Base field 3.3.733.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 7 x + 8 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([8, -7, -1, 1]))
gp: K = nfinit(Polrev([8, -7, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, -7, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1,0]),K([-4,1,1]),K([-4,1,1]),K([-5,22,-21]),K([452,-338,-30])])
gp: E = ellinit([Polrev([1,1,0]),Polrev([-4,1,1]),Polrev([-4,1,1]),Polrev([-5,22,-21]),Polrev([452,-338,-30])], K);
magma: E := EllipticCurve([K![1,1,0],K![-4,1,1],K![-4,1,1],K![-5,22,-21],K![452,-338,-30]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-2a^2-2a+19)\) | = | \((-a+3)^{2}\cdot(a^2+2a-3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 175 \) | = | \(5^{2}\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((25a^2+54a-7)\) | = | \((-a+3)^{6}\cdot(a^2+2a-3)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 765625 \) | = | \(5^{6}\cdot7^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{2353024794450}{49} a^{2} - \frac{434940018425}{49} a + \frac{16016561377497}{49} \) | ||
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: | \(2\) | |
| Generators | $\left(-\frac{109}{25} a^{2} - \frac{96}{25} a + \frac{744}{25} : \frac{2722}{125} a^{2} + \frac{549}{125} a - 138 : 1\right)$ | $\left(\frac{53}{4} a^{2} + \frac{73}{4} a - 38 : -\frac{1229}{8} a^{2} - \frac{1895}{8} a + 491 : 1\right)$ |
| Heights | \(3.7781342436710927801707368668132025885\) | \(1.0387287516048832963970216570836413859\) |
| 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);
| ||
| Torsion generators: | $\left(5 a - 4 : -3 a^{2} - a + 4 : 1\right)$ | $\left(-3 a : a^{2} + a + 2 : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| ||
BSD invariants
| Analytic rank: | \( 2 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(2\) | ||
| Regulator: | \( 0.29783268066656565891552334045505652696 \) | ||
| Period: | \( 72.401813751051594104680379014145386748 \) | ||
| Tamagawa product: | \( 8 \) = \(2^{2}\cdot2\) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 3.5841181921351919282727124687104657817 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a+3)\) | \(5\) | \(4\) | \(I_0^{*}\) | Additive | \(1\) | \(2\) | \(6\) | \(0\) |
| \((a^2+2a-3)\) | \(7\) | \(2\) | \(I_{2}\) | 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 |
|---|---|
| \(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
175.2-c
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.