Base field 4.4.9225.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 10 x^{2} + 7 x + 19 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([19, 7, -10, -1, 1]))
gp: K = nfinit(Polrev([19, 7, -10, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19, 7, -10, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1/4,-1/2,0,1/4]),K([-15/4,-3/2,1,1/4]),K([-15/4,-3/2,1,1/4]),K([1799/4,915/2,-53,-241/4]),K([-7087/2,-2799,404,707/2])])
gp: E = ellinit([Polrev([1/4,-1/2,0,1/4]),Polrev([-15/4,-3/2,1,1/4]),Polrev([-15/4,-3/2,1,1/4]),Polrev([1799/4,915/2,-53,-241/4]),Polrev([-7087/2,-2799,404,707/2])], K);
magma: E := EllipticCurve([K![1/4,-1/2,0,1/4],K![-15/4,-3/2,1,1/4],K![-15/4,-3/2,1,1/4],K![1799/4,915/2,-53,-241/4],K![-7087/2,-2799,404,707/2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((1/2a^3-3a-1/2)\) | = | \((1/2a^3-3a-1/2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 25 \) | = | \(25\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((625)\) | = | \((1/2a^3-3a-1/2)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 152587890625 \) | = | \(25^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{10463690358987}{2500} a^{3} + \frac{9678146215168}{625} a^{2} + \frac{12827568843}{250} a - \frac{14724711948631}{500} \) | ||
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 | $\left(-217 a^{3} - 236 a^{2} + 1694 a + 2022 : 11391 a^{3} + 12267 a^{2} - 88397 a - 103959 : 1\right)$ |
| Height | \(1.4698381415082935216892734860569099386\) |
| 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: | $\left(-\frac{7}{16} a^{3} + \frac{1}{4} a^{2} + \frac{37}{8} a - \frac{3}{16} : -\frac{1}{4} a^{3} - \frac{7}{4} a^{2} + \frac{1}{2} a + \frac{33}{8} : 1\right)$ |
| 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: | \( 1.4698381415082935216892734860569099386 \) | ||
| Period: | \( 5.1755588229641767930714288427852862214 \) | ||
| Tamagawa product: | \( 8 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.53450733678301 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((1/2a^3-3a-1/2)\) | \(25\) | \(8\) | \(I_{8}\) | 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 |
|---|---|
| \(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
25.1-d
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.