Base field \(\Q(\sqrt{22}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 22 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-22, 0, 1]))
gp: K = nfinit(Polrev([-22, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-22, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1]),K([0,0]),K([0,0]),K([-7457690,1589987]),K([11072187325,-2360598268])])
gp: E = ellinit([Polrev([0,1]),Polrev([0,0]),Polrev([0,0]),Polrev([-7457690,1589987]),Polrev([11072187325,-2360598268])], K);
magma: E := EllipticCurve([K![0,1],K![0,0],K![0,0],K![-7457690,1589987],K![11072187325,-2360598268]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((12)\) | = | \((-3a-14)^{4}\cdot(-a+5)\cdot(a+5)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 144 \) | = | \(2^{4}\cdot3\cdot3\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((288)\) | = | \((-3a-14)^{10}\cdot(-a+5)^{2}\cdot(a+5)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 82944 \) | = | \(2^{10}\cdot3^{2}\cdot3^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{3065617154}{9} \) | ||
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 | $\left(-162 a + 758 : 5474 a - 25671 : 1\right)$ |
Height | \(1.2510858530685689475604901713816412971\) |
Torsion structure: | \(\Z/4\Z\) |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
Torsion generator: | $\left(-259 a + 1213 : -2375 a + 11144 : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
|
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Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 1.2510858530685689475604901713816412971 \) | ||
Period: | \( 22.734034070006347604148590511773734758 \) | ||
Tamagawa product: | \( 16 \) = \(2^{2}\cdot2\cdot2\) | ||
Torsion order: | \(4\) | ||
Leading coefficient: | \( 6.0639034713704160065218406280981133828 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-3a-14)\) | \(2\) | \(4\) | \(I_{2}^{*}\) | Additive | \(-1\) | \(4\) | \(10\) | \(0\) |
\((-a+5)\) | \(3\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
\((a+5)\) | \(3\) | \(2\) | \(I_{2}\) | 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, 4 and 8.
Its isogeny class
144.1-j
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
Base field | Curve |
---|---|
\(\Q\) | 48.a1 |
\(\Q\) | 23232.do1 |