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
5.1-a1 5.1-a $2$
$3$
4.4.12400.1 $4$
$[4, 0]$
5.1
\( 5 \)
\( - 5^{9} \)
$12.16804$
$(1/2a^3-1/2a^2-7/2a+5/2)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$ ✓
$3$
3B $1$
\( 1 \)
$1$
$217.0163666$
1.948864504
\( \frac{102109}{6250} a^{3} - \frac{1063917}{3125} a^{2} - \frac{972919}{6250} a + \frac{8947842}{3125} \)
\( \bigl[\frac{1}{2} a^{3} - \frac{7}{2} a\) , \( \frac{1}{2} a^{3} - \frac{5}{2} a + 1\) , \( \frac{1}{2} a^{2} + a - \frac{5}{2}\) , \( \frac{1}{2} a^{3} - \frac{7}{2} a + 8\) , \( a^{3} - 2 a^{2} - 4 a + 9\bigr] \) ${y}^2+\left(\frac{1}{2}a^{3}-\frac{7}{2}a\right){x}{y}+\left(\frac{1}{2}a^{2}+a-\frac{5}{2}\right){y}={x}^{3}+\left(\frac{1}{2}a^{3}-\frac{5}{2}a+1\right){x}^{2}+\left(\frac{1}{2}a^{3}-\frac{7}{2}a+8\right){x}+a^{3}-2a^{2}-4a+9$
5.1-a2 5.1-a $2$
$3$
4.4.12400.1 $4$
$[4, 0]$
5.1
\( 5 \)
\( - 5^{3} \)
$12.16804$
$(1/2a^3-1/2a^2-7/2a+5/2)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$ ✓
$3$
3B $1$
\( 1 \)
$1$
$217.0163666$
1.948864504
\( \frac{462073577}{25} a^{3} - \frac{1794849939}{50} a^{2} - \frac{3800943272}{25} a + \frac{14755379619}{50} \)
\( \bigl[\frac{1}{2} a^{2} + a - \frac{7}{2}\) , \( \frac{1}{2} a^{3} - \frac{5}{2} a - 1\) , \( a + 1\) , \( \frac{3}{2} a^{3} + \frac{7}{2} a^{2} - \frac{21}{2} a - \frac{41}{2}\) , \( \frac{19}{2} a^{3} + 25 a^{2} - \frac{85}{2} a - 106\bigr] \) ${y}^2+\left(\frac{1}{2}a^{2}+a-\frac{7}{2}\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(\frac{1}{2}a^{3}-\frac{5}{2}a-1\right){x}^{2}+\left(\frac{3}{2}a^{3}+\frac{7}{2}a^{2}-\frac{21}{2}a-\frac{41}{2}\right){x}+\frac{19}{2}a^{3}+25a^{2}-\frac{85}{2}a-106$
5.1-b1 5.1-b $2$
$3$
4.4.12400.1 $4$
$[4, 0]$
5.1
\( 5 \)
\( - 5^{9} \)
$12.16804$
$(1/2a^3-1/2a^2-7/2a+5/2)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$ ✓
$3$
3B $1$
\( 3^{2} \)
$0.036476401$
$185.1864460$
2.183799909
\( \frac{102109}{6250} a^{3} - \frac{1063917}{3125} a^{2} - \frac{972919}{6250} a + \frac{8947842}{3125} \)
\( \bigl[1\) , \( a - 1\) , \( \frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{5}{2} a - \frac{5}{2}\) , \( -\frac{1}{2} a^{3} + \frac{5}{2} a + 3\) , \( -2 a^{2} + 7\bigr] \) ${y}^2+{x}{y}+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-\frac{5}{2}\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-\frac{1}{2}a^{3}+\frac{5}{2}a+3\right){x}-2a^{2}+7$
5.1-b2 5.1-b $2$
$3$
4.4.12400.1 $4$
$[4, 0]$
5.1
\( 5 \)
\( - 5^{3} \)
$12.16804$
$(1/2a^3-1/2a^2-7/2a+5/2)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$ ✓
$3$
3B $1$
\( 3 \)
$0.109429204$
$185.1864460$
2.183799909
\( \frac{462073577}{25} a^{3} - \frac{1794849939}{50} a^{2} - \frac{3800943272}{25} a + \frac{14755379619}{50} \)
\( \bigl[\frac{1}{2} a^{2} + a - \frac{5}{2}\) , \( -\frac{1}{2} a^{3} + \frac{9}{2} a + 1\) , \( \frac{1}{2} a^{2} - \frac{7}{2}\) , \( \frac{5}{2} a^{3} + \frac{11}{2} a^{2} - \frac{35}{2} a - \frac{73}{2}\) , \( -a^{3} + \frac{1}{2} a^{2} + 19 a + \frac{53}{2}\bigr] \) ${y}^2+\left(\frac{1}{2}a^{2}+a-\frac{5}{2}\right){x}{y}+\left(\frac{1}{2}a^{2}-\frac{7}{2}\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}+\frac{9}{2}a+1\right){x}^{2}+\left(\frac{5}{2}a^{3}+\frac{11}{2}a^{2}-\frac{35}{2}a-\frac{73}{2}\right){x}-a^{3}+\frac{1}{2}a^{2}+19a+\frac{53}{2}$
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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.
