| 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 |
| 55.1-a1 |
55.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( - 5 \cdot 11^{4} \) |
$0.54414$ |
$(-2a+1), (-3a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$19.86707574$ |
0.493601465 |
\( -\frac{626283905886387}{73205} a + \frac{1013348626965991}{73205} \) |
\( \bigl[1\) , \( -\phi + 1\) , \( 1\) , \( 9 \phi - 25\) , \( -6 \phi + 44\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-\phi+1\right){x}^{2}+\left(9\phi-25\right){x}-6\phi+44$ |
| 55.1-a2 |
55.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( - 5^{3} \cdot 11^{12} \) |
$0.54414$ |
$(-2a+1), (-3a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$2.207452860$ |
0.493601465 |
\( -\frac{114278307303626907}{78460709418025} a + \frac{203603378036088236}{78460709418025} \) |
\( \bigl[1\) , \( -\phi + 1\) , \( 1\) , \( 54 \phi\) , \( -374 \phi - 198\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-\phi+1\right){x}^{2}+54\phi{x}-374\phi-198$ |
| 55.1-a3 |
55.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( - 5 \cdot 11 \) |
$0.54414$ |
$(-2a+1), (-3a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$39.73415148$ |
0.493601465 |
\( \frac{45227}{55} a + \frac{26979}{55} \) |
\( \bigl[\phi + 1\) , \( 0\) , \( \phi + 1\) , \( -\phi - 1\) , \( -\phi\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+\left(\phi+1\right){y}={x}^{3}+\left(-\phi-1\right){x}-\phi$ |
| 55.1-a4 |
55.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( 5^{6} \cdot 11^{6} \) |
$0.54414$ |
$(-2a+1), (-3a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$4.414905721$ |
0.493601465 |
\( -\frac{1485675267531}{221445125} a + \frac{4152064659709}{221445125} \) |
\( \bigl[1\) , \( -\phi + 1\) , \( 1\) , \( -21 \phi - 25\) , \( -54 \phi - 58\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-\phi+1\right){x}^{2}+\left(-21\phi-25\right){x}-54\phi-58$ |
| 55.1-a5 |
55.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( 5^{12} \cdot 11^{3} \) |
$0.54414$ |
$(-2a+1), (-3a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$2.207452860$ |
0.493601465 |
\( -\frac{4560282420936767}{20796875} a + \frac{7378860561741612}{20796875} \) |
\( \bigl[1\) , \( -\phi + 1\) , \( 1\) , \( -16 \phi - 210\) , \( 1110 \phi - 534\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-\phi+1\right){x}^{2}+\left(-16\phi-210\right){x}+1110\phi-534$ |
| 55.1-a6 |
55.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( 5^{2} \cdot 11^{2} \) |
$0.54414$ |
$(-2a+1), (-3a+1)$ |
0 |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$39.73415148$ |
0.493601465 |
\( \frac{132583563}{605} a + \frac{166070482}{605} \) |
\( \bigl[\phi + 1\) , \( 0\) , \( \phi + 1\) , \( 4 \phi - 11\) , \( -9 \phi + 13\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+\left(\phi+1\right){y}={x}^{3}+\left(4\phi-11\right){x}-9\phi+13$ |
| 55.1-a7 |
55.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( - 5^{3} \cdot 11^{3} \) |
$0.54414$ |
$(-2a+1), (-3a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$4.414905721$ |
0.493601465 |
\( \frac{754904381777}{33275} a + \frac{466557150454}{33275} \) |
\( \bigl[\phi + 1\) , \( 0\) , \( \phi + 1\) , \( -6 \phi - 1\) , \( \phi - 17\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+\left(\phi+1\right){y}={x}^{3}+\left(-6\phi-1\right){x}+\phi-17$ |
| 55.1-a8 |
55.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( 5^{4} \cdot 11 \) |
$0.54414$ |
$(-2a+1), (-3a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$19.86707574$ |
0.493601465 |
\( \frac{48555143354501}{275} a + \frac{30008729421823}{275} \) |
\( \bigl[\phi + 1\) , \( 0\) , \( \phi + 1\) , \( -6 \phi - 26\) , \( 28 \phi + 8\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+\left(\phi+1\right){y}={x}^{3}+\left(-6\phi-26\right){x}+28\phi+8$ |
*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.
