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
76.1-a1
76.1-a
$2$
$5$
\(\Q(\sqrt{5}) \)
$2$
$[2, 0]$
76.1
\( 2^{2} \cdot 19 \)
\( 2^{10} \cdot 19^{5} \)
$0.58997$
$(4a-3), (2)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$5$
5B.1.4[2]
$1$
\( 1 \)
$1$
$1.294481094$
0.578909544
\( -\frac{216015168585867}{79235168} a - \frac{133460768531977}{79235168} \)
\( \bigl[1\) , \( 0\) , \( \phi\) , \( -29 \phi + 3\) , \( -28 \phi - 68\bigr] \)
${y}^2+{x}{y}+\phi{y}={x}^{3}+\left(-29\phi+3\right){x}-28\phi-68$
76.1-a2
76.1-a
$2$
$5$
\(\Q(\sqrt{5}) \)
$2$
$[2, 0]$
76.1
\( 2^{2} \cdot 19 \)
\( 2^{2} \cdot 19 \)
$0.58997$
$(4a-3), (2)$
0
$\Z/5\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$5$
5B.1.1[2]
$1$
\( 1 \)
$1$
$32.36202735$
0.578909544
\( \frac{2748165}{38} a - \frac{2190494}{19} \)
\( \bigl[\phi\) , \( 0\) , \( \phi\) , \( \phi\) , \( 0\bigr] \)
${y}^2+\phi{x}{y}+\phi{y}={x}^{3}+\phi{x}$
76.1-b1
76.1-b
$4$
$27$
\(\Q(\sqrt{5}) \)
$2$
$[2, 0]$
76.1
\( 2^{2} \cdot 19 \)
\( 2^{2} \cdot 19^{3} \)
$0.58997$
$(4a-3), (2)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3B.1.2
$9$
\( 3 \)
$1$
$0.048505770$
0.585695878
\( \frac{114103502883977596459469}{13718} a - \frac{184623348172988440508401}{13718} \)
\( \bigl[\phi + 1\) , \( 0\) , \( 1\) , \( 3364 \phi - 15986\) , \( 229016 \phi - 793226\bigr] \)
${y}^2+\left(\phi+1\right){x}{y}+{y}={x}^{3}+\left(3364\phi-15986\right){x}+229016\phi-793226$
76.1-b2
76.1-b
$4$
$27$
\(\Q(\sqrt{5}) \)
$2$
$[2, 0]$
76.1
\( 2^{2} \cdot 19 \)
\( 2^{6} \cdot 19 \)
$0.58997$
$(4a-3), (2)$
0
$\Z/9\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3B.1.1
$1$
\( 3 \)
$1$
$35.36070655$
0.585695878
\( -\frac{2000807}{152} a - \frac{1240925}{152} \)
\( \bigl[\phi + 1\) , \( 0\) , \( 1\) , \( -\phi - 1\) , \( 0\bigr] \)
${y}^2+\left(\phi+1\right){x}{y}+{y}={x}^{3}+\left(-\phi-1\right){x}$
76.1-b3
76.1-b
$4$
$27$
\(\Q(\sqrt{5}) \)
$2$
$[2, 0]$
76.1
\( 2^{2} \cdot 19 \)
\( 2^{6} \cdot 19^{9} \)
$0.58997$
$(4a-3), (2)$
0
$\Z/3\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3Cs.1.1
$1$
\( 3^{3} \)
$1$
$0.436551932$
0.585695878
\( \frac{1526212543995640211}{1290750791116} a - \frac{4951272962401030967}{2581501582232} \)
\( \bigl[\phi + 1\) , \( 0\) , \( 1\) , \( 44 \phi - 196\) , \( 264 \phi - 1122\bigr] \)
${y}^2+\left(\phi+1\right){x}{y}+{y}={x}^{3}+\left(44\phi-196\right){x}+264\phi-1122$
76.1-b4
76.1-b
$4$
$27$
\(\Q(\sqrt{5}) \)
$2$
$[2, 0]$
76.1
\( 2^{2} \cdot 19 \)
\( 2^{18} \cdot 19^{3} \)
$0.58997$
$(4a-3), (2)$
0
$\Z/9\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3Cs.1.1
$1$
\( 3^{3} \)
$1$
$3.928967394$
0.585695878
\( \frac{1183564903}{1755904} a + \frac{1562473373}{3511808} \)
\( \bigl[\phi + 1\) , \( 0\) , \( 1\) , \( 4 \phi + 4\) , \( 8 \phi - 2\bigr] \)
${y}^2+\left(\phi+1\right){x}{y}+{y}={x}^{3}+\left(4\phi+4\right){x}+8\phi-2$
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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.