Base field 6.6.1241125.1
Generator \(a\), with minimal polynomial \( x^{6} - 7 x^{4} - 2 x^{3} + 11 x^{2} + 7 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 7, 11, -2, -7, 0, 1]))
gp: K = nfinit(Polrev([1, 7, 11, -2, -7, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 7, 11, -2, -7, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-19,-52,-2,34,2,-5]),K([1,-3,-1,1,0,0]),K([1,1,0,0,0,0]),K([-3693,-21217,-19626,22801,5149,-4178]),K([-220717,-1338554,-1289635,1461510,337696,-269508])])
gp: E = ellinit([Polrev([-19,-52,-2,34,2,-5]),Polrev([1,-3,-1,1,0,0]),Polrev([1,1,0,0,0,0]),Polrev([-3693,-21217,-19626,22801,5149,-4178]),Polrev([-220717,-1338554,-1289635,1461510,337696,-269508])], K);
magma: E := EllipticCurve([K![-19,-52,-2,34,2,-5],K![1,-3,-1,1,0,0],K![1,1,0,0,0,0],K![-3693,-21217,-19626,22801,5149,-4178],K![-220717,-1338554,-1289635,1461510,337696,-269508]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a^4+a^3-4a^2-3a+1)\) | = | \((a^4+a^3-4a^2-3a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 41 \) | = | \(41\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-26a^5+42a^4+176a^3-166a^2-301a+21)\) | = | \((a^4+a^3-4a^2-3a+1)^{7}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 194754273881 \) | = | \(41^{7}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{300039257776916723031380}{194754273881} a^{5} - \frac{135201991016220514678646}{194754273881} a^{4} - \frac{1435403563400771691241019}{194754273881} a^{3} + \frac{343098620572538429827465}{194754273881} a^{2} + \frac{360645134275870330036563}{194754273881} a - \frac{34885214721449234564804}{194754273881} \) | ||
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: | \(0\) |
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: | \( 0 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(0\) | ||
Regulator: | \( 1 \) | ||
Period: | \( 0.55357546064756299660352397232241976964 \) | ||
Tamagawa product: | \( 1 \) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 1.19306 \) | ||
Analytic order of Ш: | \( 2401 \) (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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a^4+a^3-4a^2-3a+1)\) | \(41\) | \(1\) | \(I_{7}\) | Non-split multiplicative | \(1\) | \(1\) | \(7\) | \(7\) |
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
41.1-a
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.