Base field \(\Q(\sqrt{-3}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -1, 1]))
gp: K = nfinit(Polrev([1, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1]),K([0,1]),K([1,1]),K([-103681,-32401]),K([-12381240,-6545490])])
gp: E = ellinit([Polrev([0,1]),Polrev([0,1]),Polrev([1,1]),Polrev([-103681,-32401]),Polrev([-12381240,-6545490])], K);
magma: E := EllipticCurve([K![0,1],K![0,1],K![1,1],K![-103681,-32401],K![-12381240,-6545490]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((315a-75)\) | = | \((-2a+1)^{2}\cdot(-5a+3)^{2}\cdot(5)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 81225 \) | = | \(3^{2}\cdot19^{2}\cdot25\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((55112400a+30300885)\) | = | \((-2a+1)^{14}\cdot(-5a+3)^{6}\cdot(5)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 5625474760017225 \) | = | \(3^{14}\cdot19^{6}\cdot25\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{1114544804970241}{405} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Mordell-Weil group
| Rank: | \(1\) |
| Generator | $\left(-\frac{965103}{45602} a - \frac{17005511}{91204} : -\frac{157090875}{13771804} a + \frac{4990856241}{27543608} : 1\right)$ |
| Height | \(8.0554149162572018912719081815108446876\) |
| 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: | $\left(-\frac{109}{4} a - \frac{751}{4} : 107 a - \frac{113}{8} : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 8.0554149162572018912719081815108446876 \) | ||
| Period: | \( 0.074031481477691275258490823261450506658 \) | ||
| Tamagawa product: | \( 8 \) = \(2\cdot2^{2}\cdot1\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.7544425258751457899834582203215905609 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a+1)\) | \(3\) | \(2\) | \(I_{8}^{*}\) | Additive | \(-1\) | \(2\) | \(14\) | \(8\) |
| \((-5a+3)\) | \(19\) | \(4\) | \(I_0^{*}\) | Additive | \(-1\) | \(2\) | \(6\) | \(0\) |
| \((5)\) | \(25\) | \(1\) | \(I_{1}\) | Non-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, 4, 8 and 16.
Its isogeny class
81225.1-a
consists of curves linked by isogenies of
degrees dividing 16.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.