Elliptic curve with Cremona label 1122g1 (LMFDB label 1122.g2)

• Label: 1122g1
• Conductor: $1122$
• Discriminant: $3.401\times 10^{17}$
• j-invariant: \( \frac{7722211175253055152433}{340131399900069888} \)
• CM: no
• Rank: $0$
• Torsion structure: \(\Z/{2}\Z\)

Minimal Weierstrass equation

\( y^2 + x y + y = x^{3} + x^{2} - 411787 x - 97932871 \)

Mordell-Weil group structure

\(\Z/{2}\Z\)

Torsion generators

\( \left(-429, 214\right) \)

Integral points

\( \left(-429, 214\right) \)

Invariants

Conductor:	\( 1122 \)	=	\(2 \cdot 3 \cdot 11 \cdot 17\)
Discriminant:	\(340131399900069888 \)	=	\(2^{26} \cdot 3^{13} \cdot 11 \cdot 17^{2} \)
j-invariant:	\( \frac{7722211175253055152433}{340131399900069888} \)	=	\(2^{-26} \cdot 3^{-13} \cdot 11^{-1} \cdot 17^{-2} \cdot 19765777^{3}\)
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:	\(0.188978559914\)
Tamagawa product:	\( 52 \)  = \( ( 2 \cdot 13 )\cdot1\cdot1\cdot2 \)
Torsion order:	\(2\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form   1122.2.a.g

\( q + q^{2} - q^{3} + q^{4} + 2q^{5} - q^{6} - 2q^{7} + q^{8} + q^{9} + 2q^{10} + q^{11} - q^{12} + 4q^{13} - 2q^{14} - 2q^{15} + q^{16} + q^{17} + q^{18} - 2q^{19} + O(q^{20}) \)

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

Modular degree:	16224
\( \Gamma_0(N) \)-optimal:	yes
Manin constant:	1

Special L-value

\( L(E,1) \) ≈ \( 2.45672127888 \)

Local data

This elliptic curve is semistable.

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord(\(N\))	ord(\(\Delta\))	ord\((j)_{-}\)
\(2\)	\(26\)	\( I_{26} \)	Split multiplicative	-1	1	26	26
\(3\)	\(1\)	\( I_{13} \)	Non-split multiplicative	1	1	13	13
\(11\)	\(1\)	\( I_{1} \)	Split multiplicative	-1	1	1	1
\(17\)	\(2\)	\( I_{2} \)	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 X6.

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

The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.

prime	Image of Galois representation
\(2\)	B

$p$-adic data

$p$-adic regulators

All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).

Iwasawa invariants

$p$	2	3	5	7	11	13	17
Reduction type	split	nonsplit	ordinary	ordinary	split	ordinary	split
$\lambda$-invariant(s)	4	0	0	0	1	2	1
$\mu$-invariant(s)	0	0	0	0	0	0	0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 17$ of good reduction are zero.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 1122g consists of 2 curves linked by isogenies of degree 2.

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

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$2$	\(\Q(\sqrt{33}) \)	\(\Z/2\Z \times \Z/2\Z\)	Not in database
$4$	4.0.610368.2	\(\Z/4\Z\)	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.