Elliptic curve search results

Base field	Conductor norm	Rank*	Torsion order	CM discriminant
Base field degree/signature	Bad \(p\)	$\Q$-curves	Torsion structure	CM
Analytic order of Ш*	Cyclic isogeny degree	Isogeny class size	Reduction	Galois image
j-invariant	Regulator*	Curves per isogeny class	Isogeny class degree	Nonmax$\ \ell$

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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
27.1-a2	27.1-a	$2$	$3$	6.6.1387029.1	$6$	$[6, 0]$	27.1	\( 3^{3} \)	\( - 3^{11} \)	$138.50378$	$(a^5-3a^4-2a^3+8a^2+a-2)$	$1$	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$				✓	$3$	3B.1.1	$$	\( 1 \)	$1.089220952$			\( 3426222102 a^{5} - 12812570435 a^{4} + 2623723832 a^{3} + 28894116691 a^{2} - 24794656162 a + 4632631517 \)	\( \bigl[a^{4} - 2 a^{3} - 2 a^{2} + 4 a\) , \( a^{5} - 3 a^{4} - 3 a^{3} + 10 a^{2} + 3 a - 3\) , \( -a^{5} + 3 a^{4} + 3 a^{3} - 9 a^{2} - 2 a + 3\) , \( -3 a^{5} - a^{4} + 10 a^{3} + 4 a^{2} + a - 1\) , \( -41 a^{5} + 17 a^{4} + 129 a^{3} - 31 a^{2} - 56 a + 14\bigr] \)	${y}^2+\left(a^{4}-2a^{3}-2a^{2}+4a\right){x}{y}+\left(-a^{5}+3a^{4}+3a^{3}-9a^{2}-2a+3\right){y}={x}^{3}+\left(a^{5}-3a^{4}-3a^{3}+10a^{2}+3a-3\right){x}^{2}+\left(-3a^{5}-a^{4}+10a^{3}+4a^{2}+a-1\right){x}-41a^{5}+17a^{4}+129a^{3}-31a^{2}-56a+14$
27.1-p2	27.1-p	$2$	$3$	6.6.1387029.1	$6$	$[6, 0]$	27.1	\( 3^{3} \)	\( - 3^{11} \)	$138.50378$	$(a^5-3a^4-2a^3+8a^2+a-2)$	$0$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$				✓	$3$	3B	$$	\( 1 \)	$1$			\( 3426222102 a^{5} - 12812570435 a^{4} + 2623723832 a^{3} + 28894116691 a^{2} - 24794656162 a + 4632631517 \)	\( \bigl[a^{3} - 4 a + 1\) , \( 3 a^{5} - 7 a^{4} - 9 a^{3} + 18 a^{2} + 4 a - 5\) , \( -a^{5} + 3 a^{4} + 2 a^{3} - 7 a^{2} + 2\) , \( 7 a^{5} - 34 a^{4} + 40 a^{3} + 16 a^{2} - 31 a + 10\) , \( 53 a^{5} - 232 a^{4} + 239 a^{3} + 81 a^{2} - 149 a + 29\bigr] \)	${y}^2+\left(a^{3}-4a+1\right){x}{y}+\left(-a^{5}+3a^{4}+2a^{3}-7a^{2}+2\right){y}={x}^{3}+\left(3a^{5}-7a^{4}-9a^{3}+18a^{2}+4a-5\right){x}^{2}+\left(7a^{5}-34a^{4}+40a^{3}+16a^{2}-31a+10\right){x}+53a^{5}-232a^{4}+239a^{3}+81a^{2}-149a+29$

 

  *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.