Elliptic curve with LMFDB label 121.a1 (Cremona label 121a2)

• Label: 121.a1
• Conductor: $121$
• Discriminant: $-25937424601$
• j-invariant: $-121$
• CM: no
• Rank: $0$
• Torsion structure: trivial

Minimal Weierstrass equation

$y^2+xy+y=x^3+x^2-305x+7888$

(, )

Mordell-Weil group structure

trivial

Invariants

Conductor:	$N$	=	$121$	=	$11^{2}$
Discriminant:	$\Delta$	=	$-25937424601$	=	$-1 \cdot 11^{10}$
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:	$h_{\mathrm{Faltings}}$	≈	$0.68098679828350618264603233113$
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$-1.3172592623818026040722539838$
$abc$ quality:	$Q$	≈	$0.9461121308337243$
Szpiro ratio:	$\sigma_{m}$	≈	$6.5685422788728385$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$0$
Mordell-Weil rank:	$r$	=	$0$
Regulator:	$\mathrm{Reg}(E/\Q)$	=	$1$
Real period:	$\Omega$	≈	$1.0197948617829165568371172784$
Tamagawa product:	$\prod_{p}c_p$	=	$1$
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$1$
Special value:	$L(E,1)$	≈	$1.0197948617829165568371172784$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$    (exact)

BSD formula

$\begin{aligned} 1.019794862 \approx L(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.019795 \cdot 1.000000 \cdot 1}{1^2} \\ & \approx 1.019794862\end{aligned}$

Modular invariants

Modular form   121.2.a.a

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

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

Modular degree:	66
$\Gamma_0(N)$-optimal:	no
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	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$11$	$1$	$II^{*}$	additive	-1	2	10	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	4.2.0.1
$11$	11B.1.4	11.120.1.4

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

$\left(\begin{array}{rr} 1 & 84 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 0 \\ 0 & 39 \end{array}\right),\left(\begin{array}{rr} 45 & 44 \\ 44 & 45 \end{array}\right),\left(\begin{array}{rr} 45 & 33 \\ 0 & 67 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 44 & 1 \end{array}\right),\left(\begin{array}{rr} 45 & 22 \\ 22 & 1 \end{array}\right),\left(\begin{array}{rr} 12 & 55 \\ 77 & 23 \end{array}\right),\left(\begin{array}{rr} 1 & 44 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 45 & 44 \\ 66 & 1 \end{array}\right)$.

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	$32$	$1$

Isogenies

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

Twists

The minimal quadratic twist of this elliptic curve is 121.c2, its twist by $-11$.

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
$5$	$\Q(\zeta_{11})^+$	$\Z/11\Z$	5.5.14641.1-121.1-d2
$6$	6.0.937024.1	$\Z/2\Z \oplus \Z/2\Z$	not in database
$8$	8.2.3874403907.1	$\Z/3\Z$	not in database
$12$	12.2.6799340234604544.1	$\Z/4\Z$	not in database
$15$	15.5.388863829589238784.1	$\Z/22\Z$	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	6	0	-	0	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 $0$.