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
81.1-a1
81.1-a
$4$
$27$
\(\Q(\sqrt{15}) \)
$2$
$[2, 0]$
81.1
\( 3^{4} \)
\( 2^{12} \cdot 3^{10} \)
$2.07652$
$(3,a)$
$1$
$\mathsf{trivial}$
$\textsf{potential}$
$-27$
$N(\mathrm{U}(1))$
✓
✓
✓
$3$
3B.1.2
$1$
\( 1 \)
$9.354323771$
$1.040337491$
2.512702187
\( -12288000 \)
\( \bigl[0\) , \( 0\) , \( 0\) , \( -960 a - 3720\) , \( -31878 a - 123464\bigr] \)
${y}^2={x}^{3}+\left(-960a-3720\right){x}-31878a-123464$
81.1-a2
81.1-a
$4$
$27$
\(\Q(\sqrt{15}) \)
$2$
$[2, 0]$
81.1
\( 3^{4} \)
\( 3^{10} \)
$2.07652$
$(3,a)$
$1$
$\Z/3\Z$
$\textsf{potential}$
$-27$
$N(\mathrm{U}(1))$
✓
✓
✓
$3$
3B.1.1
$1$
\( 1 \)
$3.118107923$
$28.08911226$
2.512702187
\( -12288000 \)
\( \bigl[0\) , \( 0\) , \( a\) , \( -14880 a - 57630\) , \( 1944558 a + 7531237\bigr] \)
${y}^2+a{y}={x}^{3}+\left(-14880a-57630\right){x}+1944558a+7531237$
81.1-a3
81.1-a
$4$
$27$
\(\Q(\sqrt{15}) \)
$2$
$[2, 0]$
81.1
\( 3^{4} \)
\( 3^{6} \)
$2.07652$
$(3,a)$
$1$
$\Z/3\Z$
$\textsf{potential}$
$-3$
$N(\mathrm{U}(1))$
✓
✓
✓
$3$
3Cs.1.1
$1$
\( 3 \)
$1.039369307$
$28.08911226$
2.512702187
\( 0 \)
\( \bigl[0\) , \( 0\) , \( a\) , \( 0\) , \( 7686 a + 29764\bigr] \)
${y}^2+a{y}={x}^{3}+7686a+29764$
81.1-a4
81.1-a
$4$
$27$
\(\Q(\sqrt{15}) \)
$2$
$[2, 0]$
81.1
\( 3^{4} \)
\( 2^{12} \cdot 3^{6} \)
$2.07652$
$(3,a)$
$1$
$\Z/3\Z$
$\textsf{potential}$
$-3$
$N(\mathrm{U}(1))$
✓
✓
✓
$3$
3Cs.1.1
$1$
\( 3 \)
$3.118107923$
$9.363037422$
2.512702187
\( 0 \)
\( \bigl[0\) , \( 0\) , \( 0\) , \( 0\) , \( -126 a - 488\bigr] \)
${y}^2={x}^{3}-126a-488$
81.1-b1
81.1-b
$4$
$27$
\(\Q(\sqrt{15}) \)
$2$
$[2, 0]$
81.1
\( 3^{4} \)
\( 3^{10} \)
$2.07652$
$(3,a)$
$1$
$\mathsf{trivial}$
$\textsf{potential}$
$-27$
$N(\mathrm{U}(1))$
✓
✓
✓
$1$
\( 3 \)
$3.650319782$
$1.040337491$
2.941580832
\( -12288000 \)
\( \bigl[0\) , \( 0\) , \( a\) , \( -30\) , \( -67\bigr] \)
${y}^2+a{y}={x}^{3}-30{x}-67$
81.1-b2
81.1-b
$4$
$27$
\(\Q(\sqrt{15}) \)
$2$
$[2, 0]$
81.1
\( 3^{4} \)
\( 2^{12} \cdot 3^{10} \)
$2.07652$
$(3,a)$
$1$
$\mathsf{trivial}$
$\textsf{potential}$
$-27$
$N(\mathrm{U}(1))$
✓
✓
✓
$1$
\( 3 \)
$0.135197028$
$28.08911226$
2.941580832
\( -12288000 \)
\( \bigl[0\) , \( 0\) , \( 0\) , \( -960 a - 3720\) , \( 31878 a + 123464\bigr] \)
${y}^2={x}^{3}+\left(-960a-3720\right){x}+31878a+123464$
81.1-b3
81.1-b
$4$
$27$
\(\Q(\sqrt{15}) \)
$2$
$[2, 0]$
81.1
\( 3^{4} \)
\( 3^{6} \)
$2.07652$
$(3,a)$
$1$
$\mathsf{trivial}$
$\textsf{potential}$
$-3$
$N(\mathrm{U}(1))$
✓
✓
✓
$3$
3Cs
$1$
\( 1 \)
$1.216773260$
$9.363037422$
2.941580832
\( 0 \)
\( \bigl[0\) , \( 0\) , \( a\) , \( 0\) , \( -4\bigr] \)
${y}^2+a{y}={x}^{3}-4$
81.1-b4
81.1-b
$4$
$27$
\(\Q(\sqrt{15}) \)
$2$
$[2, 0]$
81.1
\( 3^{4} \)
\( 2^{12} \cdot 3^{6} \)
$2.07652$
$(3,a)$
$1$
$\mathsf{trivial}$
$\textsf{potential}$
$-3$
$N(\mathrm{U}(1))$
✓
✓
✓
$3$
3Cs
$1$
\( 1 \)
$0.405591086$
$28.08911226$
2.941580832
\( 0 \)
\( \bigl[0\) , \( 0\) , \( 0\) , \( 0\) , \( 126 a + 488\bigr] \)
${y}^2={x}^{3}+126a+488$
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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.