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
16.3-a8
16.3-a
$8$
$12$
\(\Q(\sqrt{5}, \sqrt{13})\)
$4$
$[4, 0]$
16.3
\( 2^{4} \)
\( 2^{8} \)
$8.21423$
$(1/2a^3-9/2a+1)$
0
$\Z/2\Z\oplus\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2, 3$
2Cs , 3B
$1$
\( 1 \)
$1$
$1201.988595$
1.155758264
\( \frac{2492391}{2} a^{3} + \frac{6521955}{2} a^{2} - 1430550 a - 943918 \)
\( \bigl[a\) , \( \frac{1}{4} a^{3} + \frac{1}{2} a^{2} - \frac{13}{4} a - \frac{3}{2}\) , \( a\) , \( \frac{13}{4} a^{3} + \frac{9}{2} a^{2} - \frac{125}{4} a - \frac{63}{2}\) , \( -18 a^{3} - \frac{19}{2} a^{2} + \frac{301}{2} a + 88\bigr] \)
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(\frac{1}{4}a^{3}+\frac{1}{2}a^{2}-\frac{13}{4}a-\frac{3}{2}\right){x}^{2}+\left(\frac{13}{4}a^{3}+\frac{9}{2}a^{2}-\frac{125}{4}a-\frac{63}{2}\right){x}-18a^{3}-\frac{19}{2}a^{2}+\frac{301}{2}a+88$
256.5-f3
256.5-f
$8$
$12$
\(\Q(\sqrt{5}, \sqrt{13})\)
$4$
$[4, 0]$
256.5
\( 2^{8} \)
\( 2^{8} \)
$11.61668$
$(1/2a^3-9/2a+1)$
0
$\Z/2\Z\oplus\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2, 3$
2Cs , 3B
$1$
\( 1 \)
$1$
$1979.870563$
1.903721696
\( \frac{2492391}{2} a^{3} + \frac{6521955}{2} a^{2} - 1430550 a - 943918 \)
\( \bigl[\frac{1}{2} a^{2} + \frac{1}{2} a - 1\) , \( \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - 4 a\) , \( \frac{1}{2} a^{2} + \frac{1}{2} a - 1\) , \( -\frac{95}{4} a^{3} + \frac{31}{2} a^{2} + \frac{803}{4} a - \frac{273}{2}\) , \( \frac{503}{4} a^{3} - \frac{173}{2} a^{2} - \frac{4295}{4} a + \frac{1471}{2}\bigr] \)
${y}^2+\left(\frac{1}{2}a^{2}+\frac{1}{2}a-1\right){x}{y}+\left(\frac{1}{2}a^{2}+\frac{1}{2}a-1\right){y}={x}^{3}+\left(\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-4a\right){x}^{2}+\left(-\frac{95}{4}a^{3}+\frac{31}{2}a^{2}+\frac{803}{4}a-\frac{273}{2}\right){x}+\frac{503}{4}a^{3}-\frac{173}{2}a^{2}-\frac{4295}{4}a+\frac{1471}{2}$
256.5-i3
256.5-i
$8$
$12$
\(\Q(\sqrt{5}, \sqrt{13})\)
$4$
$[4, 0]$
256.5
\( 2^{8} \)
\( 2^{8} \)
$11.61668$
$(1/2a^3-9/2a+1)$
$1$
$\Z/2\Z\oplus\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2, 3$
2Cs , 3B
$1$
\( 1 \)
$3.133744579$
$332.9329116$
4.012795028
\( \frac{2492391}{2} a^{3} + \frac{6521955}{2} a^{2} - 1430550 a - 943918 \)
\( \bigl[\frac{1}{2} a^{2} - \frac{1}{2} a - 1\) , \( \frac{1}{2} a^{3} + \frac{1}{2} a^{2} - 5 a - 3\) , \( \frac{1}{2} a^{2} - \frac{1}{2} a - 1\) , \( -\frac{5}{4} a^{3} + \frac{27}{4} a + \frac{7}{2}\) , \( -\frac{1}{4} a^{3} + a^{2} - \frac{9}{4} a - \frac{5}{2}\bigr] \)
${y}^2+\left(\frac{1}{2}a^{2}-\frac{1}{2}a-1\right){x}{y}+\left(\frac{1}{2}a^{2}-\frac{1}{2}a-1\right){y}={x}^{3}+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-5a-3\right){x}^{2}+\left(-\frac{5}{4}a^{3}+\frac{27}{4}a+\frac{7}{2}\right){x}-\frac{1}{4}a^{3}+a^{2}-\frac{9}{4}a-\frac{5}{2}$
256.5-p3
256.5-p
$8$
$12$
\(\Q(\sqrt{5}, \sqrt{13})\)
$4$
$[4, 0]$
256.5
\( 2^{8} \)
\( 2^{8} \)
$11.61668$
$(1/2a^3-9/2a+1)$
$1$
$\Z/2\Z\oplus\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2, 3$
2Cs , 3B
$1$
\( 1 \)
$1.044581526$
$998.7987350$
4.012795028
\( \frac{2492391}{2} a^{3} + \frac{6521955}{2} a^{2} - 1430550 a - 943918 \)
\( \bigl[\frac{1}{2} a^{2} - \frac{1}{2} a - 1\) , \( \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{5}{4} a + \frac{5}{2}\) , \( 0\) , \( \frac{1}{2} a^{3} - \frac{3}{2} a^{2} + 1\) , \( 0\bigr] \)
${y}^2+\left(\frac{1}{2}a^{2}-\frac{1}{2}a-1\right){x}{y}={x}^{3}+\left(\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{5}{4}a+\frac{5}{2}\right){x}^{2}+\left(\frac{1}{2}a^{3}-\frac{3}{2}a^{2}+1\right){x}$
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Pari/GP
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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.