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
4.1-a1
4.1-a
$6$
$45$
4.4.10512.1
$4$
$[4, 0]$
4.1
\( 2^{2} \)
\( 2^{10} \)
$10.89529$
$(a^3-a^2-5a)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3, 5$
3B.1.2 , 5B.4.2
$9$
\( 5 \)
$3.227465434$
$0.353820636$
2.004816315
\( -\frac{46866366404774471142406706382678508925}{8} a^{3} - 837204824113363505263545929366924088 a^{2} + \frac{81776852534182441962754365040569479719}{2} a + \frac{327944882292203164723438266897992960759}{8} \)
\( \bigl[a^{2} - a - 4\) , \( a\) , \( a^{3} - 5 a - 5\) , \( 589 a^{3} + 152 a^{2} - 4327 a - 4831\) , \( 19288 a^{3} + 3575 a^{2} - 137422 a - 142859\bigr] \)
${y}^2+\left(a^{2}-a-4\right){x}{y}+\left(a^{3}-5a-5\right){y}={x}^{3}+a{x}^{2}+\left(589a^{3}+152a^{2}-4327a-4831\right){x}+19288a^{3}+3575a^{2}-137422a-142859$
4.1-b1
4.1-b
$6$
$45$
4.4.10512.1
$4$
$[4, 0]$
4.1
\( 2^{2} \)
\( 2^{10} \)
$10.89529$
$(a^3-a^2-5a)$
$1$
$\Z/3\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3, 5$
3B.1.1 , 5B.4.2
$1$
\( 5 \)
$3.227465434$
$28.65947154$
2.004816315
\( -\frac{46866366404774471142406706382678508925}{8} a^{3} - 837204824113363505263545929366924088 a^{2} + \frac{81776852534182441962754365040569479719}{2} a + \frac{327944882292203164723438266897992960759}{8} \)
\( \bigl[a^{3} - 6 a - 4\) , \( a^{3} - 6 a - 6\) , \( a^{3} - a^{2} - 5 a - 1\) , \( 589 a^{3} + 153 a^{2} - 4324 a - 4830\) , \( -18547 a^{3} - 3626 a^{2} + 131800 a + 137439\bigr] \)
${y}^2+\left(a^{3}-6a-4\right){x}{y}+\left(a^{3}-a^{2}-5a-1\right){y}={x}^{3}+\left(a^{3}-6a-6\right){x}^{2}+\left(589a^{3}+153a^{2}-4324a-4830\right){x}-18547a^{3}-3626a^{2}+131800a+137439$
4.1-c5
4.1-c
$6$
$45$
4.4.10512.1
$4$
$[4, 0]$
4.1
\( 2^{2} \)
\( 2^{10} \)
$10.89529$
$(a^3-a^2-5a)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3, 5$
3B , 5B.4.2
$9$
\( 1 \)
$1$
$17.53722959$
1.539433102
\( -\frac{46866366404774471142406706382678508925}{8} a^{3} - 837204824113363505263545929366924088 a^{2} + \frac{81776852534182441962754365040569479719}{2} a + \frac{327944882292203164723438266897992960759}{8} \)
\( \bigl[1\) , \( -a + 1\) , \( a^{3} - a^{2} - 5 a - 1\) , \( 2253 a^{3} - 4398 a^{2} - 7394 a + 1117\) , \( -114542 a^{3} + 212243 a^{2} + 407642 a - 62220\bigr] \)
${y}^2+{x}{y}+\left(a^{3}-a^{2}-5a-1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(2253a^{3}-4398a^{2}-7394a+1117\right){x}-114542a^{3}+212243a^{2}+407642a-62220$
4.1-d3
4.1-d
$6$
$45$
4.4.10512.1
$4$
$[4, 0]$
4.1
\( 2^{2} \)
\( 2^{10} \)
$10.89529$
$(a^3-a^2-5a)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3, 5$
3B , 5B.1.2
$225$
\( 1 \)
$1$
$0.578216325$
1.268908164
\( -\frac{46866366404774471142406706382678508925}{8} a^{3} - 837204824113363505263545929366924088 a^{2} + \frac{81776852534182441962754365040569479719}{2} a + \frac{327944882292203164723438266897992960759}{8} \)
\( \bigl[a^{3} - a^{2} - 5 a - 1\) , \( a + 1\) , \( a^{3} - a^{2} - 5 a\) , \( 2253 a^{3} - 4398 a^{2} - 7392 a + 1116\) , \( 116795 a^{3} - 216641 a^{2} - 415035 a + 63335\bigr] \)
${y}^2+\left(a^{3}-a^{2}-5a-1\right){x}{y}+\left(a^{3}-a^{2}-5a\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(2253a^{3}-4398a^{2}-7392a+1116\right){x}+116795a^{3}-216641a^{2}-415035a+63335$
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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.