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
Mordell-Weil generators
Integral points
,
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: | 221184 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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-optimal: | no | |
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 | 2 | 2 | |||
additive | -1 | 2 | 10 | 4 | |||
additive | 1 | 2 | 6 | 0 |
Galois representations
The -adic Galois representation has maximal image for all primes except those listed in the table below.
prime | mod- image | -adic image |
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2Cs | 8.24.0.18 |
The image of the adelic Galois representation has level , index , genus , and generators
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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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good | |||
nonsplit multiplicative | |||
additive | |||
additive |
Isogenies
This curve has non-trivial cyclic isogenies of degree for
2 and 4.
Its isogeny class 24843d
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 21a2, its twist by .
Growth of torsion in number fields
The number fields of degree less than 24 such that is strictly larger than are as follows:
Base change curve | |||
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not in database | |||
not in database | |||
not in database | |||
8.0.1421970391296.10 | not in database | ||
8.0.364024420171776.33 | not in database | ||
8.8.4494128644096.1 | not in database | ||
8.0.12797733521664.81 | not in database | ||
deg 8 | not in database | ||
deg 16 | not in database | ||
deg 16 | not in database | ||
deg 16 | 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 | 7 | 13 | |
Reduction type | ord | nonsplit | add | add |
-invariant(s) | 5 | 0 | - | - |
-invariant(s) | 0 | 0 | - | - |
All Iwasawa and -invariants for primes of good reduction are zero.
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 .