Elliptic curve with Cremona label 468270bf2 (LMFDB label 468270.bf2)

• Label: 468270bf2
• Conductor: $468270$
• Discriminant: $5.373\times 10^{16}$
• j-invariant: \( \frac{76711450249}{41602500} \)
• CM: no
• Rank: $1$
• Torsion structure: \(\Z/{2}\Z \times \Z/{2}\Z\)

Minimal Weierstrass equation

\(y^2+xy=x^3-x^2-96399x-2864295\)  

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

\(P\)	=	\(\left(-159, 2982\right)\)
\(\hat{h}(P)\)	≈	$1.1384058885660341890258037788$

Torsion generators

\( \left(-294, 147\right) \), \( \left(-30, 15\right) \)  

Integral points

\( \left(-294, 147\right) \), \( \left(-159, 2982\right) \), \( \left(-159, -2823\right) \), \( \left(-129, 2787\right) \), \( \left(-129, -2658\right) \), \( \left(-30, 15\right) \), \( \left(331, 1022\right) \), \( \left(331, -1353\right) \), \( \left(696, 15987\right) \), \( \left(696, -16683\right) \), \( \left(916, 25557\right) \), \( \left(916, -26473\right) \), \( \left(35445, 6655125\right) \), \( \left(35445, -6690570\right) \)  

Invariants

Conductor:	\( 468270 \)	=	\(2 \cdot 3^{2} \cdot 5 \cdot 11^{2} \cdot 43\)
Discriminant:	\(53728296180322500 \)	=	\(2^{2} \cdot 3^{8} \cdot 5^{4} \cdot 11^{6} \cdot 43^{2} \)
j-invariant:	\( \frac{76711450249}{41602500} \)	=	\(2^{-2} \cdot 3^{-2} \cdot 5^{-4} \cdot 7^{3} \cdot 43^{-2} \cdot 607^{3}\)
Endomorphism ring:	\(\Z\)
Geometric endomorphism ring:	\(\Z\)	(no potential complex multiplication)
Sato-Tate group:	$\mathrm{SU}(2)$
Faltings height:	\(1.9011698513779527540067172252\dots\)
Stable Faltings height:	\(0.15291607064471263627812281776\dots\)

BSD invariants

Analytic rank:	\(1\)
Regulator:	\(1.1384058885660341890258037788\dots\)
Real period:	\(0.28888484015132798951419476365\dots\)
Tamagawa product:	\( 256 \)  = \( 2\cdot2^{2}\cdot2^{2}\cdot2^{2}\cdot2 \)
Torsion order:	\(4\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form 468270.2.a.bf

\( q - q^{2} + q^{4} + q^{5} - 4q^{7} - q^{8} - q^{10} + 2q^{13} + 4q^{14} + q^{16} + 2q^{17} - 4q^{19} + O(q^{20}) \)  

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

Modular degree:	4423680
\( \Gamma_0(N) \)-optimal:	not computed^* (one of 3 curves in this isogeny class which might be optimal)
Manin constant:	1 (conditional^*)

^* The optimal curve in each isogeny class has not been determined in all cases for conductors over 400000. The Manin constant is correct provided that curve 468270bf1 is optimal.

Special L-value

\( L'(E,1) \) ≈ \( 5.2618912503316686479850795976225445037 \)

Local data

This elliptic curve is not semistable. There are 5 primes of bad reduction:

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord(\(N\))	ord(\(\Delta\))	ord\((j)_{-}\)
\(2\)	\(2\)	\(I_{2}\)	Non-split multiplicative	1	1	2	2
\(3\)	\(4\)	\(I_2^{*}\)	Additive	-1	2	8	2
\(5\)	\(4\)	\(I_{4}\)	Split multiplicative	-1	1	4	4
\(11\)	\(4\)	\(I_0^{*}\)	Additive	-1	2	6	0
\(43\)	\(2\)	\(I_{2}\)	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.

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 468270bf consists of 2 curves linked by isogenies of degrees dividing 4.