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
47.2-a1
47.2-a
$2$
$2$
5.5.70601.1
$5$
$[5, 0]$
47.2
\( 47 \)
\( -47 \)
$34.89424$
$(-a^4+a^3+6a^2-2)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2B
$1$
\( 1 \)
$1$
$2238.811346$
2.10645450
\( \frac{1877152}{47} a^{4} + \frac{2877117}{47} a^{3} - \frac{2158825}{47} a^{2} - \frac{1887071}{47} a + \frac{717218}{47} \)
\( \bigl[a^{4} - a^{3} - 5 a^{2} + 3 a + 3\) , \( -a^{4} + a^{3} + 4 a^{2} + 1\) , \( a^{4} - a^{3} - 4 a^{2} + a\) , \( -a^{3} + 5 a + 5\) , \( a^{4} - a^{3} - 5 a^{2} + 2 a + 3\bigr] \)
${y}^2+\left(a^{4}-a^{3}-5a^{2}+3a+3\right){x}{y}+\left(a^{4}-a^{3}-4a^{2}+a\right){y}={x}^{3}+\left(-a^{4}+a^{3}+4a^{2}+1\right){x}^{2}+\left(-a^{3}+5a+5\right){x}+a^{4}-a^{3}-5a^{2}+2a+3$
47.2-a2
47.2-a
$2$
$2$
5.5.70601.1
$5$
$[5, 0]$
47.2
\( 47 \)
\( - 47^{2} \)
$34.89424$
$(-a^4+a^3+6a^2-2)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2B
$1$
\( 2 \)
$1$
$1119.405673$
2.10645450
\( \frac{194915039076075}{2209} a^{4} + \frac{294330947080514}{2209} a^{3} - \frac{235793793475384}{2209} a^{2} - \frac{202027318016940}{2209} a + \frac{77654766827483}{2209} \)
\( \bigl[a^{4} - 6 a^{2} - a + 4\) , \( a^{3} - 2 a^{2} - 4 a + 3\) , \( 0\) , \( -8 a^{4} + 48 a^{2} + 20 a - 36\) , \( -17 a^{4} - 7 a^{3} + 119 a^{2} + 66 a - 129\bigr] \)
${y}^2+\left(a^{4}-6a^{2}-a+4\right){x}{y}={x}^{3}+\left(a^{3}-2a^{2}-4a+3\right){x}^{2}+\left(-8a^{4}+48a^{2}+20a-36\right){x}-17a^{4}-7a^{3}+119a^{2}+66a-129$
47.2-b1
47.2-b
$2$
$3$
5.5.70601.1
$5$
$[5, 0]$
47.2
\( 47 \)
\( - 47^{3} \)
$34.89424$
$(-a^4+a^3+6a^2-2)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3B
$1$
\( 1 \)
$0.030642720$
$4853.403072$
2.79858346
\( \frac{9074112}{103823} a^{4} + \frac{132922005}{103823} a^{3} - \frac{356032536}{103823} a^{2} - \frac{222800132}{103823} a + \frac{342458773}{103823} \)
\( \bigl[a^{4} - a^{3} - 5 a^{2} + 3 a + 3\) , \( -2 a^{4} + a^{3} + 11 a^{2} + a - 6\) , \( a + 1\) , \( -2 a^{4} + a^{3} + 11 a^{2} + a - 7\) , \( 0\bigr] \)
${y}^2+\left(a^{4}-a^{3}-5a^{2}+3a+3\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-2a^{4}+a^{3}+11a^{2}+a-6\right){x}^{2}+\left(-2a^{4}+a^{3}+11a^{2}+a-7\right){x}$
47.2-b2
47.2-b
$2$
$3$
5.5.70601.1
$5$
$[5, 0]$
47.2
\( 47 \)
\( -47 \)
$34.89424$
$(-a^4+a^3+6a^2-2)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3B
$1$
\( 1 \)
$0.091928162$
$1617.801024$
2.79858346
\( \frac{42542580556}{47} a^{4} - \frac{21887546864}{47} a^{3} - \frac{232611726616}{47} a^{2} - \frac{13686260207}{47} a + \frac{157328368641}{47} \)
\( \bigl[3 a^{4} - 2 a^{3} - 15 a^{2} + 2 a + 5\) , \( -a^{3} + 2 a^{2} + 4 a - 4\) , \( 2 a^{4} - a^{3} - 10 a^{2} + 3\) , \( 4 a^{4} - 6 a^{3} - 10 a^{2} + 2 a - 3\) , \( 19 a^{4} - 11 a^{3} - 83 a^{2} - 29 a + 17\bigr] \)
${y}^2+\left(3a^{4}-2a^{3}-15a^{2}+2a+5\right){x}{y}+\left(2a^{4}-a^{3}-10a^{2}+3\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-4\right){x}^{2}+\left(4a^{4}-6a^{3}-10a^{2}+2a-3\right){x}+19a^{4}-11a^{3}-83a^{2}-29a+17$
47.2-c1
47.2-c
$1$
$1$
5.5.70601.1
$5$
$[5, 0]$
47.2
\( 47 \)
\( -47 \)
$34.89424$
$(-a^4+a^3+6a^2-2)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$1$
\( 1 \)
$0.010452360$
$11112.37363$
2.18567608
\( -\frac{87216945}{47} a^{4} - \frac{132154029}{47} a^{3} + \frac{104407562}{47} a^{2} + \frac{90132064}{47} a - \frac{34286979}{47} \)
\( \bigl[3 a^{4} - 2 a^{3} - 15 a^{2} + 2 a + 5\) , \( 3 a^{4} - 2 a^{3} - 15 a^{2} + a + 5\) , \( 1\) , \( 8 a^{4} - 5 a^{3} - 41 a^{2} - a + 23\) , \( 4 a^{4} - 4 a^{3} - 21 a^{2} + 8 a + 17\bigr] \)
${y}^2+\left(3a^{4}-2a^{3}-15a^{2}+2a+5\right){x}{y}+{y}={x}^{3}+\left(3a^{4}-2a^{3}-15a^{2}+a+5\right){x}^{2}+\left(8a^{4}-5a^{3}-41a^{2}-a+23\right){x}+4a^{4}-4a^{3}-21a^{2}+8a+17$
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Pari/GP
SageMath
Magma
Oscar
CSV
*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.