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
23.1-a1
23.1-a
$4$
$10$
\(\Q(\zeta_{11})^+\)
$5$
$[5, 0]$
23.1
\( 23 \)
\( - 23^{5} \)
$14.79438$
$(a^4-3a^2-1)$
0
$\Z/10\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2, 5$
2B , 5B.1.1
$1$
\( 5 \)
$1$
$1986.252133$
0.820765345
\( -\frac{47053068568}{6436343} a^{4} + \frac{46268012320}{6436343} a^{3} + \frac{141240968488}{6436343} a^{2} - \frac{112796046627}{6436343} a + \frac{7676072842}{6436343} \)
\( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( -a^{4} - a^{3} + 5 a^{2} + 4 a - 4\) , \( a^{3} - 3 a + 1\) , \( 2 a^{3} + a^{2} - 3 a + 1\) , \( 2 a^{4} - 4 a^{2} + a\bigr] \)
${y}^2+\left(a^{3}+a^{2}-2a-2\right){x}{y}+\left(a^{3}-3a+1\right){y}={x}^{3}+\left(-a^{4}-a^{3}+5a^{2}+4a-4\right){x}^{2}+\left(2a^{3}+a^{2}-3a+1\right){x}+2a^{4}-4a^{2}+a$
23.1-a2
23.1-a
$4$
$10$
\(\Q(\zeta_{11})^+\)
$5$
$[5, 0]$
23.1
\( 23 \)
\( - 23^{10} \)
$14.79438$
$(a^4-3a^2-1)$
0
$\Z/10\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2, 5$
2B , 5B.1.1
$1$
\( 2 \cdot 5 \)
$1$
$993.1260669$
0.820765345
\( \frac{74868976618071917973}{41426511213649} a^{4} - \frac{54092178455960064989}{41426511213649} a^{3} - \frac{312961789930170680193}{41426511213649} a^{2} + \frac{134824139137681173208}{41426511213649} a + \frac{261494050662338262657}{41426511213649} \)
\( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( -a^{4} - a^{3} + 5 a^{2} + 4 a - 4\) , \( a^{3} - 3 a + 1\) , \( -5 a^{4} + 7 a^{3} + 11 a^{2} - 3 a - 4\) , \( -a^{4} - 13 a^{3} + 27 a^{2} + 14 a - 8\bigr] \)
${y}^2+\left(a^{3}+a^{2}-2a-2\right){x}{y}+\left(a^{3}-3a+1\right){y}={x}^{3}+\left(-a^{4}-a^{3}+5a^{2}+4a-4\right){x}^{2}+\left(-5a^{4}+7a^{3}+11a^{2}-3a-4\right){x}-a^{4}-13a^{3}+27a^{2}+14a-8$
23.1-a3
23.1-a
$4$
$10$
\(\Q(\zeta_{11})^+\)
$5$
$[5, 0]$
23.1
\( 23 \)
\( -23 \)
$14.79438$
$(a^4-3a^2-1)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2, 5$
2B , 5B.1.2
$625$
\( 1 \)
$1$
$0.635600682$
0.820765345
\( -\frac{23939398238252891600044}{23} a^{4} + \frac{43828185875438962615638}{23} a^{3} + \frac{59344380437326987721953}{23} a^{2} - \frac{121119995485666779459359}{23} a + \frac{28810521719920268473488}{23} \)
\( \bigl[a^{3} - 3 a + 1\) , \( -a^{4} + a^{3} + 5 a^{2} - 4 a - 5\) , \( 0\) , \( -88 a^{4} + 205 a^{3} + 9 a^{2} - 233 a - 49\) , \( -1417 a^{4} + 3592 a^{3} - 564 a^{2} - 3064 a + 554\bigr] \)
${y}^2+\left(a^{3}-3a+1\right){x}{y}={x}^{3}+\left(-a^{4}+a^{3}+5a^{2}-4a-5\right){x}^{2}+\left(-88a^{4}+205a^{3}+9a^{2}-233a-49\right){x}-1417a^{4}+3592a^{3}-564a^{2}-3064a+554$
23.1-a4
23.1-a
$4$
$10$
\(\Q(\zeta_{11})^+\)
$5$
$[5, 0]$
23.1
\( 23 \)
\( - 23^{2} \)
$14.79438$
$(a^4-3a^2-1)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2, 5$
2B , 5B.1.2
$625$
\( 2 \)
$1$
$0.317800341$
0.820765345
\( \frac{1812431798583025575975116650734948165}{529} a^{4} - \frac{1296559921989620389944481933388349849}{529} a^{3} - \frac{7618766625551254351602825393413091355}{529} a^{2} + \frac{3268768315434618465890517821484807321}{529} a + \frac{6367683864076909597378964320128797635}{529} \)
\( \bigl[a^{4} - 3 a^{2} + a + 1\) , \( a^{4} + a^{3} - 4 a^{2} - 4 a + 2\) , \( a^{4} - 3 a^{2} + 1\) , \( -959 a^{4} - 683 a^{3} + 2238 a^{2} + 491 a - 1224\) , \( -40616 a^{4} - 34613 a^{3} + 92332 a^{2} + 42697 a - 29653\bigr] \)
${y}^2+\left(a^{4}-3a^{2}+a+1\right){x}{y}+\left(a^{4}-3a^{2}+1\right){y}={x}^{3}+\left(a^{4}+a^{3}-4a^{2}-4a+2\right){x}^{2}+\left(-959a^{4}-683a^{3}+2238a^{2}+491a-1224\right){x}-40616a^{4}-34613a^{3}+92332a^{2}+42697a-29653$
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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.