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

Show commands: Magma / Oscar / PariGP / SageMath

Simplified equation

$y^2+xy+y=x^3+x^2-305x+7888$ y^2+xy+y=x^3+x^2-305x+7888	(homogenize, simplify)
$y^2z+xyz+yz^2=x^3+x^2z-305xz^2+7888z^3$ y^2z+xyz+yz^2=x^3+x^2z-305xz^2+7888z^3	(dehomogenize, simplify)
$y^2=x^3-395307x+373960422$ y^2=x^3-395307x+373960422	(homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, 1, 1, -305, 7888])

gp: E = ellinit([1, 1, 1, -305, 7888])

magma: E := EllipticCurve([1, 1, 1, -305, 7888]);

oscar: E = elliptic_curve([1, 1, 1, -305, 7888])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

Invariants

Conductor:	$N$	=	$121$	=	$11^{2}$	comment: Conductor sage: E.conductor().factor() gp: ellglobalred(E)[1] magma: Conductor(E); oscar: conductor(E)
Discriminant:	$\Delta$	=	$-25937424601$	=	$-1 \cdot 11^{10}$	comment: Discriminant sage: E.discriminant().factor() gp: E.disc magma: Discriminant(E); oscar: discriminant(E)
j-invariant:	$j$	=	$-121$	=	$-1 \cdot 11^{2}$	comment: j-invariant sage: E.j_invariant().factor() gp: E.j magma: jInvariant(E); oscar: j_invariant(E)
Endomorphism ring:	$\mathrm{End}(E)$	=	$\Z$
Geometric endomorphism ring:	$\mathrm{End}(E_{\overline{\Q}})$	=	$\Z$ (no potential complex multiplication)			sage: E.has_cm() magma: HasComplexMultiplication(E);
Sato-Tate group:	$\mathrm{ST}(E)$	=	$\mathrm{SU}(2)$
Faltings height:	$h_{\mathrm{Faltings}}$	≈	$0.68098679828350618264603233113$			gp: ellheight(E) magma: FaltingsHeight(E); oscar: faltings_height(E)
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$-1.3172592623818026040722539838$			magma: StableFaltingsHeight(E); oscar: stable_faltings_height(E)
$abc$ quality:	$Q$	≈	$0.9461121308337243$
Szpiro ratio:	$\sigma_{m}$	≈	$6.5685422788728385$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$0$	sage: E.analytic_rank() gp: ellanalyticrank(E) magma: AnalyticRank(E);
Mordell-Weil rank:	$r$	=	$0$	comment: Rank sage: E.rank() gp: [lower,upper] = ellrank(E) magma: Rank(E);
Regulator:	$\mathrm{Reg}(E/\Q)$	=	$1$	comment: Regulator sage: E.regulator() G = E.gen \\ if available matdet(ellheightmatrix(E,G)) magma: Regulator(E);
Real period:	$\Omega$	≈	$1.0197948617829165568371172784$	comment: Real Period sage: E.period_lattice().omega() gp: if(E.disc>0,2,1)E.omega[1] magma: (Discriminant(E) gt 0 select 2 else 1) RealPeriod(E);
Tamagawa product:	$\prod_{p}c_p$	=	$1$	comment: Tamagawa numbers sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] \| i<-[1..#gr[4][,1]]] magma: TamagawaNumbers(E); oscar: tamagawa_numbers(E)
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$1$	comment: Torsion order sage: E.torsion_order() gp: elltors(E)[1] magma: Order(TorsionSubgroup(E)); oscar: prod(torsion_structure(E)[1])
Special value:	$L(E,1)$	≈	$1.0197948617829165568371172784$	comment: Special L-value r = E.rank(); E.lseries().dokchitser().derivative(1,r)/r.factorial() gp: [r,L1r] = ellanalyticrank(E); L1r/r! magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
Analytic order of Ш:	Ш ${}_{\mathrm{an}}$	=	$1$ (exact)	comment: Order of Sha sage: E.sha().an_numerical() magma: MordellWeilShaInformation(E);

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

# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)

E = EllipticCurve(%s); r = E.rank(); ar = E.analytic_rank(); assert r == ar;

Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical();

omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order();

assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))

/* self-contained Magma code snippet for the BSD formula (checks rank, computes analytic sha) */

E := EllipticCurve(%s); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar;

sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1);

reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E);

assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);

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})

comment: q-expansion of modular form

sage: E.q_eigenform(20)

\\ actual modular form, use for small N

[mf,F] = mffromell(E)

Ser(mfcoefs(mf,20),q)

\\ or just the series

Ser(ellan(E,20),q)*q

magma: ModularForm(E);

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

Modular degree:	66	comment: Modular degree sage: E.modular_degree() gp: ellmoddegree(E) magma: ModularDegree(E);
$\Gamma_0(N)$ -optimal:	no
Manin constant:	1	comment: Manin constant magma: ManinConstant(E);

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

comment: Local data

sage: E.local_data()

gp: ellglobalred(E)[5]

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

oscar: [(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]

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

comment: mod p Galois image

sage: rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

gens = [[1, 84, 0, 1], [15, 0, 0, 39], [45, 44, 44, 45], [45, 33, 0, 67], [1, 0, 44, 1], [45, 22, 22, 1], [12, 55, 77, 23], [1, 44, 0, 1], [45, 44, 66, 1]]

GL(2,Integers(88)).subgroup(gens)

Gens := [[1, 84, 0, 1], [15, 0, 0, 39], [45, 44, 44, 45], [45, 33, 0, 67], [1, 0, 44, 1], [45, 22, 22, 1], [12, 55, 77, 23], [1, 44, 0, 1], [45, 44, 66, 1]];

sub<GL(2,Integers(88))|Gens>;

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

gp: ellisomat(E)

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 $0$ .

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

Mordell-Weil group structure

Invariants

BSD invariants

BSD formula

Modular invariants

Local data at primes of bad reduction

Galois representations

Isogenies

Twists

Growth of torsion in number fields

Iwasawa invariants

$p$ -adic regulators

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	121.a1

Conductor	$121$
Discriminant	$-25937424601$
j-invariant	$-121$
CM	no
Rank	$0$
Torsion structure	trivial

Properties

Related objects

Downloads

Learn more

Mordell-Weil group structure

Invariants

Local data at primes of bad reduction

Growth of torsion in number fields

ppp-adic regulators

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

$p$ -adic regulators