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
27.2-a1
27.2-a
$2$
$3$
6.6.434581.1
$6$
$[6, 0]$
27.2
\( 3^{3} \)
\( 3^{9} \)
$77.52722$
$(-2a^5+4a^4+7a^3-9a^2-3a+2)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3B
$1$
\( 1 \)
$1$
$852.4945286$
1.29317
\( \frac{1286151805}{27} a^{5} - \frac{97850411}{3} a^{4} - \frac{6302966594}{27} a^{3} - \frac{1859319866}{27} a^{2} + \frac{2699241622}{27} a + \frac{977886878}{27} \)
\( \bigl[a + 1\) , \( a^{5} - 4 a^{4} + 12 a^{2} - 3 a - 4\) , \( a^{5} - 2 a^{4} - 4 a^{3} + 5 a^{2} + 4 a - 1\) , \( -2 a^{5} + 3 a^{4} + 8 a^{3} - 6 a - 1\) , \( -3 a^{5} + 6 a^{4} + 10 a^{3} - 10 a^{2} - 3 a\bigr] \)
${y}^2+\left(a+1\right){x}{y}+\left(a^{5}-2a^{4}-4a^{3}+5a^{2}+4a-1\right){y}={x}^{3}+\left(a^{5}-4a^{4}+12a^{2}-3a-4\right){x}^{2}+\left(-2a^{5}+3a^{4}+8a^{3}-6a-1\right){x}-3a^{5}+6a^{4}+10a^{3}-10a^{2}-3a$
27.2-a2
27.2-a
$2$
$3$
6.6.434581.1
$6$
$[6, 0]$
27.2
\( 3^{3} \)
\( 3^{3} \)
$77.52722$
$(-2a^5+4a^4+7a^3-9a^2-3a+2)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3B
$1$
\( 1 \)
$1$
$852.4945286$
1.29317
\( -\frac{3164945057}{3} a^{5} + 2902622828 a^{4} + \frac{6076665952}{3} a^{3} - \frac{20306097035}{3} a^{2} + \frac{2731169941}{3} a + \frac{4107624116}{3} \)
\( \bigl[2 a^{5} - 4 a^{4} - 7 a^{3} + 8 a^{2} + 5 a - 2\) , \( -a^{2} + a\) , \( 2 a^{5} - 5 a^{4} - 5 a^{3} + 12 a^{2} - 3\) , \( -10 a^{5} + 23 a^{4} + 29 a^{3} - 48 a^{2} - 13 a + 2\) , \( -18 a^{5} + 38 a^{4} + 59 a^{3} - 78 a^{2} - 35 a\bigr] \)
${y}^2+\left(2a^{5}-4a^{4}-7a^{3}+8a^{2}+5a-2\right){x}{y}+\left(2a^{5}-5a^{4}-5a^{3}+12a^{2}-3\right){y}={x}^{3}+\left(-a^{2}+a\right){x}^{2}+\left(-10a^{5}+23a^{4}+29a^{3}-48a^{2}-13a+2\right){x}-18a^{5}+38a^{4}+59a^{3}-78a^{2}-35a$
27.2-b1
27.2-b
$1$
$1$
6.6.434581.1
$6$
$[6, 0]$
27.2
\( 3^{3} \)
\( 3^{15} \)
$77.52722$
$(-2a^5+4a^4+7a^3-9a^2-3a+2)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$1$
\( 1 \)
$1$
$858.1016348$
1.30168
\( \frac{1612895}{243} a^{5} - \frac{688339}{27} a^{4} + \frac{4362505}{243} a^{3} + \frac{2436914}{243} a^{2} - \frac{72826}{27} a - \frac{317266}{81} \)
\( \bigl[a^{5} - 2 a^{4} - 3 a^{3} + 3 a^{2} + 2 a + 1\) , \( -3 a^{5} + 8 a^{4} + 6 a^{3} - 18 a^{2} + a + 6\) , \( 3 a^{5} - 6 a^{4} - 10 a^{3} + 12 a^{2} + 5 a - 2\) , \( 11 a^{5} - 13 a^{4} - 58 a^{3} + 16 a^{2} + 61 a + 21\) , \( -560 a^{5} + 744 a^{4} + 2735 a^{3} - 952 a^{2} - 2871 a - 825\bigr] \)
${y}^2+\left(a^{5}-2a^{4}-3a^{3}+3a^{2}+2a+1\right){x}{y}+\left(3a^{5}-6a^{4}-10a^{3}+12a^{2}+5a-2\right){y}={x}^{3}+\left(-3a^{5}+8a^{4}+6a^{3}-18a^{2}+a+6\right){x}^{2}+\left(11a^{5}-13a^{4}-58a^{3}+16a^{2}+61a+21\right){x}-560a^{5}+744a^{4}+2735a^{3}-952a^{2}-2871a-825$
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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.