Base field \(\Q(\sqrt{19}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 19 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-19, 0, 1]))
gp: K = nfinit(Polrev([-19, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-19, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1]),K([-1,1]),K([1,1]),K([13465,-3084]),K([343555,-78809])])
gp: E = ellinit([Polrev([1,1]),Polrev([-1,1]),Polrev([1,1]),Polrev([13465,-3084]),Polrev([343555,-78809])], K);
magma: E := EllipticCurve([K![1,1],K![-1,1],K![1,1],K![13465,-3084],K![343555,-78809]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-18a+78)\) | = | \((-3a+13)^{3}\cdot(-a-4)\cdot(-a+4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
Conductor norm: | \( 72 \) | = | \(2^{3}\cdot3\cdot3\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((96a+96)\) | = | \((-3a+13)^{11}\cdot(-a-4)^{3}\cdot(-a+4)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -165888 \) | = | \(-2^{11}\cdot3^{3}\cdot3\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{388117029814481}{27} a + \frac{1691762911268209}{27} \) | ||
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: | \(0\) |
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);
| |
Torsion generator: | $\left(\frac{5}{2} a - 14 : \frac{21}{4} a - \frac{69}{4} : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 0 \) | ||
sage: E.rank()
magma: Rank(E);
|
|||
Mordell-Weil rank: | \(0\) | ||
Regulator: | \( 1 \) | ||
Period: | \( 8.8199694773647099928253342571200501196 \) | ||
Tamagawa product: | \( 3 \) = \(1\cdot3\cdot1\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 3.0351596555483700821981386020458045639 \) | ||
Analytic order of Ш: | \( 4 \) (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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-3a+13)\) | \(2\) | \(1\) | \(II^{*}\) | Additive | \(1\) | \(3\) | \(11\) | \(0\) |
\((-a-4)\) | \(3\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
\((-a+4)\) | \(3\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
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 and 4.
Its isogeny class
72.1-e
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.