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 |
648.3-a1 |
648.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
648.3 |
\( 2^{3} \cdot 3^{4} \) |
\( 2^{10} \cdot 3^{28} \) |
$1.27520$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1.079273864$ |
$0.605891169$ |
1.849572148 |
\( \frac{207646}{6561} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 37\) , \( -607\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+37{x}-607$ |
648.3-a2 |
648.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
648.3 |
\( 2^{3} \cdot 3^{4} \) |
\( 2^{8} \cdot 3^{14} \) |
$1.27520$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$0.134909233$ |
$2.423564678$ |
1.849572148 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 6\) , \( -7\bigr] \) |
${y}^2={x}^{3}+6{x}-7$ |
648.3-a3 |
648.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
648.3 |
\( 2^{3} \cdot 3^{4} \) |
\( 2^{4} \cdot 3^{16} \) |
$1.27520$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$0.269818466$ |
$2.423564678$ |
1.849572148 |
\( \frac{35152}{9} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -8\) , \( 14\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}-8{x}+14$ |
648.3-a4 |
648.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
648.3 |
\( 2^{3} \cdot 3^{4} \) |
\( 2^{8} \cdot 3^{20} \) |
$1.27520$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$0.539636932$ |
$1.211782339$ |
1.849572148 |
\( \frac{1556068}{81} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -53\) , \( -121\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}-53{x}-121$ |
648.3-a5 |
648.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
648.3 |
\( 2^{3} \cdot 3^{4} \) |
\( 2^{11} \cdot 3^{32} \) |
$1.27520$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$2.158547729$ |
$0.302945584$ |
1.849572148 |
\( -\frac{2327042746553}{43046721} a + \frac{2081263802600}{43046721} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 630 a + 757\) , \( 2646 a - 15079\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(630a+757\right){x}+2646a-15079$ |
648.3-a6 |
648.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
648.3 |
\( 2^{3} \cdot 3^{4} \) |
\( 2^{11} \cdot 3^{32} \) |
$1.27520$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$2.158547729$ |
$0.302945584$ |
1.849572148 |
\( \frac{2327042746553}{43046721} a + \frac{2081263802600}{43046721} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -630 a + 757\) , \( -2646 a - 15079\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-630a+757\right){x}-2646a-15079$ |
648.3-a7 |
648.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
648.3 |
\( 2^{3} \cdot 3^{4} \) |
\( 2^{8} \cdot 3^{14} \) |
$1.27520$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.539636932$ |
$1.211782339$ |
1.849572148 |
\( \frac{28756228}{3} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -143\) , \( 743\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}-143{x}+743$ |
648.3-a8 |
648.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
648.3 |
\( 2^{3} \cdot 3^{4} \) |
\( 2^{10} \cdot 3^{16} \) |
$1.27520$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1.079273864$ |
$0.605891169$ |
1.849572148 |
\( \frac{3065617154}{9} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -863\) , \( -9355\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}-863{x}-9355$ |
*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.