Elliptic curve with Cremona label 377520dy2 (LMFDB label 377520.dy3)

• Label: 377520dy2
• Conductor: $377520$
• Discriminant: $1.109\times 10^{23}$
• j-invariant: \( \frac{445574312599094932036}{61129333175625} \)
• CM: no
• Rank: $1$
• Torsion structure: \(\Z/{2}\Z \times \Z/{2}\Z\)

Minimal Weierstrass equation

\(y^2=x^3+x^2-194066616x+1040389299684\)

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

$P$	=	\(\left(8214, 24300\right)\)
$\hat{h}(P)$	≈	$1.4959966809891333322589053806$

Torsion generators

\( \left(7971, 0\right) \), \( \left(8114, 0\right) \)

Integral points

\( \left(-16086, 0\right) \), \((-6186,\pm 1415700)\), \((7872,\pm 23958)\), \( \left(7971, 0\right) \), \( \left(8114, 0\right) \), \((8214,\pm 24300)\), \((13155,\pm 874152)\), \((42720,\pm 8409258)\)

Invariants

Conductor:	\( 377520 \)	=	$2^{4} \cdot 3 \cdot 5 \cdot 11^{2} \cdot 13$
Discriminant:	$110893406832582042240000 $	=	$2^{10} \cdot 3^{14} \cdot 5^{4} \cdot 11^{8} \cdot 13^{2} $
j-invariant:	\( \frac{445574312599094932036}{61129333175625} \)	=	$2^{2} \cdot 3^{-14} \cdot 5^{-4} \cdot 7^{3} \cdot 11^{-2} \cdot 13^{-2} \cdot 643^{3} \cdot 1069^{3}$
Endomorphism ring:	$\Z$
Geometric endomorphism ring:	\(\Z\)	(no potential complex multiplication)
Sato-Tate group:	$\mathrm{SU}(2)$
Faltings height:	$3.4397201800310088497922152902\dots$
Stable Faltings height:	$1.6631498931652024865802167333\dots$

BSD invariants

Analytic rank:	$1$
Regulator:	$1.4959966809891333322589053806\dots$
Real period:	$0.10172825219747958645332707703\dots$
Tamagawa product:	$ 896 $  = $ 2^{2}\cdot( 2 \cdot 7 )\cdot2\cdot2^{2}\cdot2 $
Torsion order:	$4$
Analytic order of Ш:	$1$ (exact)
Special value:	$ L'(E,1) $ ≈ $ 8.5223671484142783638433844459 $

Modular invariants

Modular form 377520.2.a.dy

\( q + q^{3} - q^{5} - 4 q^{7} + q^{9} - q^{13} - q^{15} + 2 q^{17} + 4 q^{19} + O(q^{20}) \)

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

Modular degree:	82575360
$ \Gamma_0(N) $-optimal:	no
Manin constant:	1

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$(j)_{-}$
$2$	$4$	$I_{2}^{*}$	Additive	1	4	10	0
$3$	$14$	$I_{14}$	Split multiplicative	-1	1	14	14
$5$	$2$	$I_{4}$	Non-split multiplicative	1	1	4	4
$11$	$4$	$I_{2}^{*}$	Additive	-1	2	8	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

$p$-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 377520dy 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
$4$	\(\Q(\sqrt{33}, \sqrt{-39})\)	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$4$	\(\Q(\sqrt{-2}, \sqrt{-33})\)	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$4$	\(\Q(\sqrt{2}, \sqrt{143})\)	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$8$	Deg 8	\(\Z/2\Z \times \Z/6\Z\)	Not in database
$16$	Deg 16	\(\Z/4\Z \times \Z/4\Z\)	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.