Base field 5.5.24217.1
Generator \(a\), with minimal polynomial \( x^{5} - 5 x^{3} - x^{2} + 3 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 3, -1, -5, 0, 1]))
gp: K = nfinit(Polrev([1, 3, -1, -5, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, -1, -5, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([3,4,-9,-1,2]),K([-4,-3,5,1,-1]),K([-1,1,1,0,0]),K([-113,-93,175,29,-39]),K([-280,-215,221,54,-50])])
gp: E = ellinit([Polrev([3,4,-9,-1,2]),Polrev([-4,-3,5,1,-1]),Polrev([-1,1,1,0,0]),Polrev([-113,-93,175,29,-39]),Polrev([-280,-215,221,54,-50])], K);
magma: E := EllipticCurve([K![3,4,-9,-1,2],K![-4,-3,5,1,-1],K![-1,1,1,0,0],K![-113,-93,175,29,-39],K![-280,-215,221,54,-50]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-3a^4+2a^3+14a^2-7a-7)\) | = | \((-3a^4+2a^3+14a^2-7a-7)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 83 \) | = | \(83\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((202520a^4-107605a^3-982884a^2+464813a+453634)\) | = | \((-3a^4+2a^3+14a^2-7a-7)^{14}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -736365263311636486061599129 \) | = | \(-83^{14}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{1059382013228619221572682927247070754}{736365263311636486061599129} a^{4} + \frac{2076414215077956021875977058309950102}{736365263311636486061599129} a^{3} + \frac{1227082340350165006565171062239335310}{736365263311636486061599129} a^{2} - \frac{1345722272615370392506917877231739167}{736365263311636486061599129} a - \frac{540496562267508012147668315035311817}{736365263311636486061599129} \) | ||
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 | $\left(\frac{519}{25} a^{4} - \frac{132}{25} a^{3} - \frac{2149}{25} a^{2} + \frac{503}{25} a + \frac{1348}{25} : \frac{15391}{125} a^{4} + \frac{3202}{125} a^{3} - \frac{64936}{125} a^{2} + \frac{1417}{125} a + \frac{33297}{125} : 1\right)$ |
Height | \(0.90166935191560727332295204418535167903\) |
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: | \( 0.90166935191560727332295204418535167903 \) | ||
Period: | \( 0.68528975793727198880701026472024965958 \) | ||
Tamagawa product: | \( 2 \) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 1.94561747 \) | ||
Analytic order of Ш: | \( 49 \) (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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-3a^4+2a^3+14a^2-7a-7)\) | \(83\) | \(2\) | \(I_{14}\) | Non-split multiplicative | \(1\) | \(1\) | \(14\) | \(14\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(7\) | 7B.1.3 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
7.
Its isogeny class
83.1-c
consists of curves linked by isogenies of
degree 7.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.