Elliptic curve with LMFDB label 369600.ov2 (Cremona label 369600ov3)

• Label: 369600.ov2
• Conductor: $369600$
• Discriminant: $3.751\times 10^{23}$
• j-invariant: \( \frac{72313087342699809269}{11447096545640448} \)
• CM: no
• Rank: $1$
• Torsion structure: \(\Z/{2}\Z\)

Minimal Weierstrass equation

\(y^2=x^3+x^2-27774593x-48029603457\)

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

$P$	=	\(\left(-2427, 71340\right)\)
$\hat{h}(P)$	≈	$6.8010137110704829670270165444$

Torsion generators

\( \left(5983, 0\right) \)

Integral points

\((-2427,\pm 71340)\), \( \left(5983, 0\right) \)

Invariants

Conductor:	\( 369600 \)	=	$2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11$
Discriminant:	$375098459607546200064000 $	=	$2^{38} \cdot 3^{10} \cdot 5^{3} \cdot 7^{5} \cdot 11 $
j-invariant:	\( \frac{72313087342699809269}{11447096545640448} \)	=	$2^{-20} \cdot 3^{-10} \cdot 7^{-5} \cdot 11^{-1} \cdot 149^{3} \cdot 27961^{3}$
Endomorphism ring:	$\Z$
Geometric endomorphism ring:	\(\Z\)	(no potential complex multiplication)
Sato-Tate group:	$\mathrm{SU}(2)$
Faltings height:	$3.2461493054106191070324265870\dots$
Stable Faltings height:	$1.8040690564621760492563885715\dots$

BSD invariants

Analytic rank:	$1$
Regulator:	$6.8010137110704829670270165444\dots$
Real period:	$0.066466378049373054442577858905\dots$
Tamagawa product:	$ 80 $  = $ 2^{2}\cdot( 2 \cdot 5 )\cdot2\cdot1\cdot1 $
Torsion order:	$2$
Analytic order of Ш:	$1$ (exact)
Special value:	$ L'(E,1) $ ≈ $ 9.0407749687796065150076131564 $

Modular invariants

Modular form 369600.2.a.ov

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

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

Modular degree:	36864000
$ \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$(j)_{-}$
$2$	$4$	$I_{28}^{*}$	Additive	-1	6	38	20
$3$	$10$	$I_{10}$	Split multiplicative	-1	1	10	10
$5$	$2$	$III$	Additive	-1	2	3	0
$7$	$1$	$I_{5}$	Non-split multiplicative	1	1	5	5
$11$	$1$	$I_{1}$	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
$5$	5B.4.1	5.12.0.1

$p$-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, 5 and 10.
Its isogeny class 369600.ov consists of 4 curves linked by isogenies of degrees dividing 10.

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
$2$	\(\Q(\sqrt{385}) \)	\(\Z/2\Z \times \Z/2\Z\)	Not in database
$2$	\(\Q(\sqrt{-2}) \)	\(\Z/10\Z\)	Not in database
$4$	4.4.22176000.4	\(\Z/4\Z\)	Not in database
$4$	\(\Q(\sqrt{-2}, \sqrt{385})\)	\(\Z/2\Z \times \Z/10\Z\)	Not in database
$8$	Deg 8	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$8$	Deg 8	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$8$	Deg 8	\(\Z/6\Z\)	Not in database
$8$	Deg 8	\(\Z/20\Z\)	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
$16$	Deg 16	\(\Z/2\Z \times \Z/20\Z\)	Not in database
$16$	Deg 16	\(\Z/2\Z \times \Z/20\Z\)	Not in database
$16$	Deg 16	\(\Z/30\Z\)	Not in database
$20$	20.4.1505680748169532571648000000000000000.1	\(\Z/10\Z\)	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.