Properties

Label 4.4.9909.1-21.1-a2
Base field 4.4.9909.1
Conductor norm \( 21 \)
CM no
Base change no
Q-curve no
Torsion order \( 4 \)
Rank \( 1 \)

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Base field 4.4.9909.1

Generator \(a\), with minimal polynomial \( x^{4} - 6 x^{2} - 3 x + 3 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, -3, -6, 0, 1]))
 
gp: K = nfinit(Polrev([3, -3, -6, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -3, -6, 0, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+a{x}^{2}+\left(-13750a^{3}+16600a^{2}+62447a-34158\right){x}+1313070a^{3}-1585569a^{2}-5963840a+3262240\)
sage: E = EllipticCurve([K([1,0,0,0]),K([0,1,0,0]),K([-2,0,1,0]),K([-34158,62447,16600,-13750]),K([3262240,-5963840,-1585569,1313070])])
 
gp: E = ellinit([Polrev([1,0,0,0]),Polrev([0,1,0,0]),Polrev([-2,0,1,0]),Polrev([-34158,62447,16600,-13750]),Polrev([3262240,-5963840,-1585569,1313070])], K);
 
magma: E := EllipticCurve([K![1,0,0,0],K![0,1,0,0],K![-2,0,1,0],K![-34158,62447,16600,-13750],K![3262240,-5963840,-1585569,1313070]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a-3)\) = \((-a^3+a^2+4a)\cdot(a^3-a^2-4a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 21 \) = \(3\cdot7\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-2487a^3+2531a^2+9684a-1314)\) = \((-a^3+a^2+4a)^{2}\cdot(a^3-a^2-4a+1)^{16}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 299096375126409 \) = \(3^{2}\cdot7^{16}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{3075730896093395126238965002}{99698791708803} a^{3} - \frac{3714007662409537173380188813}{99698791708803} a^{2} - \frac{4656548508714567651606714255}{33232930569601} a + \frac{2547146210989604914629326570}{33232930569601} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \(1\)
Generator $\left(\frac{35588434}{1113025} a^{3} - \frac{41646111}{1113025} a^{2} - \frac{32178412}{222605} a + \frac{88379063}{1113025} : -\frac{20256966794}{1174241375} a^{3} + \frac{19955954476}{1174241375} a^{2} + \frac{17108767582}{234848275} a - \frac{45430592508}{1174241375} : 1\right)$
Height \(4.2563373253223029300708455110830687319\)
Torsion structure: \(\Z/2\Z\oplus\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(32 a^{3} - 38 a^{2} - 145 a + \frac{319}{4} : -16 a^{3} + \frac{37}{2} a^{2} + \frac{145}{2} a - \frac{311}{8} : 1\right)$ $\left(32 a^{3} - 39 a^{2} - 146 a + 80 : -16 a^{3} + 19 a^{2} + 73 a - 39 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 4.2563373253223029300708455110830687319 \)
Period: \( 87.105030326823751242452985106402193751 \)
Tamagawa product: \( 4 \)  =  \(2\cdot2\)
Torsion order: \(4\)
Leading coefficient: \( 3.72446898122785 \)
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\((j)_{-}\)
\((-a^3+a^2+4a)\) \(3\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((a^3-a^2-4a+1)\) \(7\) \(2\) \(I_{16}\) Non-split multiplicative \(1\) \(1\) \(16\) \(16\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 21.1-a consists of curves linked by isogenies of degrees dividing 8.

Base change

This elliptic curve is not a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.