Base field \(\Q(\sqrt{-51}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 13 \); class number \(2\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([13, -1, 1]))
gp: K = nfinit(Polrev([13, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1]),K([1,-1]),K([1,0]),K([-219,43]),K([1107,-327])])
gp: E = ellinit([Polrev([0,1]),Polrev([1,-1]),Polrev([1,0]),Polrev([-219,43]),Polrev([1107,-327])], K);
magma: E := EllipticCurve([K![0,1],K![1,-1],K![1,0],K![-219,43],K![1107,-327]]);
This is not a global minimal model: it is minimal at all primes except \((3,a+1)\). No global minimal model exists.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((15,3a+3)\) | = | \((3,a+1)^{2}\cdot(5,a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 45 \) | = | \(3^{2}\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-67493007a+123825753)\) | = | \((3,a+1)^{25}\cdot(5,a+1)^{7}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 66194422612734375 \) | = | \(3^{25}\cdot5^{7}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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Minimal discriminant: | \((92583a-169857)\) | = | \((3,a+1)^{13}\cdot(5,a+1)^{7}\) |
Minimal discriminant norm: | \( 124556484375 \) | = | \(3^{13}\cdot5^{7}\) |
j-invariant: | \( -\frac{7882339931}{6328125} a + \frac{34505684782}{6328125} \) | ||
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 | $\left(11 : -a + 4 : 1\right)$ |
Height | \(0.24706161659105216266330749637646001684\) |
Torsion structure: | trivial |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
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BSD invariants
Analytic rank: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 0.24706161659105216266330749637646001684 \) | ||
Period: | \( 1.5241049719395121891071563899854996905 \) | ||
Tamagawa product: | \( 14 \) = \(2\cdot7\) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 2.9527256558566931437274600640928852133 \) | ||
Analytic order of Ш: | \( 1 \) (rounded) |
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
Primes of good reduction for the curve but which divide the
discriminant of the model above (if any) are included.
prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((3,a+1)\) | \(3\) | \(2\) | \(I_{7}^{*}\) | Additive | \(-1\) | \(2\) | \(13\) | \(7\) |
\((5,a+1)\) | \(5\) | \(7\) | \(I_{7}\) | Split multiplicative | \(-1\) | \(1\) | \(7\) | \(7\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 45.1-b consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.