{2} - \frac{233924111}{512} a + \frac{486084887}{512} \) |
\( \bigl[a^{2} + a - 5\) , \( -a^{2} + a + 5\) , \( a + 1\) , \( 8 a^{2} - a - 47\) , \( -14 a^{2} + 2 a + 85\bigr] \) |
${y}^2+\left(a^{2}+a-5\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+a+5\right){x}^{2}+\left(8a^{2}-a-47\right){x}-14a^{2}+2a+85$ |
16.3-a1 |
16.3-a |
$2$ |
$3$ |
3.3.733.1 |
$3$ |
$[3, 0]$ |
16.3 |
\( 2^{4} \) |
\( - 2^{21} \) |
$3.84041$ |
$(a^2-6)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$27.77354004$ |
2.051678216 |
\( -\frac{153560745}{512} a^{2} - \frac{233924111}{512} a + \frac{486084887}{512} \) |
\( \bigl[0\) , \( a + 1\) , \( a^{2} - 4\) , \( -8 a^{2} - 16 a + 16\) , \( -55 a^{2} - 79 a + 184\bigr] \) |
${y}^2+\left(a^{2}-4\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-8a^{2}-16a+16\right){x}-55a^{2}-79a+184$ |
50.2-a1 |
50.2-a |
$2$ |
$3$ |
3.3.733.1 |
$3$ |
$[3, 0]$ |
50.2 |
\( 2 \cdot 5^{2} \) |
\( - 2^{9} \cdot 5^{6} \) |
$4.64357$ |
$(a^2-6), (-a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 3^{2} \) |
$1$ |
$8.977725141$ |
2.984398597 |
\( -\frac{153560745}{512} a^{2} - \frac{233924111}{512} a + \frac{486084887}{512} \) |
\( \bigl[a^{2} + a - 5\) , \( a^{2} + a - 5\) , \( a + 1\) , \( 6 a^{2} + 3 a - 34\) , \( 11 a^{2} + 4 a - 69\bigr] \) |
${y}^2+\left(a^{2}+a-5\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}+a-5\right){x}^{2}+\left(6a^{2}+3a-34\right){x}+11a^{2}+4a-69$ |
64.4-a2 |
64.4-a |
$2$ |
$3$ |
3.3.733.1 |
$3$ |
$[3, 0]$ |
64.4 |
\( 2^{6} \) |
\( - 2^{27} \) |
$4.83861$ |
$(a^2-6)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$32.89239640$ |
2.429816763 |
\( -\frac{153560745}{512} a^{2} - \frac{233924111}{512} a + \frac{486084887}{512} \) |
\( \bigl[0\) , \( -a - 1\) , \( a^{2} - 4\) , \( 31 a^{2} - 111 a + 87\) , \( 358 a^{2} - 1324 a + 1061\bigr] \) |
${y}^2+\left(a^{2}-4\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(31a^{2}-111a+87\right){x}+358a^{2}-1324a+1061$ |
64.4-n1 |
64.4-n |
$2$ |
$3$ |
3.3.733.1 |
$3$ |
$[3, 0]$ |
64.4 |
\( 2^{6} \) |
\( - 2^{27} \) |
$4.83861$ |
$(a^2-6)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$0.355900500$ |
$25.87488915$ |
4.081655584 |
\( -\frac{153560745}{512} a^{2} - \frac{233924111}{512} a + \frac{486084887}{512} \) |
\( \bigl[0\) , \( -a^{2} + 6\) , \( a^{2} - 4\) , \( 15 a^{2} - 6 a - 80\) , \( 67 a^{2} + 42 a - 532\bigr] \) |
${y}^2+\left(a^{2}-4\right){y}={x}^{3}+\left(-a^{2}+6\right){x}^{2}+\left(15a^{2}-6a-80\right){x}+67a^{2}+42a-532$ |
98.2-a1 |
98.2-a |
$2$ |
$3$ |
3.3.733.1 |
$3$ |
$[3, 0]$ |
98.2 |
\( 2 \cdot 7^{2} \) |
\( - 2^{9} \cdot 7^{6} \) |
$5.19471$ |
$(a^2-6), (a^2+2a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$20.99482285$ |
0.775461475 |
\( -\frac{153560745}{512} a^{2} - \frac{233924111}{512} a + \frac{486084887}{512} \) |
\( \bigl[a^{2} + a - 5\) , \( -a^{2} + a + 6\) , \( a + 1\) , \( 3 a^{2} - a - 12\) , \( -2 a^{2} + a + 14\bigr] \) |
${y}^2+\left(a^{2}+a-5\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+a+6\right){x}^{2}+\left(3a^{2}-a-12\right){x}-2a^{2}+a+14$ |
*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.