Elliptic curve with Cremona label 462d2 (LMFDB label 462.d1)

• Label: 462d2
• Conductor: $462$
• Discriminant: $1.670\times 10^{17}$
• j-invariant: \( \frac{10228636028672744397625}{167006381634183168} \)
• CM: no
• Rank: $0$
• Torsion structure: \(\Z/{2}\Z\)

Minimal Weierstrass equation

\(y^2+xy+y=x^3-452236x+115355594\)

Mordell-Weil group structure

$\Z/{2}\Z$

Torsion generators

\( \left(\frac{1399}{4}, -\frac{1403}{8}\right) \)

Integral points

None

Invariants

Conductor:	\( 462 \)	=	$2 \cdot 3 \cdot 7 \cdot 11$
Discriminant:	$167006381634183168 $	=	$2^{13} \cdot 3^{8} \cdot 7^{10} \cdot 11 $
j-invariant:	\( \frac{10228636028672744397625}{167006381634183168} \)	=	$2^{-13} \cdot 3^{-8} \cdot 5^{3} \cdot 7^{-10} \cdot 11^{-1} \cdot 4341461^{3}$
Endomorphism ring:	$\Z$
Geometric endomorphism ring:	\(\Z\)	(no potential complex multiplication)
Sato-Tate group:	$\mathrm{SU}(2)$
Faltings height:	$2.1039415247472178684061861606\dots$
Stable Faltings height:	$2.1039415247472178684061861606\dots$

BSD invariants

Analytic rank:	$0$
Regulator:	$1$
Real period:	$0.32293215924508325422980079615\dots$
Tamagawa product:	$ 16 $  = $ 1\cdot2^{3}\cdot2\cdot1 $
Torsion order:	$2$
Analytic order of Ш:	$1$ (exact)
Special value:	$ L(E,1) $ ≈ $ 1.2917286369803330169192031846 $

Modular invariants

Modular form   462.2.a.d

\( q - q^{2} + q^{3} + q^{4} - q^{6} - q^{7} - q^{8} + q^{9} - q^{11} + q^{12} + 6 q^{13} + q^{14} + q^{16} + 4 q^{17} - q^{18} + 6 q^{19} + O(q^{20}) \)

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

Modular degree:	8320
$ \Gamma_0(N) $-optimal:	no
Manin constant:	1

Local data

This elliptic curve is semistable. There are 4 primes of bad reduction:

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord($N$)	ord($\Delta$)	ord$(j)_{-}$
$2$	$1$	$I_{13}$	Non-split multiplicative	1	1	13	13
$3$	$8$	$I_{8}$	Split multiplicative	-1	1	8	8
$7$	$2$	$I_{10}$	Non-split multiplicative	1	1	10	10
$11$	$1$	$I_{1}$	Non-split multiplicative	1	1	1	1

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
$2$	2B	2.3.0.1

$p$-adic regulators

All $p$-adic regulators are identically $1$ since the rank is $0$.

Iwasawa invariants

$p$	2	3	7	11
Reduction type	nonsplit	split	nonsplit	nonsplit
$\lambda$-invariant(s)	2	3	0	0
$\mu$-invariant(s)	0	0	0	0

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

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 462d consists of 2 curves linked by isogenies of degree 2.

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

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$2$	\(\Q(\sqrt{22}) \)	\(\Z/2\Z \times \Z/2\Z\)	Not in database
$4$	4.0.17248.3	\(\Z/4\Z\)	Not in database
$8$	8.0.2303789694976.18	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$8$	8.4.364092804038656.25	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$8$	8.2.99636092064432.5	\(\Z/6\Z\)	Not in database
$16$	Deg 16	\(\Z/8\Z\)	Not in database
$16$	Deg 16	\(\Z/2\Z \times \Z/6\Z\)	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.