Base field \(\Q(\sqrt{65}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 16 \); class number \(2\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-16, -1, 1]))
gp: K = nfinit(Polrev([-16, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-16, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1]),K([1,1]),K([0,1]),K([-4312,-1211]),K([-179456,-50801])])
gp: E = ellinit([Polrev([0,1]),Polrev([1,1]),Polrev([0,1]),Polrev([-4312,-1211]),Polrev([-179456,-50801])], K);
magma: E := EllipticCurve([K![0,1],K![1,1],K![0,1],K![-4312,-1211],K![-179456,-50801]]);
This is not a global minimal model: it is minimal at all primes except \((2,a)\). No global minimal model exists.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((15,3a+6)\) | = | \((5,a+2)\cdot(3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 45 \) | = | \(5\cdot9\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((2788425a+2624400)\) | = | \((2,a)^{12}\cdot(5,a+2)^{4}\cdot(3)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 110199605760000 \) | = | \(2^{12}\cdot5^{4}\cdot9^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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Minimal discriminant: | \((164025)\) | = | \((5,a+2)^{4}\cdot(3)^{8}\) |
Minimal discriminant norm: | \( 26904200625 \) | = | \(5^{4}\cdot9^{8}\) |
j-invariant: | \( \frac{272223782641}{164025} \) | ||
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 | $\left(\frac{2713}{144} a + \frac{2713}{36} : -\frac{413057}{1728} a - \frac{182921}{216} : 1\right)$ | |
Height | \(8.2265602054962857211249709802820999455\) | |
Torsion structure: | \(\Z/2\Z\oplus\Z/2\Z\) | |
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 generators: | $\left(-\frac{29}{4} a - 29 : \frac{141}{8} a + 58 : 1\right)$ | $\left(-7 a - 28 : 17 a + 56 : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
|
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Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 8.2265602054962857211249709802820999455 \) | ||
Period: | \( 1.9616888821913405749936556941151444563 \) | ||
Tamagawa product: | \( 16 \) = \(1\cdot2\cdot2^{3}\) | ||
Torsion order: | \(4\) | ||
Leading coefficient: | \( 4.0033331103078439801750466443065028204 \) | ||
Analytic order of Ш: | \( 1 \) (rounded) |
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
Primes of good reduction for the curve but which divide the
discriminant of the model above (if any) are included.
prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2,a)\) | \(2\) | \(1\) | \(I_0\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
\((5,a+2)\) | \(5\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
\((3)\) | \(9\) | \(8\) | \(I_{8}\) | Split multiplicative | \(-1\) | \(1\) | \(8\) | \(8\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
45.1-a
consists of curves linked by isogenies of
degrees dividing 16.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
Base field | Curve |
---|---|
\(\Q\) | 75.b2 |
\(\Q\) | 2535.j2 |