Elliptic curve with Cremona label 35131c2 (LMFDB label 35131.b2)

• Label: 35131c2
• Conductor: $35131$
• Discriminant: $-4.336\times 10^{13}$
• j-invariant: \( -\frac{89915392}{6859} \)
• CM: no
• Rank: $0$
• Torsion structure: trivial

Minimal Weierstrass equation

\(y^2+y=x^3-x^2-17257x+934070\)  

Mordell-Weil group structure

trivial

Integral points

\(\)  

Invariants

Conductor:	\( 35131 \)	=	\(19 \cdot 43^{2}\)
Discriminant:	\(-43358229153091 \)	=	\(-1 \cdot 19^{3} \cdot 43^{6} \)
j-invariant:	\( -\frac{89915392}{6859} \)	=	\(-1 \cdot 2^{18} \cdot 7^{3} \cdot 19^{-3}\)
Endomorphism ring:	\(\Z\)
Geometric endomorphism ring:	\(\Z\)	(no potential complex multiplication)
Sato-Tate group:	$\mathrm{SU}(2)$
Faltings height:	\(1.3647330705040441695588935218\dots\)
Stable Faltings height:	\(-0.51586698734273704217752773487\dots\)

BSD invariants

Analytic rank:	\(0\)
Regulator:	\(1\)
Real period:	\(0.62937568936744920308418551671\dots\)
Tamagawa product:	\( 1 \)
Torsion order:	\(1\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form 35131.2.a.b

\( q + 2q^{3} - 2q^{4} - 3q^{5} + q^{7} + q^{9} + 3q^{11} - 4q^{12} - 4q^{13} - 6q^{15} + 4q^{16} - 3q^{17} - q^{19} + O(q^{20}) \)  

For more coefficients, see the Downloads section to the right.

Modular degree:	81270
\( \Gamma_0(N) \)-optimal:	no
Manin constant:	1

Special L-value

\( L(E,1) \) ≈ \( 0.62937568936744920308418551670890529080 \)

Local data

This elliptic curve is not semistable. There are 2 primes of bad reduction:

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord(\(N\))	ord(\(\Delta\))	ord\((j)_{-}\)
\(19\)	\(1\)	\(I_{3}\)	Non-split multiplicative	1	1	3	3
\(43\)	\(1\)	\(I_0^{*}\)	Additive	-1	2	6	0

Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.

prime	Image of Galois representation
\(3\)	Cs

$p$-adic data

$p$-adic regulators

All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).

Iwasawa invariants

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47
Reduction type	ss	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	nonsplit	ss	ordinary	ordinary	ordinary	ordinary	add	ordinary
$\lambda$-invariant(s)	0,3	0	0	2	0	0	0	0	0,0	0	0	0	0	-	0
$\mu$-invariant(s)	0,0	0	0	0	0	0	0	0	0,0	0	0	0	0	-	0

An entry - indicates that the invariants are not computed because the reduction is additive.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 35131c consists of 2 curves linked by isogenies of degrees dividing 9.

Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$2$	\(\Q(\sqrt{129}) \)	\(\Z/3\Z\)	Not in database
$2$	\(\Q(\sqrt{-43}) \)	\(\Z/3\Z\)	Not in database
$3$	3.1.76.1	\(\Z/2\Z\)	Not in database
$4$	\(\Q(\sqrt{-3}, \sqrt{-43})\)	\(\Z/3\Z \times \Z/3\Z\)	Not in database
$6$	6.0.109744.2	\(\Z/2\Z \times \Z/2\Z\)	Not in database
$6$	6.2.12399275664.2	\(\Z/6\Z\)	Not in database
$6$	6.0.459232432.4	\(\Z/6\Z\)	Not in database
$12$	Deg 12	\(\Z/4\Z\)	Not in database
$12$	Deg 12	\(\Z/3\Z \times \Z/6\Z\)	Not in database
$12$	Deg 12	\(\Z/2\Z \times \Z/6\Z\)	Not in database
$12$	Deg 12	\(\Z/2\Z \times \Z/6\Z\)	Not in database
$18$	18.6.8482682052114376250117148040700509530548601.2	\(\Z/9\Z\)	Not in database
$18$	18.0.810936484408725968917662372428067.1	\(\Z/9\Z\)	Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.