Elliptic Curve with LMFDB label 40425.p2 (Cremona label 40425y2)

• Label: 40425.p2
• Conductor: 40425
• Discriminant: 162151572515625
• j-invariant: \( \frac{169112377}{88209} \)
• CM: no
• Rank: 1
• Torsion Structure: \(\Z/{2}\Z \times \Z/{2}\Z\)

Minimal Weierstrass equation

\( y^2 + x y + y = x^{3} + x^{2} - 14113 x - 208594 \)

Mordell-Weil group structure

\(\Z\times \Z/{2}\Z \times \Z/{2}\Z\)

Infinite order Mordell-Weil generator and height

\(P\)	=	\( \left(-64, 693\right) \)
\(\hat{h}(P)\)	≈	1.30634595814

Torsion generators

\( \left(-15, 7\right) \), \( \left(125, -63\right) \)

Integral points

\( \left(-64, 693\right) \), \( \left(-50, 637\right) \), \( \left(-15, 7\right) \), \( \left(125, -63\right) \), \( \left(129, 307\right) \), \( \left(260, 3582\right) \), \( \left(370, 6552\right) \)

Note: only one of each pair $\pm P$ is listed.

Invariants

Conductor:	\( 40425 \)	=	\(3 \cdot 5^{2} \cdot 7^{2} \cdot 11\)
Discriminant:	\(162151572515625 \)	=	\(3^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{2} \)
j-invariant:	\( \frac{169112377}{88209} \)	=	\(3^{-6} \cdot 7^{3} \cdot 11^{-2} \cdot 79^{3}\)
Endomorphism ring:	\(\Z\)		(no Complex Multiplication)
Sato-Tate Group:	$\mathrm{SU}(2)$

BSD invariants

Rank:	\(1\)
Regulator:	\(1.30634595814\)
Real period:	\(0.463927711946\)
Tamagawa product:	\( 64 \)  = \( 2\cdot2^{2}\cdot2^{2}\cdot2 \)
Torsion order:	\(4\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form 40425.2.a.p

\( q - q^{2} - q^{3} - q^{4} + q^{6} + 3q^{8} + q^{9} + q^{11} + q^{12} - 2q^{13} - q^{16} - 2q^{17} - q^{18} + O(q^{20}) \)

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

Modular degree:	110592
\( \Gamma_0(N) \)-optimal:	no
Manin constant:	1

Special L-value

\( L'(E,1) \) ≈ \( 2.42420036547 \)

Local data

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord(\(N\))	ord(\(\Delta\))	ord\((j)_{-}\)
\(3\)	\(2\)	\( I_{6} \)	Non-split multiplicative	1	1	6	6
\(5\)	\(4\)	\( I_0^{*} \)	Additive	1	2	6	0
\(7\)	\(4\)	\( I_0^{*} \)	Additive	-1	2	6	0
\(11\)	\(2\)	\( I_{2} \)	Split multiplicative	-1	1	2	2

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X8.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by $$ and has index 6.

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

prime	Image of Galois representation
\(2\)	Cs

$p$-adic data

$p$-adic regulators

\(p\)-adic regulators are not yet computed for curves that are not \(\Gamma_0\)-optimal.

Iwasawa invariants

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47
Reduction type	ordinary	nonsplit	add	add	split	ordinary	ordinary	ss	ordinary	ordinary	ordinary	ordinary	ordinary	ss	ordinary
$\lambda$-invariant(s)	8	1	-	-	2	3	1	1,1	1	1	1	1	1	1,1	1
$\mu$-invariant(s)	1	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 \(d=\) 2.
Its isogeny class 40425.p consists of 4 curves linked by isogenies of degrees dividing 4.

Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base-change curve
4	\(\Q(\sqrt{3}, \sqrt{35})\)	\(\Z/2\Z \times \Z/4\Z\)	Not in database
	\(\Q(\sqrt{33}, \sqrt{-105})\)	\(\Z/2\Z \times \Z/4\Z\)	Not in database
	\(\Q(\sqrt{-11}, \sqrt{-35})\)	\(\Z/2\Z \times \Z/4\Z\)	Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.