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
82.1-a1
82.1-a
$2$
$5$
\(\Q(\sqrt{2}) \)
$2$
$[2, 0]$
82.1
\( 2 \cdot 41 \)
\( 2 \cdot 41^{5} \)
$0.76057$
$(a), (2a+7)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$5$
5B.1.2
$1$
\( 5 \)
$1$
$0.651371891$
1.151473703
\( -\frac{5814814559175147}{231712402} a - \frac{4128881686911505}{115856201} \)
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( -15 a - 79\) , \( -47 a - 251\bigr] \)
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-15a-79\right){x}-47a-251$
82.1-a2
82.1-a
$2$
$5$
\(\Q(\sqrt{2}) \)
$2$
$[2, 0]$
82.1
\( 2 \cdot 41 \)
\( 2^{5} \cdot 41 \)
$0.76057$
$(a), (2a+7)$
0
$\Z/5\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$5$
5B.1.1
$1$
\( 5 \)
$1$
$16.28429728$
1.151473703
\( \frac{89373}{328} a + \frac{148955}{164} \)
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 1\) , \( -a - 1\bigr] \)
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+{x}-a-1$
82.1-b1
82.1-b
$2$
$3$
\(\Q(\sqrt{2}) \)
$2$
$[2, 0]$
82.1
\( 2 \cdot 41 \)
\( 2^{21} \cdot 41^{3} \)
$0.76057$
$(a), (2a+7)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3B.1.2
$1$
\( 1 \)
$1$
$1.872522881$
0.662036813
\( -\frac{25086047287643}{141150208} a - \frac{17749977458361}{70575104} \)
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( -7 a - 33\) , \( -97 a + 10\bigr] \)
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-7a-33\right){x}-97a+10$
82.1-b2
82.1-b
$2$
$3$
\(\Q(\sqrt{2}) \)
$2$
$[2, 0]$
82.1
\( 2 \cdot 41 \)
\( 2^{7} \cdot 41 \)
$0.76057$
$(a), (2a+7)$
0
$\Z/3\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3B.1.1
$1$
\( 1 \)
$1$
$16.85270592$
0.662036813
\( \frac{64288010347}{656} a - \frac{45458394669}{328} \)
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( 13 a - 18\) , \( -30 a + 42\bigr] \)
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(13a-18\right){x}-30a+42$
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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.