Elliptic curve with Cremona label 6240ba1 (LMFDB label 6240.u3)

• Label: 6240ba1
• Conductor: $6240$
• Discriminant: $2433600$
• j-invariant: \( \frac{504358336}{38025} \)
• CM: no
• Rank: $1$
• Torsion structure: \(\Z/{2}\Z \times \Z/{2}\Z\)

Minimal Weierstrass equation

\(y^2=x^3+x^2-66x-216\)  

Mordell-Weil group structure

$\Z\times \Z/{2}\Z \times \Z/{2}\Z$

Infinite order Mordell-Weil generator and height

$P$	=	\(\left(21, 90\right)\)
$\hat{h}(P)$	≈	$2.7077321322842255603795325494$

Torsion generators

\( \left(-4, 0\right) \), \( \left(9, 0\right) \)  

Integral points

\( \left(-6, 0\right) \), \( \left(-4, 0\right) \), \( \left(9, 0\right) \), \((21,\pm 90)\)  

Invariants

Conductor:	\( 6240 \)	=	$2^{5} \cdot 3 \cdot 5 \cdot 13$
Discriminant:	$2433600 $	=	$2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2} $
j-invariant:	\( \frac{504358336}{38025} \)	=	$2^{6} \cdot 3^{-2} \cdot 5^{-2} \cdot 13^{-2} \cdot 199^{3}$
Endomorphism ring:	$\Z$
Geometric endomorphism ring:	\(\Z\)	(no potential complex multiplication)
Sato-Tate group:	$\mathrm{SU}(2)$
Faltings height:	$-0.029371143300508013491975127215\dots$
Stable Faltings height:	$-0.37594473358048066820059118794\dots$

BSD invariants

Analytic rank:	$1$
Regulator:	$2.7077321322842255603795325494\dots$
Real period:	$1.6808628335731643632398454780\dots$
Tamagawa product:	$ 16 $  = $ 2\cdot2\cdot2\cdot2 $
Torsion order:	$4$
Analytic order of Ш:	$1$ (exact)
Special value:	$ L'(E,1) $ ≈ $ 4.5513263044283697000553108219321639564 $

Modular invariants

Modular form   6240.2.a.u

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

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

Modular degree:	768
$ \Gamma_0(N) $-optimal:	yes
Manin constant:	1

Local data

This elliptic curve is not semistable. There are 4 primes of bad reduction:

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord($N$)	ord($\Delta$)	ord$(j)_{-}$
$2$	$2$	$III$	Additive	-1	5	6	0
$3$	$2$	$I_{2}$	Split multiplicative	-1	1	2	2
$5$	$2$	$I_{2}$	Non-split multiplicative	1	1	2	2
$13$	$2$	$I_{2}$	Non-split multiplicative	1	1	2	2

Galois representations

The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$	mod-$\ell$ image	$\ell$-adic image
$2$	2Cs	8.12.0.1

$p$-adic regulators

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.

Iwasawa invariants

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47
Reduction type	add	split	nonsplit	ss	ss	nonsplit	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ss
$\lambda$-invariant(s)	-	2	1	1,1	1,1	1	1	1	1	1	3	1	3	1	1,1
$\mu$-invariant(s)	-	0	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 $d=$ 2.
Its isogeny class 6240ba consists of 4 curves linked by isogenies of degrees dividing 4.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$4$	\(\Q(\sqrt{-2}, \sqrt{-15})\)	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$4$	\(\Q(\sqrt{13}, \sqrt{15})\)	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$4$	\(\Q(\sqrt{2}, \sqrt{-13})\)	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$8$	Deg 8	\(\Z/2\Z \times \Z/6\Z\)	Not in database
$16$	16.0.8979181539709000089600000000.18	\(\Z/4\Z \times \Z/4\Z\)	Not in database
$16$	Deg 16	\(\Z/2\Z \times \Z/8\Z\)	Not in database
$16$	Deg 16	\(\Z/2\Z \times \Z/8\Z\)	Not in database
$16$	Deg 16	\(\Z/2\Z \times \Z/8\Z\)	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.