Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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(homogenize, simplify) |
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(dehomogenize, simplify) |
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
Conductor: | = | = | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | = | = | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | = | = | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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Endomorphism ring: | = | |||||
Geometric endomorphism ring: | = | (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | = | |||||
Faltings height: | ≈ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | ≈ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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quality: | ≈ | |||||
Szpiro ratio: | ≈ |
BSD invariants
Analytic rank: | = | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Mordell-Weil rank: | = | comment: Rank
sage: E.rank()
gp: [lower,upper] = ellrank(E)
magma: Rank(E);
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Regulator: | = | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | ≈ | 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);
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Tamagawa product: | = | 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)
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Torsion order: | = | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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Special value: | ≈ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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Analytic order of Ш: | Ш | = | (exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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BSD formula
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 132 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 3 primes of bad reduction:
Tamagawa number | Kodaira symbol | Reduction type | Root number | ||||
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nonsplit multiplicative | 1 | 1 | 11 | 11 | |||
additive | 1 | 2 | 2 | 0 | |||
split multiplicative | -1 | 1 | 1 | 1 |
Galois representations
The -adic Galois representation has maximal image for all primes .
The image of the adelic Galois representation has level , index , genus , and generators
.
The torsion field is a degree- Galois extension of with isomorphic to the projection of to .
The table below list all primes for which the Serre invariants associated to the mod- Galois representation are exceptional.
Reduction type | Serre weight | Serre conductor | |
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nonsplit multiplicative | |||
additive | |||
split multiplicative |
Isogenies
This curve has no rational isogenies. Its isogeny class 550c consists of this curve only.
Twists
This elliptic curve is its own minimal quadratic twist.
Growth of torsion in number fields
The number fields of degree less than 24 such that is strictly larger than (which is trivial) are as follows:
Base change curve | |||
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3.1.2200.1 | not in database | ||
6.0.425920000.1 | not in database | ||
8.2.8004966750000.9 | not in database | ||
deg 12 | 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.
Iwasawa invariants
2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | |
Reduction type | nonsplit | ord | add | ss | split | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
-invariant(s) | 1 | 6 | - | 0,0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
-invariant(s) | 0 | 0 | - | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
-adic regulators
All -adic regulators are identically since the rank is .