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-a3 |
54.1-a |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
54.1 |
\( 2 \cdot 3^{3} \) |
\( 2^{27} \cdot 3^{11} \) |
$0.68514$ |
$(a), (-a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$0.911840236$ |
0.644768414 |
\( \frac{241123607}{16384} a + \frac{59710933}{8192} \) |
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( -51 a + 69\) , \( 62 a + 339\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-51a+69\right){x}+62a+339$ |
432.1-a3 |
432.1-a |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
432.1 |
\( 2^{4} \cdot 3^{3} \) |
\( 2^{39} \cdot 3^{11} \) |
$1.15227$ |
$(a), (-a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$0.555327402$ |
$0.455920118$ |
1.432230274 |
\( \frac{241123607}{16384} a + \frac{59710933}{8192} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( -201 a + 277\) , \( 974 a + 2839\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-201a+277\right){x}+974a+2839$ |
486.3-a3 |
486.3-a |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
486.3 |
\( 2 \cdot 3^{5} \) |
\( 2^{27} \cdot 3^{11} \) |
$1.18670$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{3} \) |
$1$ |
$0.911840236$ |
1.934305242 |
\( \frac{241123607}{16384} a + \frac{59710933}{8192} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 70 a - 11\) , \( -205 a - 287\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(70a-11\right){x}-205a-287$ |
3888.3-a3 |
3888.3-a |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3888.3 |
\( 2^{4} \cdot 3^{5} \) |
\( 2^{39} \cdot 3^{11} \) |
$1.99579$ |
$(a), (-a-1), (a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$0.455920118$ |
1.934305242 |
\( \frac{241123607}{16384} a + \frac{59710933}{8192} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 279 a - 35\) , \( -1881 a - 1696\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(279a-35\right){x}-1881a-1696$ |
6534.1-c3 |
6534.1-c |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6534.1 |
\( 2 \cdot 3^{3} \cdot 11^{2} \) |
\( 2^{27} \cdot 3^{5} \cdot 11^{6} \) |
$2.27237$ |
$(a), (-a-1), (a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 3 \) |
$1$ |
$0.476193032$ |
1.010157967 |
\( \frac{241123607}{16384} a + \frac{59710933}{8192} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -248 a - 92\) , \( -1568 a + 1072\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-248a-92\right){x}-1568a+1072$ |
6534.3-c3 |
6534.3-c |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6534.3 |
\( 2 \cdot 3^{3} \cdot 11^{2} \) |
\( 2^{27} \cdot 3^{5} \cdot 11^{6} \) |
$2.27237$ |
$(a), (-a-1), (a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \cdot 3^{4} \) |
$0.040466699$ |
$0.476193032$ |
4.414797924 |
\( \frac{241123607}{16384} a + \frac{59710933}{8192} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( 111 a - 328\) , \( -1204 a + 2020\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(111a-328\right){x}-1204a+2020$ |
6912.1-c3 |
6912.1-c |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6912.1 |
\( 2^{8} \cdot 3^{3} \) |
\( 2^{51} \cdot 3^{5} \) |
$2.30454$ |
$(a), (-a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$1.345277405$ |
$0.394838404$ |
3.004735343 |
\( \frac{241123607}{16384} a + \frac{59710933}{8192} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -156 a - 480\) , \( -2284 a - 3628\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-156a-480\right){x}-2284a-3628$ |
6912.1-d3 |
6912.1-d |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6912.1 |
\( 2^{8} \cdot 3^{3} \) |
\( 2^{51} \cdot 3^{5} \) |
$2.30454$ |
$(a), (-a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$0.394838404$ |
1.675157478 |
\( \frac{241123607}{16384} a + \frac{59710933}{8192} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -156 a - 480\) , \( 2284 a + 3628\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-156a-480\right){x}+2284a+3628$ |
15606.1-a3 |
15606.1-a |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
15606.1 |
\( 2 \cdot 3^{3} \cdot 17^{2} \) |
\( 2^{27} \cdot 3^{11} \cdot 17^{6} \) |
$2.82492$ |
$(a), (-a-1), (-2a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 3^{3} \) |
$1$ |
$0.221153741$ |
4.222241379 |
