Base field \(\Q(\sqrt{-2}) \)
Generator \(a\), with minimal polynomial \( x^{2} + 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 0, 1]))
gp: K = nfinit(Polrev([2, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0]),K([-1,0]),K([0,0]),K([-8316,-38745]),K([3437478,-3718953])])
gp: E = ellinit([Polrev([1,0]),Polrev([-1,0]),Polrev([0,0]),Polrev([-8316,-38745]),Polrev([3437478,-3718953])], K);
magma: E := EllipticCurve([K![1,0],K![-1,0],K![0,0],K![-8316,-38745],K![3437478,-3718953]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((153a)\) | = | \((a)\cdot(-a-1)^{2}\cdot(a-1)^{2}\cdot(-2a+3)\cdot(2a+3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 46818 \) | = | \(2\cdot3^{2}\cdot3^{2}\cdot17\cdot17\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((4136899013169237a+2156459775493680)\) | = | \((a)\cdot(-a-1)^{7}\cdot(a-1)^{7}\cdot(-2a+3)^{4}\cdot(2a+3)^{16}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 38878185653643466599797326266738 \) | = | \(2\cdot3^{7}\cdot3^{7}\cdot17^{4}\cdot17^{16}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{61437923106397764191713}{291967151254001210886} a - \frac{146204680417750243067160}{48661191875666868481} \) | ||
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: | \(1\) |
Generator | $\left(-\frac{56677}{33282} a - \frac{53838098399}{12014802} : \frac{12490350017994377}{58896559404} a + \frac{1859277270754}{774954729} : 1\right)$ |
Height | \(13.650030979053254270129658757837746817\) |
Torsion structure: | \(\Z/2\Z\) |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
Torsion generator: | $\left(-36 a - \frac{573}{4} : 18 a + \frac{573}{8} : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
|
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Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 13.650030979053254270129658757837746817 \) | ||
Period: | \( 0.030625691201991347690030299437781183420 \) | ||
Tamagawa product: | \( 32 \) = \(1\cdot2^{2}\cdot2\cdot2\cdot2\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 4.7296011836923811625945918455771001461 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a)\) | \(2\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
\((-a-1)\) | \(3\) | \(4\) | \(I_{1}^{*}\) | Additive | \(-1\) | \(2\) | \(7\) | \(1\) |
\((a-1)\) | \(3\) | \(2\) | \(I_{1}^{*}\) | Additive | \(-1\) | \(2\) | \(7\) | \(1\) |
\((-2a+3)\) | \(17\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
\((2a+3)\) | \(17\) | \(2\) | \(I_{16}\) | Non-split multiplicative | \(1\) | \(1\) | \(16\) | \(16\) |
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 and 16.
Its isogeny class
46818.8-k
consists of curves linked by isogenies of
degrees dividing 16.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.