Base field \(\Q(\sqrt{249}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 62 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-62, -1, 1]))
gp: K = nfinit(Polrev([-62, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-62, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0]),K([0,0]),K([1,1]),K([-661075726,78794542]),K([-8972223402422,1069411885670])])
gp: E = ellinit([Polrev([1,0]),Polrev([0,0]),Polrev([1,1]),Polrev([-661075726,78794542]),Polrev([-8972223402422,1069411885670])], K);
magma: E := EllipticCurve([K![1,0],K![0,0],K![1,1],K![-661075726,78794542],K![-8972223402422,1069411885670]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((2)\) | = | \((-59a-436)\cdot(59a-495)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 4 \) | = | \(2\cdot2\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-3220a-23816)\) | = | \((-59a-436)^{18}\cdot(59a-495)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 1048576 \) | = | \(2^{18}\cdot2^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{34191163052375}{262144} a - \frac{286859170558125}{262144} \) | ||
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 | $\left(-\frac{9504361}{3675} a + \frac{79740019}{3675} : -\frac{13977906461}{385875} a + \frac{117270979369}{385875} : 1\right)$ |
Height | \(4.8882223547075563786282171364960051647\) |
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: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
|
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Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 4.8882223547075563786282171364960051647 \) | ||
Period: | \( 1.5208709471417341950278911352731982295 \) | ||
Tamagawa product: | \( 4 \) = \(2\cdot2\) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 3.7690650241577018902455414318911057089 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-59a-436)\) | \(2\) | \(2\) | \(I_{18}\) | Non-split multiplicative | \(1\) | \(1\) | \(18\) | \(18\) |
\((59a-495)\) | \(2\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3 and 9.
Its isogeny class
4.1-a
consists of curves linked by isogenies of
degrees dividing 9.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.