\( \frac{241123607}{16384} a + \frac{59710933}{8192} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -880 a - 1135\) , \( -17954 a - 7025\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-880a-1135\right){x}-17954a-7025$ |
15606.3-c3 |
15606.3-c |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
15606.3 |
\( 2 \cdot 3^{3} \cdot 17^{2} \) |
\( 2^{27} \cdot 3^{11} \cdot 17^{6} \) |
$2.82492$ |
$(a), (-a-1), (2a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \cdot 3^{3} \) |
$0.297670427$ |
$0.221153741$ |
5.027345582 |
\( \frac{241123607}{16384} a + \frac{59710933}{8192} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 779 a + 1274\) , \( 7756 a - 23803\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(779a+1274\right){x}+7756a-23803$ |
19494.1-a3 |
19494.1-a |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
19494.1 |
\( 2 \cdot 3^{3} \cdot 19^{2} \) |
\( 2^{27} \cdot 3^{5} \cdot 19^{6} \) |
$2.98647$ |
$(a), (-a-1), (-3a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$2.294665826$ |
$0.362328569$ |
2.351619324 |
\( \frac{241123607}{16384} a + \frac{59710933}{8192} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( 346 a + 391\) , \( -637 a + 5674\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(346a+391\right){x}-637a+5674$ |
19494.3-e3 |
19494.3-e |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
19494.3 |
\( 2 \cdot 3^{3} \cdot 19^{2} \) |
\( 2^{27} \cdot 3^{5} \cdot 19^{6} \) |
$2.98647$ |
$(a), (-a-1), (3a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \cdot 3^{4} \) |
$0.093027807$ |
$0.362328569$ |
7.722277008 |
\( \frac{241123607}{16384} a + \frac{59710933}{8192} \) |
\( \bigl[a + 1\) , \( 0\) , \( a\) , \( -14 a + 628\) , \( -4122 a - 92\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-14a+628\right){x}-4122a-92$ |
27648.1-d3 |
27648.1-d |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
27648.1 |
\( 2^{10} \cdot 3^{3} \) |
\( 2^{57} \cdot 3^{5} \) |
$3.25911$ |
$(a), (-a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$0.279192913$ |
1.184515212 |
\( \frac{241123607}{16384} a + \frac{59710933}{8192} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 314 a + 959\) , \( 7569 a - 8177\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(314a+959\right){x}+7569a-8177$ |
27648.1-g3 |
27648.1-g |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
27648.1 |
\( 2^{10} \cdot 3^{3} \) |
\( 2^{57} \cdot 3^{11} \) |
$3.25911$ |
$(a), (-a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$9$ |
\( 2 \) |
$1$ |
$0.161192103$ |
2.051640531 |
\( \frac{241123607}{16384} a + \frac{59710933}{8192} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 1606 a - 2213\) , \( 43809 a - 28957\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(1606a-2213\right){x}+43809a-28957$ |
27648.1-j3 |
27648.1-j |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
27648.1 |
\( 2^{10} \cdot 3^{3} \) |
\( 2^{57} \cdot 3^{11} \) |
$3.25911$ |
$(a), (-a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$6.113405851$ |
$0.161192103$ |
5.574449434 |
\( \frac{241123607}{16384} a + \frac{59710933}{8192} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 1606 a - 2213\) , \( -43809 a + 28957\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(1606a-2213\right){x}-43809a+28957$ |
27648.1-m3 |
27648.1-m |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
27648.1 |
\( 2^{10} \cdot 3^{3} \) |
\( 2^{57} \cdot 3^{5} \) |
$3.25911$ |
$(a), (-a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$3.726666584$ |
$0.279192913$ |
5.885724349 |
\( \frac{241123607}{16384} a + \frac{59710933}{8192} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 314 a + 959\) , \( -7569 a + 8177\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(314a+959\right){x}-7569a+8177$ |
33750.1-f3 |
33750.1-f |
$4$ |
$27$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
33750.1 |
\( 2 \cdot 3^{3} \cdot 5^{4} \) |
\( 2^{27} \cdot 3^{11} \cdot 5^{12} \) |
$3.42572$ |
$(a), (-a-1), (5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 3^{3} \) |
$1$ |
$0.182368047$ |
3.481749436 |
\( \frac{241123607}{16384} a + \frac{59710933}{8192} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( -1254 a + 1728\) , \( 13728 a + 43960\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-1254a+1728\right){x}+13728a+43960$ |
*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.