Base field \(\Q(\sqrt{165}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 41 \); class number \(2\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-41, -1, 1]))
gp: K = nfinit(Polrev([-41, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-41, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1]),K([1,-1]),K([1,0]),K([-5232,757]),K([-227110,32808])])
gp: E = ellinit([Polrev([0,1]),Polrev([1,-1]),Polrev([1,0]),Polrev([-5232,757]),Polrev([-227110,32808])], K);
magma: E := EllipticCurve([K![0,1],K![1,-1],K![1,0],K![-5232,757],K![-227110,32808]]);
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: | \((33,a+16)\) | = | \((3,a+1)\cdot(a+5)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 33 \) | = | \(3\cdot11\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((216513)\) | = | \((3,a+1)^{18}\cdot(a+5)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 46877879169 \) | = | \(3^{18}\cdot11^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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Minimal discriminant: | \((297)\) | = | \((3,a+1)^{6}\cdot(a+5)^{2}\) |
Minimal discriminant norm: | \( 88209 \) | = | \(3^{6}\cdot11^{2}\) |
j-invariant: | \( \frac{30664297}{297} \) | ||
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 | $\left(-34 a + 233 : -797 a + 5525 : 1\right)$ | $\left(\frac{217}{44} a - \frac{1607}{44} : \frac{6349}{484} a - \frac{80423}{968} : 1\right)$ |
Heights | \(0.21360872783383510091355361365922622416\) | \(2.9577010044811001071227173060498724674\) |
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);
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Torsion generator: | $\left(5 a - 37 : 16 a - 103 : 1\right)$ | |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 2 \) | ||
sage: E.rank()
magma: Rank(E);
|
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Mordell-Weil rank: | \(2\) | ||
Regulator: | \( 0.63179074888006400498566427322609661190 \) | ||
Period: | \( 8.9362528279816060089443988973145184433 \) | ||
Tamagawa product: | \( 12 \) = \(( 2 \cdot 3 )\cdot2\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 5.2743383182483119264001647528896136961 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((3,a+1)\) | \(3\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
\((a+5)\) | \(11\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
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 and 4.
Its isogeny class
33.1-d
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
Base field | Curve |
---|---|
\(\Q\) | 825.a3 |
\(\Q\) | 1089.j3 |