Base field \(\Q(\sqrt{14}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 14 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-14, 0, 1]))
gp: K = nfinit(Polrev([-14, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-14, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1]),K([1,-1]),K([0,0]),K([-2720,-733]),K([73447,19624])])
gp: E = ellinit([Polrev([0,1]),Polrev([1,-1]),Polrev([0,0]),Polrev([-2720,-733]),Polrev([73447,19624])], K);
magma: E := EllipticCurve([K![0,1],K![1,-1],K![0,0],K![-2720,-733],K![73447,19624]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((12)\) | = | \((-a+4)^{4}\cdot(3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 144 \) | = | \(2^{4}\cdot9\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((1296)\) | = | \((-a+4)^{8}\cdot(3)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 1679616 \) | = | \(2^{8}\cdot9^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{1556068}{81} \) | ||
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 | $\left(-2 a - \frac{41}{4} : -\frac{481}{8} a - \frac{1841}{8} : 1\right)$ | $\left(\frac{19}{4} a + \frac{241}{16} : -\frac{251}{16} a - \frac{4081}{64} : 1\right)$ |
| Heights | \(1.3242647311834403315159924736656325481\) | \(2.0649752367953244401137254400794604351\) |
| Torsion structure: | \(\Z/2\Z\oplus\Z/4\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(5 a + 16 : -8 a - 35 : 1\right)$ | $\left(a + 1 : -47 a - 181 : 1\right)$ |
| 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: | \( 2.5929144509875476566860852402002271316 \) | ||
| Period: | \( 22.734034070006347604148590511773734758 \) | ||
| Tamagawa product: | \( 8 \) = \(2\cdot2^{2}\) | ||
| Torsion order: | \(8\) | ||
| Leading coefficient: | \( 3.9385891983142790376412885342643247013 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a+4)\) | \(2\) | \(2\) | \(I_0^{*}\) | Additive | \(-1\) | \(4\) | \(8\) | \(0\) |
| \((3)\) | \(9\) | \(4\) | \(I_{4}\) | 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
144.1-f
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 48.a3 |
| \(\Q\) | 9408.h3 |