Base field \(\Q(\sqrt{-11}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, -1, 1]))
gp: K = nfinit(Polrev([3, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,0]),K([0,0]),K([0,0]),K([-3,9]),K([2,72])])
gp: E = ellinit([Polrev([0,0]),Polrev([0,0]),Polrev([0,0]),Polrev([-3,9]),Polrev([2,72])], K);
magma: E := EllipticCurve([K![0,0],K![0,0],K![0,0],K![-3,9],K![2,72]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((144)\) | = | \((-a)^{2}\cdot(a-1)^{2}\cdot(2)^{4}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 20736 \) | = | \(3^{2}\cdot3^{2}\cdot4^{4}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-2239488a+6718464)\) | = | \((-a)^{9}\cdot(a-1)^{7}\cdot(2)^{10}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 45137758519296 \) | = | \(3^{9}\cdot3^{7}\cdot4^{10}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{868}{27} a - \frac{856}{27} \) | ||
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: | \(2\) | |
Generators | $\left(-9 a + 10 : -36 a : 1\right)$ | $\left(-a + 2 : -4 a - 8 : 1\right)$ |
Heights | \(0.41492476847495909822142395296041658098\) | \(0.40397743389035277158599728707464740070\) |
Torsion structure: | \(\Z/2\Z\) | |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
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Torsion generator: | $\left(3 a - 2 : 0 : 1\right)$ | |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 2 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(2\) | ||
Regulator: | \( 0.16012248343827691944589199831701501594 \) | ||
Period: | \( 1.0183519027736842006933883902337597063 \) | ||
Tamagawa product: | \( 64 \) = \(2^{2}\cdot2^{2}\cdot2^{2}\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 6.2930882711032314279153798430520167240 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a)\) | \(3\) | \(4\) | \(I_{3}^{*}\) | Additive | \(-1\) | \(2\) | \(9\) | \(3\) |
\((a-1)\) | \(3\) | \(4\) | \(I_{1}^{*}\) | Additive | \(-1\) | \(2\) | \(7\) | \(1\) |
\((2)\) | \(4\) | \(4\) | \(I_{2}^{*}\) | Additive | \(-1\) | \(4\) | \(10\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
20736.3-d
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.