| 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 |
| 54.1-a2 |
54.1-a |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
54.1 |
\( 2 \cdot 3^{3} \) |
\( 2 \cdot 3^{5} \) |
$0.68514$ |
$(a), (-a-1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$8.206562124$ |
0.644768414 |
\( -\frac{269}{2} a + 1619 \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( -a - 1\) , \( 0\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a-1\right){x}$ |
| 432.1-a2 |
432.1-a |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
432.1 |
\( 2^{4} \cdot 3^{3} \) |
\( 2^{13} \cdot 3^{5} \) |
$1.15227$ |
$(a), (-a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.020567681$ |
$4.103281062$ |
1.432230274 |
\( -\frac{269}{2} a + 1619 \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -2 a - 2\) , \( 2\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(-2a-2\right){x}+2$ |
| 486.3-a2 |
486.3-a |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
486.3 |
\( 2 \cdot 3^{5} \) |
\( 2 \cdot 3^{17} \) |
$1.18670$ |
$(a), (-a-1), (a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$2.735520708$ |
1.934305242 |
\( -\frac{269}{2} a + 1619 \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -4 a - 4\) , \( -4 a + 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-4a-4\right){x}-4a+1$ |
| 3888.3-a2 |
3888.3-a |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3888.3 |
\( 2^{4} \cdot 3^{5} \) |
\( 2^{13} \cdot 3^{17} \) |
$1.99579$ |
$(a), (-a-1), (a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$1.367760354$ |
1.934305242 |
\( -\frac{269}{2} a + 1619 \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -15 a - 11\) , \( a - 8\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-15a-11\right){x}+a-8$ |
| 6534.1-c2 |
6534.1-c |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6534.1 |
\( 2 \cdot 3^{3} \cdot 11^{2} \) |
\( 2 \cdot 3^{11} \cdot 11^{6} \) |
$2.27237$ |
$(a), (-a-1), (a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$1.428579098$ |
1.010157967 |
\( -\frac{269}{2} a + 1619 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( 10 a + 19\) , \( -18 a - 3\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(10a+19\right){x}-18a-3$ |
| 6534.3-c2 |
6534.3-c |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6534.3 |
\( 2 \cdot 3^{3} \cdot 11^{2} \) |
\( 2 \cdot 3^{11} \cdot 11^{6} \) |
$2.27237$ |
$(a), (-a-1), (a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1.092600876$ |
$1.428579098$ |
4.414797924 |
\( -\frac{269}{2} a + 1619 \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( -14 a + 10\) , \( -7 a + 18\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-14a+10\right){x}-7a+18$ |
| 6912.1-c2 |
6912.1-c |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6912.1 |
\( 2^{8} \cdot 3^{3} \) |
\( 2^{25} \cdot 3^{11} \) |
$2.30454$ |
$(a), (-a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$0.448425801$ |
$1.184515212$ |
3.004735343 |
\( -\frac{269}{2} a + 1619 \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -4 a + 32\) , \( 4 a - 12\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-4a+32\right){x}+4a-12$ |
| 6912.1-d2 |
6912.1-d |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6912.1 |
\( 2^{8} \cdot 3^{3} \) |
\( 2^{25} \cdot 3^{11} \) |
$2.30454$ |
$(a), (-a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$1.184515212$ |
1.675157478 |
\( -\frac{269}{2} a + 1619 \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -4 a + 32\) , \( -4 a + 12\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-4a+32\right){x}-4a+12$ |
| 15606.1-a2 |
15606.1-a |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
15606.1 |
\( 2 \cdot 3^{3} \cdot 17^{2} \) |
\( 2 \cdot 3^{5} \cdot 17^{6} \) |
$2.82492$ |
$(a), (-a-1), (-2a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 3 \) |
$1$ |
$1.990383674$ |
4.222241379 |
