Base field \(\Q(\sqrt{-1}) \)
Generator \(i\), with minimal polynomial \( x^{2} + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 0, 1]))
gp: K = nfinit(Polrev([1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,0]),K([0,-1]),K([1,1]),K([-33,0]),K([0,-5])])
gp: E = ellinit([Polrev([0,0]),Polrev([0,-1]),Polrev([1,1]),Polrev([-33,0]),Polrev([0,-5])], K);
magma: E := EllipticCurve([K![0,0],K![0,-1],K![1,1],K![-33,0],K![0,-5]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((294)\) | = | \((i+1)^{2}\cdot(3)\cdot(7)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 86436 \) | = | \(2^{2}\cdot9\cdot49^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((2333772)\) | = | \((i+1)^{4}\cdot(3)^{5}\cdot(7)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 5446491747984 \) | = | \(2^{4}\cdot9^{5}\cdot49^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{401408}{243} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
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Endomorphism ring: | \(\Z\) | ||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $\mathrm{SU}(2)$ |
Mordell-Weil group
Rank: | \(1\) |
Generator | $\left(12 i : 31 i - 32 : 1\right)$ |
Height | \(0.098152931708763517818763474540274697602\) |
Torsion structure: | trivial |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
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BSD invariants
Analytic rank: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 0.098152931708763517818763474540274697602 \) | ||
Period: | \( 1.1944626429087876324112110988813295404 \) | ||
Tamagawa product: | \( 15 \) = \(1\cdot5\cdot3\) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 3.5172003065428624940652234865609456347 \) | ||
Analytic order of Ш: | \( 1 \) (rounded) |
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((i+1)\) | \(2\) | \(1\) | \(IV\) | Additive | \(-1\) | \(2\) | \(4\) | \(0\) |
\((3)\) | \(9\) | \(5\) | \(I_{5}\) | Split multiplicative | \(-1\) | \(1\) | \(5\) | \(5\) |
\((7)\) | \(49\) | \(3\) | \(IV\) | Additive | \(1\) | \(2\) | \(4\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 86436.1-b consists of this curve only.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
Base field | Curve |
---|---|
\(\Q\) | 588.a1 |
\(\Q\) | 2352.o1 |