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
31.1-a1
31.1-a
$6$
$8$
\(\Q(\sqrt{5}) \)
$2$
$[2, 0]$
31.1
\( 31 \)
\( -31 \)
$0.47148$
$(5a-2)$
0
$\Z/8\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2B
$1$
\( 1 \)
$1$
$51.50883971$
0.359928959
\( -\frac{106208}{31} a + \frac{51455}{31} \)
\( \bigl[1\) , \( \phi + 1\) , \( \phi\) , \( \phi\) , \( 0\bigr] \)
${y}^2+{x}{y}+\phi{y}={x}^{3}+\left(\phi+1\right){x}^{2}+\phi{x}$
775.1-a1
775.1-a
$6$
$8$
\(\Q(\sqrt{5}) \)
$2$
$[2, 0]$
775.1
\( 5^{2} \cdot 31 \)
\( - 5^{6} \cdot 31 \)
$1.05426$
$(-2a+1), (5a-2)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
$2$
2B
$1$
\( 2 \)
$1$
$6.746456922$
1.508553628
\( -\frac{106208}{31} a + \frac{51455}{31} \)
\( \bigl[\phi + 1\) , \( \phi - 1\) , \( 1\) , \( -\phi - 2\) , \( -3 \phi - 2\bigr] \)
${y}^2+\left(\phi+1\right){x}{y}+{y}={x}^{3}+\left(\phi-1\right){x}^{2}+\left(-\phi-2\right){x}-3\phi-2$
961.2-c1
961.2-c
$6$
$8$
\(\Q(\sqrt{5}) \)
$2$
$[2, 0]$
961.2
\( 31^{2} \)
\( - 31^{7} \)
$1.11251$
$(5a-2)$
$1$
$\Z/4\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
$2$
2B
$1$
\( 2^{2} \)
$1.774176701$
$3.447362312$
1.367630581
\( -\frac{106208}{31} a + \frac{51455}{31} \)
\( \bigl[\phi + 1\) , \( -\phi\) , \( 0\) , \( 7 \phi - 23\) , \( -24 \phi + 20\bigr] \)
${y}^2+\left(\phi+1\right){x}{y}={x}^{3}-\phi{x}^{2}+\left(7\phi-23\right){x}-24\phi+20$
2511.1-f1
2511.1-f
$6$
$8$
\(\Q(\sqrt{5}) \)
$2$
$[2, 0]$
2511.1
\( 3^{4} \cdot 31 \)
\( - 3^{12} \cdot 31 \)
$1.41444$
$(5a-2), (3)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
$2$
2B
$1$
\( 2 \)
$1$
$5.028512095$
1.124409487
\( -\frac{106208}{31} a + \frac{51455}{31} \)
\( \bigl[1\) , \( -1\) , \( 1\) , \( 3 \phi - 8\) , \( 2 \phi - 5\bigr] \)
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(3\phi-8\right){x}+2\phi-5$
3751.4-b1
3751.4-b
$6$
$8$
\(\Q(\sqrt{5}) \)
$2$
$[2, 0]$
3751.4
\( 11^{2} \cdot 31 \)
\( - 11^{6} \cdot 31 \)
$1.56372$
$(-3a+2), (5a-2)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
$2$
2B
$1$
\( 2 \)
$1$
$12.20614154$
2.729376224
\( -\frac{106208}{31} a + \frac{51455}{31} \)
\( \bigl[\phi + 1\) , \( \phi\) , \( \phi + 1\) , \( 6 \phi - 12\) , \( 6 \phi - 11\bigr] \)
${y}^2+\left(\phi+1\right){x}{y}+\left(\phi+1\right){y}={x}^{3}+\phi{x}^{2}+\left(6\phi-12\right){x}+6\phi-11$
3751.6-a1
3751.6-a
$6$
$8$
\(\Q(\sqrt{5}) \)
$2$
$[2, 0]$
3751.6
\( 11^{2} \cdot 31 \)
\( - 11^{6} \cdot 31 \)
$1.56372$
$(-3a+1), (5a-2)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
$2$
2B
$1$
\( 2^{2} \)
$1$
$5.787239209$
2.588132054
\( -\frac{106208}{31} a + \frac{51455}{31} \)
\( \bigl[\phi\) , \( 1\) , \( 0\) , \( 2 \phi - 7\) , \( -5 \phi + 1\bigr] \)
${y}^2+\phi{x}{y}={x}^{3}+{x}^{2}+\left(2\phi-7\right){x}-5\phi+1$
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Pari/GP
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*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.