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
27.1-a1
27.1-a
$2$
$13$
\(\Q(\zeta_{21})^+\)
$6$
$[6, 0]$
27.1
\( 3^{3} \)
\( 3^{78} \)
$79.22201$
$(a^3+a^2-2a-1)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
✓
✓
$13$
13B.12.2
$1$
\( 2 \)
$1$
$350.7343913$
1.04131
\( -\frac{1713910976512}{1594323} \)
\( \bigl[0\) , \( a^{5} - a^{4} - 5 a^{3} + 4 a^{2} + 5 a - 2\) , \( a^{3} - 2 a\) , \( 652 a^{5} + 912 a^{4} - 2478 a^{3} - 3387 a^{2} + 131 a - 130\) , \( -26388 a^{5} - 27297 a^{4} + 115694 a^{3} + 96838 a^{2} - 65135 a + 8321\bigr] \)
${y}^2+\left(a^{3}-2a\right){y}={x}^{3}+\left(a^{5}-a^{4}-5a^{3}+4a^{2}+5a-2\right){x}^{2}+\left(652a^{5}+912a^{4}-2478a^{3}-3387a^{2}+131a-130\right){x}-26388a^{5}-27297a^{4}+115694a^{3}+96838a^{2}-65135a+8321$
27.1-a2
27.1-a
$2$
$13$
\(\Q(\zeta_{21})^+\)
$6$
$[6, 0]$
27.1
\( 3^{3} \)
\( 3^{6} \)
$79.22201$
$(a^3+a^2-2a-1)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
✓
✓
$13$
13B.12.1
$1$
\( 2 \)
$1$
$350.7343913$
1.04131
\( -\frac{28672}{3} \)
\( \bigl[0\) , \( -a^{5} + 4 a^{3} - 2 a^{2} - 2 a + 5\) , \( a^{4} - 3 a^{2} + 1\) , \( -2 a^{5} + 7 a^{4} + 4 a^{3} - 28 a^{2} + 17 a + 4\) , \( -6 a^{5} + 13 a^{4} + 15 a^{3} - 52 a^{2} + 33 a - 3\bigr] \)
${y}^2+\left(a^{4}-3a^{2}+1\right){y}={x}^{3}+\left(-a^{5}+4a^{3}-2a^{2}-2a+5\right){x}^{2}+\left(-2a^{5}+7a^{4}+4a^{3}-28a^{2}+17a+4\right){x}-6a^{5}+13a^{4}+15a^{3}-52a^{2}+33a-3$
27.1-b1
27.1-b
$2$
$13$
\(\Q(\zeta_{21})^+\)
$6$
$[6, 0]$
27.1
\( 3^{3} \)
\( 3^{78} \)
$79.22201$
$(a^3+a^2-2a-1)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
✓
✓
$13$
13B.1.2
$169$
\( 2 \cdot 13 \)
$0.991510447$
$0.046982890$
1.82314
\( -\frac{1713910976512}{1594323} \)
\( \bigl[0\) , \( -a^{5} + 5 a^{3} - a^{2} - 5 a + 2\) , \( a^{5} - 5 a^{3} + a^{2} + 5 a - 2\) , \( -391 a^{5} + 2607 a^{3} - 391 a^{2} - 3911 a + 522\) , \( -5979 a^{5} + 40423 a^{3} - 5979 a^{2} - 61479 a + 8968\bigr] \)
${y}^2+\left(a^{5}-5a^{3}+a^{2}+5a-2\right){y}={x}^{3}+\left(-a^{5}+5a^{3}-a^{2}-5a+2\right){x}^{2}+\left(-391a^{5}+2607a^{3}-391a^{2}-3911a+522\right){x}-5979a^{5}+40423a^{3}-5979a^{2}-61479a+8968$
27.1-b2
27.1-b
$2$
$13$
\(\Q(\zeta_{21})^+\)
$6$
$[6, 0]$
27.1
\( 3^{3} \)
\( 3^{6} \)
$79.22201$
$(a^3+a^2-2a-1)$
$1$
$\Z/13\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
✓
✓
$13$
13B.1.1
$1$
\( 2 \)
$0.076270034$
$226777.4369$
1.82314
\( -\frac{28672}{3} \)
\( \bigl[0\) , \( -a^{5} + 5 a^{3} - a^{2} - 5 a + 2\) , \( a^{5} - 5 a^{3} + a^{2} + 5 a - 2\) , \( -a^{5} + 7 a^{3} - a^{2} - 11 a + 2\) , \( a^{5} - 7 a^{3} + a^{2} + 11 a - 2\bigr] \)
${y}^2+\left(a^{5}-5a^{3}+a^{2}+5a-2\right){y}={x}^{3}+\left(-a^{5}+5a^{3}-a^{2}-5a+2\right){x}^{2}+\left(-a^{5}+7a^{3}-a^{2}-11a+2\right){x}+a^{5}-7a^{3}+a^{2}+11a-2$
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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.