Base field \(\Q(\sqrt{3}, \sqrt{5})\)
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 8, -7, -2, 1]))
gp: K = nfinit(Polrev([1, 8, -7, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 8, -7, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0,0,0]),K([1,0,0,0]),K([1,0,0,0]),K([-80,0,0,0]),K([242,0,0,0])])
gp: E = ellinit([Polrev([1,0,0,0]),Polrev([1,0,0,0]),Polrev([1,0,0,0]),Polrev([-80,0,0,0]),Polrev([242,0,0,0])], K);
magma: E := EllipticCurve([K![1,0,0,0],K![1,0,0,0],K![1,0,0,0],K![-80,0,0,0],K![242,0,0,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^2+a+4)\) | = | \((-2/7a^3+3/7a^2+19/7a-10/7)\cdot(4/7a^3-6/7a^2-24/7a+13/7)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 225 \) | = | \(9\cdot25\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((15)\) | = | \((-2/7a^3+3/7a^2+19/7a-10/7)^{2}\cdot(4/7a^3-6/7a^2-24/7a+13/7)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 50625 \) | = | \(9^{2}\cdot25^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{56667352321}{15} \) | ||
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 | $\left(\frac{75}{7} a^{3} - \frac{158}{7} a^{2} - \frac{1019}{14} a + \frac{1387}{14} : \frac{1557}{14} a^{3} - \frac{6575}{28} a^{2} - \frac{21085}{28} a + \frac{27267}{28} : 1\right)$ | $\left(-\frac{75}{7} a^{3} + \frac{67}{7} a^{2} + \frac{1201}{14} a + \frac{101}{7} : \frac{1707}{14} a^{3} - \frac{3035}{28} a^{2} - \frac{27295}{28} a - \frac{3153}{28} : 1\right)$ |
| Heights | \(0.92532204965083266630722942221207397146\) | \(0.92532204965083266630722942221207397146\) |
| Torsion structure: | \(\Z/2\Z\oplus\Z/8\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(2 a^{2} - 2 a - 11 : -a^{2} + a + 5 : 1\right)$ | $\left(\frac{4}{7} a^{3} - \frac{6}{7} a^{2} - \frac{24}{7} a + \frac{62}{7} : \frac{16}{7} a^{3} - \frac{24}{7} a^{2} - \frac{96}{7} a + \frac{94}{7} : 1\right)$ |
| 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: | \( 0.78050477292764989331613120471349248443 \) | ||
| Period: | \( 985.14515725135648801732976819257502568 \) | ||
| Tamagawa product: | \( 4 \) = \(2\cdot2\) | ||
| Torsion order: | \(16\) | ||
| Leading coefficient: | \( 3.20379373858852 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-2/7a^3+3/7a^2+19/7a-10/7)\) | \(9\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
| \((4/7a^3-6/7a^2-24/7a+13/7)\) | \(25\) | \(2\) | \(I_{2}\) | 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, 4, 8, 16 and 32.
Its isogeny class
225.1-e
consists of curves linked by isogenies of
degrees dividing 64.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 10 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 15.a4 |
| \(\Q\) | 75.b4 |
| \(\Q\) | 720.c4 |
| \(\Q\) | 3600.u4 |
| \(\Q(\sqrt{3}) \) | 2.2.12.1-75.1-b7 |
| \(\Q(\sqrt{3}) \) | 2.2.12.1-1875.1-d7 |
| \(\Q(\sqrt{5}) \) | 2.2.5.1-45.1-a8 |
| \(\Q(\sqrt{5}) \) | a curve with conductor norm 103680 (not in the database) |
| \(\Q(\sqrt{15}) \) | 2.2.60.1-15.1-d8 |
| \(\Q(\sqrt{15}) \) | 2.2.60.1-15.1-c8 |