| 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-a4 |
54.1-a |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
54.1 |
\( 2 \cdot 3^{3} \) |
\( 2^{9} \cdot 3^{9} \) |
$0.68514$ |
$(a), (-a-1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$1$ |
$2.735520708$ |
0.644768414 |
\( -\frac{2628365}{32} a + \frac{183347}{16} \) |
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( -11 a + 4\) , \( -26\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-11a+4\right){x}-26$ |
| 432.1-a4 |
432.1-a |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
432.1 |
\( 2^{4} \cdot 3^{3} \) |
\( 2^{21} \cdot 3^{9} \) |
$1.15227$ |
$(a), (-a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs |
$1$ |
\( 2^{2} \) |
$0.185109134$ |
$1.367760354$ |
1.432230274 |
\( -\frac{2628365}{32} a + \frac{183347}{16} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( -41 a + 17\) , \( 58 a - 141\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-41a+17\right){x}+58a-141$ |
| 486.3-a4 |
486.3-a |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
486.3 |
\( 2 \cdot 3^{5} \) |
\( 2^{9} \cdot 3^{9} \) |
$1.18670$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$2.735520708$ |
1.934305242 |
\( -\frac{2628365}{32} a + \frac{183347}{16} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 10 a + 4\) , \( 8 a + 13\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(10a+4\right){x}+8a+13$ |
| 3888.3-a4 |
3888.3-a |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3888.3 |
\( 2^{4} \cdot 3^{5} \) |
\( 2^{21} \cdot 3^{9} \) |
$1.99579$ |
$(a), (-a-1), (a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs |
$1$ |
\( 2 \) |
$1$ |
$1.367760354$ |
1.934305242 |
\( -\frac{2628365}{32} a + \frac{183347}{16} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 39 a + 25\) , \( 3 a + 164\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(39a+25\right){x}+3a+164$ |
| 6534.1-c4 |
6534.1-c |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6534.1 |
\( 2 \cdot 3^{3} \cdot 11^{2} \) |
\( 2^{9} \cdot 3^{3} \cdot 11^{6} \) |
$2.27237$ |
$(a), (-a-1), (a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs |
$1$ |
\( 1 \) |
$1$ |
$1.428579098$ |
1.010157967 |
\( -\frac{2628365}{32} a + \frac{183347}{16} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -28 a - 37\) , \( 122 a + 59\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-28a-37\right){x}+122a+59$ |
| 6534.3-c4 |
6534.3-c |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6534.3 |
\( 2 \cdot 3^{3} \cdot 11^{2} \) |
\( 2^{9} \cdot 3^{3} \cdot 11^{6} \) |
$2.27237$ |
$(a), (-a-1), (a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs |
$1$ |
\( 2 \cdot 3^{2} \) |
$0.121400097$ |
$1.428579098$ |
4.414797924 |
\( -\frac{2628365}{32} a + \frac{183347}{16} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( 31 a - 33\) , \( 101 a - 47\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(31a-33\right){x}+101a-47$ |
| 6912.1-c4 |
6912.1-c |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6912.1 |
\( 2^{8} \cdot 3^{3} \) |
\( 2^{33} \cdot 3^{3} \) |
$2.30454$ |
$(a), (-a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs |
$1$ |
\( 2^{2} \) |
$0.448425801$ |
$1.184515212$ |
3.004735343 |
\( -\frac{2628365}{32} a + \frac{183347}{16} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 4 a - 80\) , \( -44 a + 244\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(4a-80\right){x}-44a+244$ |
| 6912.1-d4 |
6912.1-d |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6912.1 |
\( 2^{8} \cdot 3^{3} \) |
\( 2^{33} \cdot 3^{3} \) |
$2.30454$ |
$(a), (-a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs |
$1$ |
\( 2 \) |
$1$ |
$1.184515212$ |
1.675157478 |
\( -\frac{2628365}{32} a + \frac{183347}{16} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 4 a - 80\) , \( 44 a - 244\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(4a-80\right){x}+44a-244$ |
| 15606.1-a4 |
15606.1-a |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
15606.1 |
\( 2 \cdot 3^{3} \cdot 17^{2} \) |
\( 2^{9} \cdot 3^{9} \cdot 17^{6} \) |
$2.82492$ |
$(a), (-a-1), (-2a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs |
$1$ |
\( 3^{2} \) |
$1$ |
$0.663461224$ |
4.222241379 |
