| 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 |
| 49.1-a1 |
49.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{9} \) |
$1.72118$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$2.700428314$ |
1.669195602 |
\( -\frac{3825287113585893}{117649} a + \frac{15836899268209340}{117649} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 392 a - 1636\) , \( 9025 a - 37367\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(392a-1636\right){x}+9025a-37367$ |
| 49.1-a2 |
49.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
49.1 |
\( 7^{2} \) |
\( - 7^{9} \) |
$1.72118$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$2.700428314$ |
1.669195602 |
\( \frac{8183558401}{117649} a - \frac{31455098491}{117649} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 22 a - 101\) , \( 180 a - 751\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(22a-101\right){x}+180a-751$ |
| 49.1-a3 |
49.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
49.1 |
\( 7^{2} \) |
\( - 7^{3} \) |
$1.72118$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$9$ |
\( 2 \) |
$1$ |
$24.30385482$ |
1.669195602 |
\( \frac{1063343}{49} a + \frac{3353382}{49} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -3 a + 4\) , \( -2 a + 4\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-3a+4\right){x}-2a+4$ |
| 49.1-a4 |
49.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{3} \) |
$1.72118$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$9$ |
\( 2 \) |
$1$ |
$24.30385482$ |
1.669195602 |
\( \frac{167034552579}{49} a + \frac{524498013905}{49} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 2 a - 21\) , \( 20 a - 85\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(2a-21\right){x}+20a-85$ |
| 49.1-b1 |
49.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
49.1 |
\( 7^{2} \) |
\( - 7^{9} \) |
$1.72118$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$2.700428314$ |
1.669195602 |
\( -\frac{8183558401}{117649} a - \frac{23271540090}{117649} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -24 a - 79\) , \( -181 a - 571\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-24a-79\right){x}-181a-571$ |
| 49.1-b2 |
49.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
49.1 |
\( 7^{2} \) |
\( - 7^{3} \) |
$1.72118$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$9$ |
\( 2 \) |
$1$ |
$24.30385482$ |
1.669195602 |
\( -\frac{1063343}{49} a + \frac{4416725}{49} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( a + 1\) , \( a + 2\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}+a+2$ |
| 49.1-b3 |
49.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{3} \) |
$1.72118$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$9$ |
\( 2 \) |
$1$ |
$24.30385482$ |
1.669195602 |
\( -\frac{167034552579}{49} a + \frac{691532566484}{49} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -4 a - 19\) , \( -21 a - 65\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-4a-19\right){x}-21a-65$ |
| 49.1-b4 |
49.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{9} \) |
$1.72118$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$2.700428314$ |
1.669195602 |
\( \frac{3825287113585893}{117649} a + \frac{12011612154623447}{117649} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -394 a - 1244\) , \( -9026 a - 28342\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-394a-1244\right){x}-9026a-28342$ |
| 49.1-c1 |
49.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{8} \) |
$1.72118$ |
$(-a-2), (-a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.442032891$ |
$5.921998271$ |
2.876569790 |
\( \frac{48788531}{117649} a - \frac{201772345}{117649} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( a + 5\) , \( 4 a + 10\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a+5\right){x}+4a+10$ |
| 49.1-d1 |
49.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{8} \) |
$1.72118$ |
$(-a-2), (-a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.442032891$ |
$5.921998271$ |
2.876569790 |
\( -\frac{48788531}{117649} a - \frac{152983814}{117649} \) |
\( \bigl[1\) , \( -a - 1\) , \( a\) , \( 5\) , \( -4 a + 9\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+5{x}-4a+9$ |
*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.
