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([-19/4,-3/2,1,1/4]),K([-5,1,1,0]),K([1/4,-1/2,0,1/4]),K([-5657/4,-2085/2,172,539/4]),K([79337/4,31729/2,-2362,-8175/4])])
gp: E = ellinit([Polrev([-19/4,-3/2,1,1/4]),Polrev([-5,1,1,0]),Polrev([1/4,-1/2,0,1/4]),Polrev([-5657/4,-2085/2,172,539/4]),Polrev([79337/4,31729/2,-2362,-8175/4])], K);
magma: E := EllipticCurve([K![-19/4,-3/2,1,1/4],K![-5,1,1,0],K![1/4,-1/2,0,1/4],K![-5657/4,-2085/2,172,539/4],K![79337/4,31729/2,-2362,-8175/4]]);
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: | \((25)\) | = | \((1/2a^3-3a-1/2)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 390625 \) | = | \(25^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{506072727552}{25} a^{3} - \frac{1113403817359}{25} a^{2} - \frac{3724555649524}{25} a + \frac{8012299584804}{25} \) | ||
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 | $\left(\frac{1}{4} a^{3} - a^{2} - \frac{5}{2} a + \frac{21}{4} : -13 a^{3} - 16 a^{2} + 102 a + 138 : 1\right)$ | |
| Height | \(0.18893879309904039000581947663408335139\) | |
| 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: | $\left(\frac{17}{4} a^{3} + 3 a^{2} - \frac{67}{2} a - \frac{115}{4} : \frac{7}{2} a^{3} + 3 a^{2} - 26 a - \frac{43}{2} : 1\right)$ | $\left(-\frac{29}{16} a^{3} - \frac{7}{2} a^{2} + \frac{109}{8} a + \frac{427}{16} : -\frac{9}{16} a^{3} - \frac{3}{2} a^{2} + \frac{41}{8} a + \frac{245}{16} : 1\right)$ |
| 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: | \( 0.18893879309904039000581947663408335139 \) | ||
| Period: | \( 1163.2308612750028385903980308518879911 \) | ||
| Tamagawa product: | \( 4 \) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 2.28825207690562 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((1/2a^3-3a-1/2)\) | \(25\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
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
25.1-a
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.