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
\( -\frac{106208}{31} a + \frac{51455}{31} \)
\( \bigl[1\) , \( \phi + 1\) , \( \phi\) , \( \phi\) , \( 0\bigr] \)
${y}^2+{x}{y}+\phi{y}={x}^{3}+\left(\phi+1\right){x}^{2}+\phi{x}$
31.1-a2
31.1-a
$6$
$8$
\(\Q(\sqrt{5}) \)
$2$
$[2, 0]$
31.1
\( 31 \)
\( - 31^{2} \)
$0.47148$
$(5a-2)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2B
$1$
\( 2 \)
$1$
$1.609651241$
0.359928959
\( -\frac{61725871986044215714}{961} a + \frac{99874558858644938523}{961} \)
\( \bigl[\phi\) , \( -1\) , \( \phi + 1\) , \( 133 \phi - 141\) , \( 737 \phi - 764\bigr] \)
${y}^2+\phi{x}{y}+\left(\phi+1\right){y}={x}^{3}-{x}^{2}+\left(133\phi-141\right){x}+737\phi-764$
31.1-a3
31.1-a
$6$
$8$
\(\Q(\sqrt{5}) \)
$2$
$[2, 0]$
31.1
\( 31 \)
\( 31^{4} \)
$0.47148$
$(5a-2)$
0
$\Z/2\Z\oplus\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2Cs
$1$
\( 2 \)
$1$
$6.438604964$
0.359928959
\( -\frac{156520379364360}{923521} a + \frac{253260256463213}{923521} \)
\( \bigl[\phi\) , \( -1\) , \( \phi + 1\) , \( -12 \phi - 21\) , \( 42 \phi + 10\bigr] \)
${y}^2+\phi{x}{y}+\left(\phi+1\right){y}={x}^{3}-{x}^{2}+\left(-12\phi-21\right){x}+42\phi+10$
31.1-a4
31.1-a
$6$
$8$
\(\Q(\sqrt{5}) \)
$2$
$[2, 0]$
31.1
\( 31 \)
\( - 31^{8} \)
$0.47148$
$(5a-2)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2B
$1$
\( 2 \)
$1$
$1.609651241$
0.359928959
\( \frac{11889611722383394}{852891037441} a - \frac{8629385062119691}{852891037441} \)
\( \bigl[1\) , \( \phi + 1\) , \( \phi\) , \( 31 \phi - 75\) , \( 141 \phi - 303\bigr] \)
${y}^2+{x}{y}+\phi{y}={x}^{3}+\left(\phi+1\right){x}^{2}+\left(31\phi-75\right){x}+141\phi-303$
31.1-a5
31.1-a
$6$
$8$
\(\Q(\sqrt{5}) \)
$2$
$[2, 0]$
31.1
\( 31 \)
\( 31^{2} \)
$0.47148$
$(5a-2)$
0
$\Z/2\Z\oplus\Z/4\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2Cs
$1$
\( 2 \)
$1$
$25.75441985$
0.359928959
\( \frac{9029272560}{961} a + \frac{5599830233}{961} \)
\( \bigl[1\) , \( \phi + 1\) , \( \phi\) , \( \phi - 5\) , \( 3 \phi - 5\bigr] \)
${y}^2+{x}{y}+\phi{y}={x}^{3}+\left(\phi+1\right){x}^{2}+\left(\phi-5\right){x}+3\phi-5$
31.1-a6
31.1-a
$6$
$8$
\(\Q(\sqrt{5}) \)
$2$
$[2, 0]$
31.1
\( 31 \)
\( -31 \)
$0.47148$
$(5a-2)$
0
$\Z/4\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2B
$1$
\( 1 \)
$1$
$12.87720992$
0.359928959
\( \frac{6130703730739448}{31} a + \frac{3788983280553597}{31} \)
\( \bigl[\phi + 1\) , \( -\phi + 1\) , \( \phi\) , \( -18 \phi + 15\) , \( 171 \phi - 265\bigr] \)
${y}^2+\left(\phi+1\right){x}{y}+\phi{y}={x}^{3}+\left(-\phi+1\right){x}^{2}+\left(-18\phi+15\right){x}+171\phi-265$
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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.