Base field 4.4.6125.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 9 x^{2} + 9 x + 11 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([11, 9, -9, -1, 1]))
gp: K = nfinit(Polrev([11, 9, -9, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, 9, -9, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,0,0,0]),K([-3,-6,1,1]),K([1,0,0,0]),K([-149,-180,30,30]),K([-492,-612,102,102])])
gp: E = ellinit([Polrev([0,0,0,0]),Polrev([-3,-6,1,1]),Polrev([1,0,0,0]),Polrev([-149,-180,30,30]),Polrev([-492,-612,102,102])], K);
magma: E := EllipticCurve([K![0,0,0,0],K![-3,-6,1,1],K![1,0,0,0],K![-149,-180,30,30],K![-492,-612,102,102]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-a^3-2a^2+5a+12)\) | = | \((-a^3-2a^2+5a+12)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 49 \) | = | \(49\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-16807)\) | = | \((-a^3-2a^2+5a+12)^{10}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 79792266297612001 \) | = | \(49^{10}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{2887553024}{16807} \) | ||
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 | $\left(-\frac{2339}{4} a^{3} - \frac{2339}{4} a^{2} + \frac{7017}{2} a + \frac{7015}{2} : \frac{128157}{4} a^{3} + 42719 a^{2} - \frac{897099}{4} a - \frac{1238855}{8} : 1\right)$ | $\left(-10 a^{3} - 10 a^{2} + 60 a + 59 : 98 a^{3} + 147 a^{2} - 637 a - 564 : 1\right)$ |
Heights | \(8.5640330262502875900809453554505545886\) | \(0.77696068475826206156102008416506032866\) |
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: | \( 2 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(2\) | ||
Regulator: | \( 6.6539169643677947389472076178142794581 \) | ||
Period: | \( 1.0929619239614507396819991504797728308 \) | ||
Tamagawa product: | \( 2 \) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 2.97357804265446 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a^3-2a^2+5a+12)\) | \(49\) | \(2\) | \(I_{10}\) | Non-split multiplicative | \(1\) | \(1\) | \(10\) | \(10\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(5\) | 5B.1.4[2] |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5.
Its isogeny class
49.1-b
consists of curves linked by isogenies of
degree 5.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 4 elliptic curves:
Base field | Curve |
---|---|
\(\Q\) | 1225.i1 |
\(\Q\) | 1225.a1 |
\(\Q(\sqrt{5}) \) | a curve with conductor norm 60025 (not in the database) |
\(\Q(\sqrt{5}) \) | 2.2.5.1-49.1-a1 |