\( -\frac{269}{2} a + 1619 \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( 3 a - 11\) , \( a - 3\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(3a-11\right){x}+a-3$ |
| 15606.3-c2 |
15606.3-c |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
15606.3 |
\( 2 \cdot 3^{3} \cdot 17^{2} \) |
\( 2 \cdot 3^{5} \cdot 17^{6} \) |
$2.82492$ |
$(a), (-a-1), (2a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$0.893011281$ |
$1.990383674$ |
5.027345582 |
\( -\frac{269}{2} a + 1619 \) |
\( \bigl[1\) , \( a\) , \( a\) , \( -5 a + 9\) , \( 4 a - 1\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-5a+9\right){x}+4a-1$ |
| 19494.1-a2 |
19494.1-a |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
19494.1 |
\( 2 \cdot 3^{3} \cdot 19^{2} \) |
\( 2 \cdot 3^{11} \cdot 19^{6} \) |
$2.98647$ |
$(a), (-a-1), (-3a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$0.764888608$ |
$1.086985707$ |
2.351619324 |
\( -\frac{269}{2} a + 1619 \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( -7 a - 39\) , \( -27 a + 30\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-7a-39\right){x}-27a+30$ |
| 19494.3-e2 |
19494.3-e |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
19494.3 |
\( 2 \cdot 3^{3} \cdot 19^{2} \) |
\( 2 \cdot 3^{11} \cdot 19^{6} \) |
$2.98647$ |
$(a), (-a-1), (3a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$2.511750799$ |
$1.086985707$ |
7.722277008 |
\( -\frac{269}{2} a + 1619 \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 17 a - 31\) , \( -29 a - 17\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(17a-31\right){x}-29a-17$ |
| 27648.1-d2 |
27648.1-d |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
27648.1 |
\( 2^{10} \cdot 3^{3} \) |
\( 2^{31} \cdot 3^{11} \) |
$3.25911$ |
$(a), (-a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$0.837578739$ |
1.184515212 |
\( -\frac{269}{2} a + 1619 \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 10 a - 65\) , \( 33 a - 49\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(10a-65\right){x}+33a-49$ |
| 27648.1-g2 |
27648.1-g |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
27648.1 |
\( 2^{10} \cdot 3^{3} \) |
\( 2^{31} \cdot 3^{5} \) |
$3.25911$ |
$(a), (-a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$1.450728932$ |
2.051640531 |
\( -\frac{269}{2} a + 1619 \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 14 a + 11\) , \( 17 a - 13\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(14a+11\right){x}+17a-13$ |
| 27648.1-j2 |
27648.1-j |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
27648.1 |
\( 2^{10} \cdot 3^{3} \) |
\( 2^{31} \cdot 3^{5} \) |
$3.25911$ |
$(a), (-a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.226422438$ |
$1.450728932$ |
5.574449434 |
\( -\frac{269}{2} a + 1619 \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 14 a + 11\) , \( -17 a + 13\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(14a+11\right){x}-17a+13$ |
| 27648.1-m2 |
27648.1-m |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
27648.1 |
\( 2^{10} \cdot 3^{3} \) |
\( 2^{31} \cdot 3^{11} \) |
$3.25911$ |
$(a), (-a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$1.242222194$ |
$0.837578739$ |
5.885724349 |
\( -\frac{269}{2} a + 1619 \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 10 a - 65\) , \( -33 a + 49\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(10a-65\right){x}-33a+49$ |
| 33750.1-f2 |
33750.1-f |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
33750.1 |
\( 2 \cdot 3^{3} \cdot 5^{4} \) |
\( 2 \cdot 3^{5} \cdot 5^{12} \) |
$3.42572$ |
$(a), (-a-1), (5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 3 \) |
$1$ |
$1.641312424$ |
3.481749436 |
\( -\frac{269}{2} a + 1619 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -11 a - 8\) , \( 9 a + 9\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-11a-8\right){x}+9a+9$ |
*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.
