Elliptic curve with LMFDB label 1089.c2 (Cremona label 1089h1)

• Label: 1089.c2
• Conductor: $1089$
• Discriminant: $-10673289$
• j-invariant: \( -121 \)
• CM: no
• Rank: $1$
• Torsion structure: trivial

Minimal Weierstrass equation

\(y^2+xy+y=x^3-x^2-23x+168\)

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

\(P\)	=	\( \left(14, 42\right) \)
\(\hat{h}(P)\)	≈	$0.13276237501049809530571116873$

Integral points

\( \left(-4, 15\right) \), \( \left(-4, -12\right) \), \( \left(3, 9\right) \), \( \left(3, -13\right) \), \( \left(14, 42\right) \), \( \left(14, -57\right) \)

Invariants

Conductor:	\( 1089 \)	=	\(3^{2} \cdot 11^{2}\)
Discriminant:	\(-10673289 \)	=	\(-1 \cdot 3^{6} \cdot 11^{4} \)
j-invariant:	\( -121 \)	=	\(-1 \cdot 11^{2}\)
Endomorphism ring:	\(\Z\)
Geometric endomorphism ring:	\(\Z\)	(no potential complex multiplication)
Sato-Tate group:	$\mathrm{SU}(2)$

BSD invariants

Analytic rank:	\(1\)
Regulator:	\(0.13276237501049809530571116873\)
Real period:	\(1.9527584900431847587163473241\)
Tamagawa product:	\( 6 \)  = \( 2\cdot3 \)
Torsion order:	\(1\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form   1089.2.a.c

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

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

Modular degree:	144
\( \Gamma_0(N) \)-optimal:	yes
Manin constant:	1

Special L-value

\( L'(E,1) \) ≈ \( 1.5555171297602838352109652600424443272 \)

Local data

This elliptic curve is not semistable. There are 2 primes of bad reduction:

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord(\(N\))	ord(\(\Delta\))	ord\((j)_{-}\)
\(3\)	\(2\)	\(I_0^{*}\)	Additive	-1	2	6	0
\(11\)	\(3\)	\(IV\)	Additive	-1	2	4	0

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 X3.

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

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

prime	Image of Galois representation
\(11\)	B.10.4

$p$-adic data

$p$-adic regulators

Note: \(p\)-adic regulator data only exists for primes \(p\ge 5\) of good ordinary reduction.

Iwasawa invariants

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47
Reduction type	ordinary	add	ordinary	ordinary	add	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ss	ordinary
$\lambda$-invariant(s)	?	-	1	1	-	1	1	1	1	1	1	1	1	1,1	1
$\mu$-invariant(s)	?	-	0	0	-	0	0	0	0	0	0	0	0	0,0	0

An entry ? indicates that the invariants have not yet been computed.

An entry - indicates that the invariants are not computed because the reduction is additive.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 11.
Its isogeny class 1089.c consists of 2 curves linked by isogenies of degree 11.

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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$3$	3.1.484.1	\(\Z/2\Z\)	Not in database
$6$	6.0.937024.1	\(\Z/2\Z \times \Z/2\Z\)	Not in database
$8$	8.2.32019867.2	\(\Z/3\Z\)	Not in database
$10$	\(\Q(\zeta_{33})^+\)	\(\Z/11\Z\)	Not in database
$12$	Deg 12	\(\Z/4\Z\)	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.