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
$4$
$27$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
81.1
\( 3^{4} \)
\( 3^{4} \)
$1.22848$
$(-a+2)$
0
$\mathsf{trivial}$
$\textsf{potential}$
$-27$
$N(\mathrm{U}(1))$
✓
✓
$7$
7Ns.2.1
$1$
\( 1 \)
$1$
$9.363037422$
2.043182272
\( -12288000 \)
\( \bigl[0\) , \( -a - 1\) , \( 1\) , \( 11 a - 28\) , \( 31 a - 85\bigr] \)
${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(11a-28\right){x}+31a-85$
81.1-a2
81.1-a
$4$
$27$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
81.1
\( 3^{4} \)
\( 3^{4} \)
$1.22848$
$(-a+2)$
0
$\mathsf{trivial}$
$\textsf{potential}$
$-27$
$N(\mathrm{U}(1))$
✓
✓
$7$
7Ns.2.1
$1$
\( 1 \)
$1$
$9.363037422$
2.043182272
\( -12288000 \)
\( \bigl[0\) , \( a + 1\) , \( 1\) , \( -9 a - 18\) , \( -41 a - 72\bigr] \)
${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-9a-18\right){x}-41a-72$
81.1-a3
81.1-a
$4$
$27$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
81.1
\( 3^{4} \)
\( 3^{12} \)
$1.22848$
$(-a+2)$
0
$\mathsf{trivial}$
$\textsf{potential}$
$-3$
$N(\mathrm{U}(1))$
✓
✓
$3, 7$
3Cs , 7Ns.2.1
$1$
\( 1 \)
$1$
$9.363037422$
2.043182272
\( 0 \)
\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( 3 a + 5\bigr] \)
${y}^2+{y}={x}^{3}+3a+5$
81.1-a4
81.1-a
$4$
$27$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
81.1
\( 3^{4} \)
\( 3^{12} \)
$1.22848$
$(-a+2)$
0
$\mathsf{trivial}$
$\textsf{potential}$
$-3$
$N(\mathrm{U}(1))$
✓
✓
$3, 7$
3Cs , 7Ns.2.1
$1$
\( 1 \)
$1$
$9.363037422$
2.043182272
\( 0 \)
\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( -3 a + 8\bigr] \)
${y}^2+{y}={x}^{3}-3a+8$
81.1-b1
81.1-b
$4$
$27$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
81.1
\( 3^{4} \)
\( 3^{10} \)
$1.22848$
$(-a+2)$
0
$\mathsf{trivial}$
$\textsf{potential}$
$-27$
$N(\mathrm{U}(1))$
✓
✓
✓
$3, 7$
3B.1.2 , 7Ns.3.1
$1$
\( 3 \)
$1$
$1.040337491$
0.681060757
\( -12288000 \)
\( \bigl[0\) , \( 0\) , \( 1\) , \( -150 a - 270\) , \( -1518 a - 2720\bigr] \)
${y}^2+{y}={x}^{3}+\left(-150a-270\right){x}-1518a-2720$
81.1-b2
81.1-b
$4$
$27$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
81.1
\( 3^{4} \)
\( 3^{10} \)
$1.22848$
$(-a+2)$
0
$\Z/3\Z$
$\textsf{potential}$
$-27$
$N(\mathrm{U}(1))$
✓
✓
✓
$3, 7$
3B.1.1 , 7Ns.3.1
$1$
\( 1 \)
$1$
$28.08911226$
0.681060757
\( -12288000 \)
\( \bigl[0\) , \( 0\) , \( 1\) , \( -30\) , \( 63\bigr] \)
${y}^2+{y}={x}^{3}-30{x}+63$
81.1-b3
81.1-b
$4$
$27$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
81.1
\( 3^{4} \)
\( 3^{6} \)
$1.22848$
$(-a+2)$
0
$\Z/3\Z$
$\textsf{potential}$
$-3$
$N(\mathrm{U}(1))$
✓
✓
✓
$3, 7$
3Cs.1.1 , 7Ns.3.1
$1$
\( 1 \)
$1$
$28.08911226$
0.681060757
\( 0 \)
\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( 0\bigr] \)
${y}^2+{y}={x}^{3}$
81.1-b4
81.1-b
$4$
$27$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
81.1
\( 3^{4} \)
\( 3^{6} \)
$1.22848$
$(-a+2)$
0
$\Z/3\Z$
$\textsf{potential}$
$-3$
$N(\mathrm{U}(1))$
✓
✓
✓
$3, 7$
3Cs.1.1 , 7Ns.3.1
$1$
\( 3 \)
$1$
$9.363037422$
0.681060757
\( 0 \)
\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( 6 a - 17\bigr] \)
${y}^2+{y}={x}^{3}+6a-17$
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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.