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

\( y^2 + x y + y = x^{3} - 195459 x + 33048382 \)

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:	$\mathrm{SU}(2)$

BSD invariants

Rank:	\(0\)
Regulator:	\(1\)
Real period:	\(0.429987605265\)
Tamagawa product:	\( 64 \)  = \( 2\cdot2^{3}\cdot2\cdot2 \)
Torsion order:	\(4\)
Analytic order of Ш:	\(1\) (exact)

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

Special L-value

\( L(E,1) \) ≈ \( 1.71995042106 \)

Local data

This elliptic curve is semistable.

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 image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X8c.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 6 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 5 \end{array}\right)$ and has index 12.

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	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 4 curves linked by isogenies of degrees dividing 4.

Growth of torsion in number fields

The number fields $K$ of degree up to 7 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
$4$	\(\Q(\sqrt{2}, \sqrt{5})\)	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$4$	\(\Q(\sqrt{-2}, \sqrt{37})\)	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$4$	\(\Q(\sqrt{-5}, \sqrt{-37})\)	\(\Z/2\Z \times \Z/4\Z\)	Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.