\( -\frac{2628365}{32} a + \frac{183347}{16} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -60 a - 240\) , \( 496 a + 1408\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-60a-240\right){x}+496a+1408$ |
| 15606.3-c4 |
15606.3-c |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
15606.3 |
\( 2 \cdot 3^{3} \cdot 17^{2} \) |
\( 2^{9} \cdot 3^{9} \cdot 17^{6} \) |
$2.82492$ |
$(a), (-a-1), (2a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs |
$1$ |
\( 2 \cdot 3^{3} \) |
$0.099223475$ |
$0.663461224$ |
5.027345582 |
\( -\frac{2628365}{32} a + \frac{183347}{16} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 39 a + 249\) , \( -1104 a + 409\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(39a+249\right){x}-1104a+409$ |
| 19494.1-a4 |
19494.1-a |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
19494.1 |
\( 2 \cdot 3^{3} \cdot 19^{2} \) |
\( 2^{9} \cdot 3^{3} \cdot 19^{6} \) |
$2.98647$ |
$(a), (-a-1), (-3a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs |
$1$ |
\( 2 \) |
$0.764888608$ |
$1.086985707$ |
2.351619324 |
\( -\frac{2628365}{32} a + \frac{183347}{16} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( 26 a + 86\) , \( 238 a - 210\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(26a+86\right){x}+238a-210$ |
| 19494.3-e4 |
19494.3-e |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
19494.3 |
\( 2 \cdot 3^{3} \cdot 19^{2} \) |
\( 2^{9} \cdot 3^{3} \cdot 19^{6} \) |
$2.98647$ |
$(a), (-a-1), (3a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs |
$1$ |
\( 2 \cdot 3^{2} \) |
$0.279083422$ |
$1.086985707$ |
7.722277008 |
\( -\frac{2628365}{32} a + \frac{183347}{16} \) |
\( \bigl[a + 1\) , \( 0\) , \( a\) , \( -34 a + 83\) , \( 193 a + 257\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-34a+83\right){x}+193a+257$ |
| 27648.1-d4 |
27648.1-d |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
27648.1 |
\( 2^{10} \cdot 3^{3} \) |
\( 2^{39} \cdot 3^{3} \) |
$3.25911$ |
$(a), (-a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs |
$1$ |
\( 2 \) |
$1$ |
$0.837578739$ |
1.184515212 |
\( -\frac{2628365}{32} a + \frac{183347}{16} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -6 a + 159\) , \( -495 a - 17\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-6a+159\right){x}-495a-17$ |
| 27648.1-g4 |
27648.1-g |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
27648.1 |
\( 2^{10} \cdot 3^{3} \) |
\( 2^{39} \cdot 3^{9} \) |
$3.25911$ |
$(a), (-a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$0.483576310$ |
2.051640531 |
\( -\frac{2628365}{32} a + \frac{183347}{16} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 326 a - 133\) , \( -2591 a - 1725\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(326a-133\right){x}-2591a-1725$ |
| 27648.1-j4 |
27648.1-j |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
27648.1 |
\( 2^{10} \cdot 3^{3} \) |
\( 2^{39} \cdot 3^{9} \) |
$3.25911$ |
$(a), (-a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs |
$1$ |
\( 2^{2} \) |
$2.037801950$ |
$0.483576310$ |
5.574449434 |
\( -\frac{2628365}{32} a + \frac{183347}{16} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 326 a - 133\) , \( 2591 a + 1725\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(326a-133\right){x}+2591a+1725$ |
| 27648.1-m4 |
27648.1-m |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
27648.1 |
\( 2^{10} \cdot 3^{3} \) |
\( 2^{39} \cdot 3^{3} \) |
$3.25911$ |
$(a), (-a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs |
$1$ |
\( 2^{2} \) |
$1.242222194$ |
$0.837578739$ |
5.885724349 |
\( -\frac{2628365}{32} a + \frac{183347}{16} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -6 a + 159\) , \( 495 a + 17\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-6a+159\right){x}+495a+17$ |
| 33750.1-f4 |
33750.1-f |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
33750.1 |
\( 2 \cdot 3^{3} \cdot 5^{4} \) |
\( 2^{9} \cdot 3^{9} \cdot 5^{12} \) |
$3.42572$ |
$(a), (-a-1), (5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs |
$1$ |
\( 3^{2} \) |
$1$ |
$0.547104141$ |
3.481749436 |
\( -\frac{2628365}{32} a + \frac{183347}{16} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( -254 a + 103\) , \( 728 a - 2415\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-254a+103\right){x}+728a-2415$ |
*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.
