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
21.3-a1
21.3-a
$1$
$1$
\(\Q(\sqrt{109}) \)
$2$
$[2, 0]$
21.3
\( 3 \cdot 7 \)
\( 3^{6} \cdot 7^{2} \)
$1.99713$
$(a+5), (-a-4)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$1$
\( 2^{2} \)
$0.113177823$
$11.34187913$
0.983610348
\( -\frac{542764456}{35721} a - \frac{100926547}{1323} \)
\( \bigl[a\) , \( 1\) , \( a + 1\) , \( -a\) , \( -a\bigr] \)
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}-a{x}-a$
21.3-b1
21.3-b
$2$
$2$
\(\Q(\sqrt{109}) \)
$2$
$[2, 0]$
21.3
\( 3 \cdot 7 \)
\( - 3^{6} \cdot 7^{7} \)
$1.99713$
$(a+5), (-a-4)$
$1$
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2B
$1$
\( 2 \cdot 7 \)
$0.510026831$
$6.845658691$
2.340954958
\( -\frac{79043995497128}{600362847} a + \frac{16746812154661}{22235661} \)
\( \bigl[a\) , \( 1\) , \( 0\) , \( 6094 a + 28779\) , \( -4182709 a - 19743017\bigr] \)
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(6094a+28779\right){x}-4182709a-19743017$
21.3-b2
21.3-b
$2$
$2$
\(\Q(\sqrt{109}) \)
$2$
$[2, 0]$
21.3
\( 3 \cdot 7 \)
\( - 3^{3} \cdot 7^{14} \)
$1.99713$
$(a+5), (-a-4)$
$1$
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2B
$1$
\( 2 \cdot 7 \)
$1.020053663$
$3.422829345$
2.340954958
\( \frac{34066465075516948}{18312022966923} a + \frac{5561486509237749}{678223072849} \)
\( \bigl[a\) , \( 1\) , \( 0\) , \( -147901 a - 698101\) , \( -67871043 a - 320361714\bigr] \)
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(-147901a-698101\right){x}-67871043a-320361714$
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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.