Elliptic curve with LMFDB label 23520.g3 (Cremona label 23520bd1)

• Label: 23520.g3
• Conductor: $23520$
• Discriminant: $1694145600$
• j-invariant: \( \frac{1906624}{225} \)
• CM: no
• Rank: $1$
• Torsion structure: \(\Z/{2}\Z \times \Z/{2}\Z\)

Minimal Weierstrass equation

\(y^2=x^3-x^2-506x-3744\)

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

\(P\)	=	\( \left(-14, 20\right) \)
\(\hat{h}(P)\)	≈	$1.6656952149485303250539869558$

Torsion generators

\( \left(-9, 0\right) \), \( \left(26, 0\right) \)

Integral points

\( \left(-16, 0\right) \), \((-14,\pm 20)\), \( \left(-9, 0\right) \), \( \left(26, 0\right) \), \((40,\pm 196)\), \((131,\pm 1470)\)

Invariants

Conductor:	\( 23520 \)	=	\(2^{5} \cdot 3 \cdot 5 \cdot 7^{2}\)
Discriminant:	\(1694145600 \)	=	\(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7^{6} \)
j-invariant:	\( \frac{1906624}{225} \)	=	\(2^{6} \cdot 3^{-2} \cdot 5^{-2} \cdot 31^{3}\)
Endomorphism ring:	\(\Z\)
Geometric endomorphism ring:	\(\Z\)	(no potential complex multiplication)
Sato-Tate group:	$\mathrm{SU}(2)$

BSD invariants

Analytic rank:	\(1\)
Regulator:	\(1.6656952149485303250539869558\)
Real period:	\(1.0142032875025477262762693755\)
Tamagawa product:	\( 32 \)  = \( 2\cdot2\cdot2\cdot2^{2} \)
Torsion order:	\(4\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form 23520.2.a.g

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

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

Modular degree:	12288
\( \Gamma_0(N) \)-optimal:	yes
Manin constant:	1

Special L-value

\( L'(E,1) \) ≈ \( 3.3787071259561246688611554435384536049 \)

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
\(3\)	\(2\)	\(I_{2}\)	Non-split multiplicative	1	1	2	2
\(5\)	\(2\)	\(I_{2}\)	Non-split multiplicative	1	1	2	2
\(7\)	\(4\)	\(I_0^{*}\)	Additive	-1	2	6	0

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

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	nonsplit	nonsplit	add	ss	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary
$\lambda$-invariant(s)	-	5	5	-	1,3	1	1	3	1	1	1	1	1	1	1
$\mu$-invariant(s)	-	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 23520.g 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
$4$	\(\Q(\sqrt{-5}, \sqrt{7})\)	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$4$	\(\Q(\sqrt{30}, \sqrt{35})\)	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$4$	\(\Q(\sqrt{6}, \sqrt{-7})\)	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$8$	Deg 8	\(\Z/2\Z \times \Z/6\Z\)	Not in database
$16$	16.0.63456228123711897600000000.13	\(\Z/4\Z \times \Z/4\Z\)	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.