Elliptic curve with LMFDB label 121.c2 (Cremona label 121c1)

• Label: 121.c2
• Conductor: $121$
• Discriminant: $-14641$
• j-invariant: $-121$
• CM: no
• Rank: $0$
• Torsion structure: trivial

Minimal Weierstrass equation

$y^2+xy=x^3+x^2-2x-7$

(, )

Mordell-Weil group structure

trivial

Invariants

Conductor:	$N$	=	$121$	=	$11^{2}$
Discriminant:	$\Delta$	=	$-14641$	=	$-1 \cdot 11^{4}$
j-invariant:	$j$	=	$-121$	=	$-1 \cdot 11^{2}$
Endomorphism ring:	$\mathrm{End}(E)$	=	$\Z$
Geometric endomorphism ring:	$\mathrm{End}(E_{\overline{\Q}})$	=	$\Z$    (no potential complex multiplication)
Sato-Tate group:	$\mathrm{ST}(E)$	=	$\mathrm{SU}(2)$
Faltings height:	$h_{\mathrm{Faltings}}$	≈	$-0.51796083811567908938493945785$
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$-1.3172592623818026040722539838$
$abc$ quality:	$Q$	≈	$0.9461121308337243$
Szpiro ratio:	$\sigma_{m}$	≈	$3.5685422788728385$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$0$
Mordell-Weil rank:	$r$	=	$0$
Regulator:	$\mathrm{Reg}(E/\Q)$	=	$1$
Real period:	$\Omega$	≈	$1.6661569203942160899376902986$
Tamagawa product:	$\prod_{p}c_p$	=	$1$
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$1$
Special value:	$L(E,1)$	≈	$1.6661569203942160899376902986$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$    (exact)

BSD formula

$\begin{aligned} 1.666156920 \approx L(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.666157 \cdot 1.000000 \cdot 1}{1^2} \\ & \approx 1.666156920\end{aligned}$

Modular invariants

Modular form   121.2.a.c

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

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

Modular degree:	6
$\Gamma_0(N)$-optimal:	yes
Manin constant:	1

Local data at primes of bad reduction

This elliptic curve is not semistable. There is only one prime $p$ of bad reduction:

$p$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$11$	$1$	$IV$	additive	-1	2	4	0

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
$2$	2G	8.4.0.2
$11$	11B.1.7	11.120.1.9

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $88 = 2^{3} \cdot 11$, index $480$, genus $16$, and generators

$\left(\begin{array}{rr} 1 & 0 \\ 44 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 44 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 84 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 45 & 44 \\ 44 & 45 \end{array}\right),\left(\begin{array}{rr} 45 & 22 \\ 22 & 1 \end{array}\right),\left(\begin{array}{rr} 12 & 55 \\ 77 & 23 \end{array}\right),\left(\begin{array}{rr} 45 & 44 \\ 66 & 1 \end{array}\right),\left(\begin{array}{rr} 45 & 33 \\ 0 & 67 \end{array}\right),\left(\begin{array}{rr} 73 & 0 \\ 0 & 49 \end{array}\right)$.

The torsion field $K:=\Q(E[88])$ is a degree-$42240$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/88\Z)$.

The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.

$\ell$	Reduction type	Serre weight	Serre conductor
$11$	additive	$52$	$1$

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 11.
Its isogeny class 121.c consists of 2 curves linked by isogenies of degree 11.

Twists

This elliptic curve is its own minimal quadratic twist.

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	$\Z/2\Z \oplus \Z/2\Z$	not in database
$8$	8.2.32019867.1	$\Z/3\Z$	not in database
$10$	$\Q(\zeta_{11})$	$\Z/11\Z$	not in database
$12$	12.2.56192894500864.1	$\Z/4\Z$	not in database

We only show fields where the torsion growth is primitive.

Iwasawa invariants

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

$p$-adic regulators

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