| 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 |
| 28.2-a1 |
28.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( - 2^{8} \cdot 7^{3} \) |
$0.58140$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 3 \) |
$1$ |
$2.155441053$ |
0.571547619 |
\( -\frac{1545435312128}{343} a + \frac{2185574023168}{343} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -18 a - 37\) , \( -68 a - 108\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(-18a-37\right){x}-68a-108$ |
| 28.2-a2 |
28.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( - 2^{8} \cdot 7 \) |
$0.58140$ |
$(a), (2a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 3 \) |
$1$ |
$19.39896948$ |
0.571547619 |
\( -\frac{4096}{7} a + \frac{16384}{7} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 2 a + 3\) , \( 0\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(2a+3\right){x}$ |
| 28.2-a3 |
28.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{4} \cdot 7^{2} \) |
$0.58140$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$38.79793896$ |
0.571547619 |
\( -\frac{435744}{49} a + \frac{712688}{49} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -2 a - 4\) , \( a + 1\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-2a-4\right){x}+a+1$ |
| 28.2-a4 |
28.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( - 2^{8} \cdot 7^{12} \) |
$0.58140$ |
$(a), (2a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$2.155441053$ |
0.571547619 |
\( \frac{29518306565684}{13841287201} a + \frac{41622722395132}{13841287201} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 9 a - 18\) , \( 320 a - 467\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(9a-18\right){x}+320a-467$ |
| 28.2-a5 |
28.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{8} \cdot 7 \) |
$0.58140$ |
$(a), (2a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 3 \) |
$1$ |
$19.39896948$ |
0.571547619 |
\( -\frac{1720664028}{7} a + \frac{2434028852}{7} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -17 a - 29\) , \( -49 a - 66\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-17a-29\right){x}-49a-66$ |
| 28.2-a6 |
28.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( - 2^{8} \cdot 7^{4} \) |
$0.58140$ |
$(a), (2a+1)$ |
0 |
$\Z/12\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$19.39896948$ |
0.571547619 |
\( \frac{861093316}{2401} a + \frac{1217791012}{2401} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( -a + 2\) , \( -12 a + 17\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-a+2\right){x}-12a+17$ |
| 28.2-a7 |
28.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{4} \cdot 7^{6} \) |
$0.58140$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2Cs, 3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$4.310882107$ |
0.571547619 |
\( \frac{1137747277344}{117649} a + \frac{1622386617968}{117649} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 34 a - 53\) , \( 133 a - 203\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(34a-53\right){x}+133a-203$ |
| 28.2-a8 |
28.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{8} \cdot 7^{3} \) |
$0.58140$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 3 \) |
$1$ |
$2.155441053$ |
0.571547619 |
\( \frac{91481168031853524}{343} a + \frac{129373908533024396}{343} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 19 a - 148\) , \( -318 a - 201\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(19a-148\right){x}-318a-201$ |
*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.
