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
729.1-a1
729.1-a
$2$
$3$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
729.1
\( 3^{6} \)
\( 3^{12} \)
$2.12780$
$(-a+2)$
$1$
$\mathsf{trivial}$
$\textsf{potential}$
$-3$
$N(\mathrm{U}(1))$
✓
✓
$7$
7Ns.2.1
$1$
\( 1 \)
$0.690454106$
$9.363037422$
2.821447178
\( 0 \)
\( \bigl[0\) , \( 0\) , \( a + 1\) , \( 0\) , \( -15 a - 27\bigr] \)
${y}^2+\left(a+1\right){y}={x}^{3}-15a-27$
729.1-a2
729.1-a
$2$
$3$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
729.1
\( 3^{6} \)
\( 3^{12} \)
$2.12780$
$(-a+2)$
$1$
$\mathsf{trivial}$
$\textsf{potential}$
$-3$
$N(\mathrm{U}(1))$
✓
✓
$7$
7Ns.2.1
$1$
\( 3 \)
$0.230151368$
$9.363037422$
2.821447178
\( 0 \)
\( \bigl[0\) , \( 0\) , \( a + 1\) , \( 0\) , \( -3\bigr] \)
${y}^2+\left(a+1\right){y}={x}^{3}-3$
729.1-b1
729.1-b
$2$
$3$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
729.1
\( 3^{6} \)
\( 3^{12} \)
$2.12780$
$(-a+2)$
$1$
$\mathsf{trivial}$
$\textsf{potential}$
$-3$
$N(\mathrm{U}(1))$
✓
✓
$7$
7Ns.2.1
$1$
\( 1 \)
$0.690454106$
$9.363037422$
2.821447178
\( 0 \)
\( \bigl[0\) , \( 0\) , \( a\) , \( 0\) , \( 14 a - 41\bigr] \)
${y}^2+a{y}={x}^{3}+14a-41$
729.1-b2
729.1-b
$2$
$3$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
729.1
\( 3^{6} \)
\( 3^{12} \)
$2.12780$
$(-a+2)$
$1$
$\mathsf{trivial}$
$\textsf{potential}$
$-3$
$N(\mathrm{U}(1))$
✓
✓
$7$
7Ns.2.1
$1$
\( 3 \)
$0.230151368$
$9.363037422$
2.821447178
\( 0 \)
\( \bigl[0\) , \( 0\) , \( a\) , \( 0\) , \( -a - 2\bigr] \)
${y}^2+a{y}={x}^{3}-a-2$
729.1-c1
729.1-c
$2$
$3$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
729.1
\( 3^{6} \)
\( 3^{6} \)
$2.12780$
$(-a+2)$
$1$
$\mathsf{trivial}$
$\textsf{potential}$
$-3$
$N(\mathrm{U}(1))$
✓
✓
$3, 7$
3B.1.2 , 7Ns.3.1
$1$
\( 1 \)
$0.977127928$
$9.363037422$
3.992900922
\( 0 \)
\( \bigl[0\) , \( 0\) , \( a\) , \( 0\) , \( -2\bigr] \)
${y}^2+a{y}={x}^{3}-2$
729.1-c2
729.1-c
$2$
$3$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
729.1
\( 3^{6} \)
\( 3^{6} \)
$2.12780$
$(-a+2)$
$1$
$\Z/3\Z$
$\textsf{potential}$
$-3$
$N(\mathrm{U}(1))$
✓
✓
$3, 7$
3B.1.1 , 7Ns.3.1
$1$
\( 1 \)
$2.931383785$
$28.08911226$
3.992900922
\( 0 \)
\( \bigl[0\) , \( 0\) , \( a\) , \( 0\) , \( a + 1\bigr] \)
${y}^2+a{y}={x}^{3}+a+1$
729.1-d1
729.1-d
$2$
$3$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
729.1
\( 3^{6} \)
\( 3^{6} \)
$2.12780$
$(-a+2)$
$1$
$\mathsf{trivial}$
$\textsf{potential}$
$-3$
$N(\mathrm{U}(1))$
✓
✓
$3, 7$
3B.1.2 , 7Ns.3.1
$1$
\( 1 \)
$0.977127928$
$9.363037422$
3.992900922
\( 0 \)
\( \bigl[0\) , \( 0\) , \( a + 1\) , \( 0\) , \( -a - 2\bigr] \)
${y}^2+\left(a+1\right){y}={x}^{3}-a-2$
729.1-d2
729.1-d
$2$
$3$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
729.1
\( 3^{6} \)
\( 3^{6} \)
$2.12780$
$(-a+2)$
$1$
$\Z/3\Z$
$\textsf{potential}$
$-3$
$N(\mathrm{U}(1))$
✓
✓
$3, 7$
3B.1.1 , 7Ns.3.1
$1$
\( 1 \)
$2.931383785$
$28.08911226$
3.992900922
\( 0 \)
\( \bigl[0\) , \( 0\) , \( a + 1\) , \( 0\) , \( -2 a + 2\bigr] \)
${y}^2+\left(a+1\right){y}={x}^{3}-2a+2$
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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.