Elliptic curve with LMFDB label 1110.g2 (Cremona label 1110f2)

• Label: 1110.g2
• Conductor: $1110$
• Discriminant: $5.614\times 10^{15}$
• j-invariant: \( \frac{825824067562227826729}{5613755625000000} \)
• CM: no
• Rank: $0$
• Torsion structure: \(\Z/{2}\Z \times \Z/{2}\Z\)

Minimal Weierstrass equation

\(y^2+xy+y=x^3-195459x+33048382\)  

Mordell-Weil group structure

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

Torsion generators

\( \left(239, -120\right) \), \( \left(271, -136\right) \)  

Integral points

\( \left(239, -120\right) \), \( \left(271, -136\right) \)  

Invariants

Conductor:	\( 1110 \)	=	$2 \cdot 3 \cdot 5 \cdot 37$
Discriminant:	$5613755625000000 $	=	$2^{6} \cdot 3^{8} \cdot 5^{10} \cdot 37^{2} $
j-invariant:	\( \frac{825824067562227826729}{5613755625000000} \)	=	$2^{-6} \cdot 3^{-8} \cdot 5^{-10} \cdot 7^{3} \cdot 13^{3} \cdot 37^{-2} \cdot 103099^{3}$
Endomorphism ring:	$\Z$
Geometric endomorphism ring:	\(\Z\)	(no potential complex multiplication)
Sato-Tate group:	$\mathrm{SU}(2)$
Faltings height:	$1.8562177484520060938226589496\dots$
Stable Faltings height:	$1.8562177484520060938226589496\dots$

BSD invariants

Analytic rank:	$0$
Regulator:	$1$
Real period:	$0.42998760526459914054667926505\dots$
Tamagawa product:	$ 64 $  = $ 2\cdot2^{3}\cdot2\cdot2 $
Torsion order:	$4$
Analytic order of Ш:	$1$ (exact)
Special value:	$ L(E,1) $ ≈ $ 1.7199504210583965621867170602046509292 $

Modular invariants

Modular form   1110.2.a.g

\( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + 4q^{7} - q^{8} + q^{9} + q^{10} + 4q^{11} + q^{12} + 2q^{13} - 4q^{14} - q^{15} + q^{16} - 2q^{17} - q^{18} + O(q^{20}) \)  

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

Modular degree:	11520
$ \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$(j)_{-}$
$2$	$2$	$I_{6}$	Non-split multiplicative	1	1	6	6
$3$	$8$	$I_{8}$	Split multiplicative	-1	1	8	8
$5$	$2$	$I_{10}$	Non-split multiplicative	1	1	10	10
$37$	$2$	$I_{2}$	Non-split multiplicative	1	1	2	2

Galois representations

The $\ell$-adic Galois representation has maximal image $\GL(2,\Z_\ell)$ 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

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

Iwasawa invariants

$p$	2	3	5	37
Reduction type	nonsplit	split	nonsplit	nonsplit
$\lambda$-invariant(s)	3	1	0	0
$\mu$-invariant(s)	1	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 1110.g 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
$4$	\(\Q(\sqrt{-5}, \sqrt{-37})\)	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$4$	\(\Q(\sqrt{2}, \sqrt{5})\)	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$4$	\(\Q(\sqrt{-2}, \sqrt{37})\)	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$8$	8.2.207501249166875.1	\(\Z/2\Z \times \Z/6\Z\)	Not in database
$16$	16.0.5892962649399466393600000000.2	\(\Z/4\Z \times \Z/4\Z\)	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.