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([0,0,0,0]),K([0,0,0,0])])
gp: E = ellinit([Polrev([1,0,0,0]),Polrev([1,0,0,0]),Polrev([1,0,0,0]),Polrev([0,0,0,0]),Polrev([0,0,0,0])], K);
magma: E := EllipticCurve([K![1,0,0,0],K![1,0,0,0],K![1,0,0,0],K![0,0,0,0],K![0,0,0,0]]);
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);
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Conductor norm: | \( 225 \) | = | \(9\cdot25\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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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);
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Discriminant norm: | \( 50625 \) | = | \(9^{2}\cdot25^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{1}{15} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
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Endomorphism ring: | \(\Z\) | ||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $\mathrm{SU}(2)$ |
Mordell-Weil group
Rank: | \(2\) | |
Generators | $\left(1 : -\frac{2}{7} a^{3} + \frac{3}{7} a^{2} + \frac{19}{7} a - \frac{17}{7} : 1\right)$ | $\left(-\frac{3}{14} a^{3} + \frac{9}{28} a^{2} + \frac{9}{7} a - \frac{23}{28} : \frac{11}{56} a^{3} - \frac{3}{28} a^{2} - \frac{47}{28} a - \frac{11}{28} : 1\right)$ |
Heights | \(0.32507821099828153138705828574587282951\) | \(0.92532204965083266630722942221207397146\) |
Torsion structure: | \(\Z/8\Z\) | |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
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Torsion generator: | $\left(\frac{2}{7} a^{3} - \frac{3}{7} a^{2} - \frac{12}{7} a + \frac{10}{7} : \frac{2}{7} a^{3} - \frac{3}{7} a^{2} - \frac{12}{7} a + \frac{10}{7} : 1\right)$ | |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
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BSD invariants
Analytic rank: | \( 2 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(2\) | ||
Regulator: | \( 0.19512619323191247332903280117837312110 \) | ||
Period: | \( 985.14515725135648801732976819257502568 \) | ||
Tamagawa product: | \( 4 \) = \(2\cdot2\) | ||
Torsion order: | \(8\) | ||
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 |
---|---|
\(2\) | 2B |
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.a7 |
\(\Q\) | 75.b7 |
\(\Q\) | 720.c7 |
\(\Q\) | 3600.u7 |
\(\Q(\sqrt{3}) \) | 2.2.12.1-75.1-b2 |
\(\Q(\sqrt{3}) \) | 2.2.12.1-1875.1-d2 |
\(\Q(\sqrt{5}) \) | 2.2.5.1-45.1-a3 |
\(\Q(\sqrt{5}) \) | a curve with conductor norm 103680 (not in the database) |
\(\Q(\sqrt{15}) \) | 2.2.60.1-15.1-c3 |
\(\Q(\sqrt{15}) \) | 2.2.60.1-15.1-d3 |