Base field \(\Q(\sqrt{3}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-3, 0, 1]))
gp: K = nfinit(Polrev([-3, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1]),K([1,1]),K([0,1]),K([-33,-18]),K([-145,-83])])
gp: E = ellinit([Polrev([0,1]),Polrev([1,1]),Polrev([0,1]),Polrev([-33,-18]),Polrev([-145,-83])], K);
magma: E := EllipticCurve([K![0,1],K![1,1],K![0,1],K![-33,-18],K![-145,-83]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a-5)\) | = | \((a+1)\cdot(2a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 22 \) | = | \(2\cdot11\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((896a+6784)\) | = | \((a+1)^{15}\cdot(2a+1)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 43614208 \) | = | \(2^{15}\cdot11^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{7452136447}{340736} a - \frac{12920117437}{340736} \) | ||
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: | \(0\) |
Torsion structure: | trivial |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
|
BSD invariants
Analytic rank: | \( 0 \) | ||
sage: E.rank()
magma: Rank(E);
|
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Mordell-Weil rank: | \(0\) | ||
Regulator: | \( 1 \) | ||
Period: | \( 2.1651376205882341480148399063132972675 \) | ||
Tamagawa product: | \( 1 \) = \(1\cdot1\) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 0.62502139403960140008145842238576467309 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a+1)\) | \(2\) | \(1\) | \(I_{15}\) | Non-split multiplicative | \(1\) | \(1\) | \(15\) | \(15\) |
\((2a+1)\) | \(11\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(3\) | 3B.1.2 |
\(5\) | 5B.4.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3, 5 and 15.
Its isogeny class
22.2-a
consists of curves linked by isogenies of
degrees dividing 15.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.