Elliptic curve with LMFDB label 62400.r3 (Cremona label 62400bh2)

• Label: 62400.r3
• Conductor: $62400$
• Discriminant: $1.046\times 10^{19}$
• j-invariant: \( \frac{242702053576633}{2554695936} \)
• CM: no
• Rank: $0$
• Torsion structure: \(\Z/{2}\Z \times \Z/{2}\Z\)

Minimal Weierstrass equation

\(y^2=x^3-x^2-2079233x+1144142337\)

Mordell-Weil group structure

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

Torsion generators

\( \left(767, 0\right) \), \( \left(897, 0\right) \)

Integral points

\( \left(-1663, 0\right) \), \( \left(767, 0\right) \), \( \left(897, 0\right) \)

Invariants

Conductor:	\( 62400 \)	=	\(2^{6} \cdot 3 \cdot 5^{2} \cdot 13\)
Discriminant:	\(10464034553856000000 \)	=	\(2^{26} \cdot 3^{10} \cdot 5^{6} \cdot 13^{2} \)
j-invariant:	\( \frac{242702053576633}{2554695936} \)	=	\(2^{-8} \cdot 3^{-10} \cdot 7^{6} \cdot 13^{-2} \cdot 19^{3} \cdot 67^{3}\)
Endomorphism ring:	\(\Z\)
Geometric endomorphism ring:	\(\Z\)	(no potential complex multiplication)
Sato-Tate group:	$\mathrm{SU}(2)$

BSD invariants

Analytic rank:	\(0\)
Regulator:	\(1\)
Real period:	\(0.22933405634269361036298908069\)
Tamagawa product:	\( 64 \)  = \( 2^{2}\cdot2\cdot2^{2}\cdot2 \)
Torsion order:	\(4\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form 62400.2.a.r

\( q - q^{3} - 4q^{7} + q^{9} + 4q^{11} + q^{13} - 2q^{17} + 8q^{19} + O(q^{20}) \)

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

Modular degree:	1966080
\( \Gamma_0(N) \)-optimal:	no
Manin constant:	1

Special L-value

\( L(E,1) \) ≈ \( 0.91733622537077444145195632276951511207 \)

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\((j)_{-}\)
\(2\)	\(4\)	\(I_{16}^{*}\)	Additive	1	6	26	8
\(3\)	\(2\)	\(I_{10}\)	Non-split multiplicative	1	1	10	10
\(5\)	\(4\)	\(I_0^{*}\)	Additive	1	2	6	0
\(13\)	\(2\)	\(I_{2}\)	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

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

Iwasawa invariants

$p$	2	3	5	13
Reduction type	add	nonsplit	add	split
$\lambda$-invariant(s)	-	0	-	1
$\mu$-invariant(s)	-	0	-	0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.

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 62400.r consists of 2 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
$4$	\(\Q(\sqrt{3}, \sqrt{-10})\)	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$4$	\(\Q(\sqrt{30}, \sqrt{-39})\)	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$4$	\(\Q(\sqrt{10}, \sqrt{13})\)	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$8$	Deg 8	\(\Z/2\Z \times \Z/6\Z\)	Not in database
$16$	16.0.8979181539709000089600000000.10	\(\Z/4\Z \times \Z/4\Z\)	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.