Properties

Label 4.4.6125.1-49.1-b1
Base field 4.4.6125.1
Conductor norm \( 49 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 1 \)
Rank \( 2 \)

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Base field 4.4.6125.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 9 x^{2} + 9 x + 11 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([11, 9, -9, -1, 1]))
 
gp: K = nfinit(Polrev([11, 9, -9, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, 9, -9, -1, 1]);
 

Weierstrass equation

\({y}^2+{y}={x}^{3}+\left(a^{3}+a^{2}-6a-3\right){x}^{2}+\left(30a^{3}+30a^{2}-180a-149\right){x}+102a^{3}+102a^{2}-612a-492\)
sage: E = EllipticCurve([K([0,0,0,0]),K([-3,-6,1,1]),K([1,0,0,0]),K([-149,-180,30,30]),K([-492,-612,102,102])])
 
gp: E = ellinit([Polrev([0,0,0,0]),Polrev([-3,-6,1,1]),Polrev([1,0,0,0]),Polrev([-149,-180,30,30]),Polrev([-492,-612,102,102])], K);
 
magma: E := EllipticCurve([K![0,0,0,0],K![-3,-6,1,1],K![1,0,0,0],K![-149,-180,30,30],K![-492,-612,102,102]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^3-2a^2+5a+12)\) = \((-a^3-2a^2+5a+12)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 49 \) = \(49\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-16807)\) = \((-a^3-2a^2+5a+12)^{10}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 79792266297612001 \) = \(49^{10}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{2887553024}{16807} \)
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: \(2\)
Generators $\left(-\frac{2339}{4} a^{3} - \frac{2339}{4} a^{2} + \frac{7017}{2} a + \frac{7015}{2} : \frac{128157}{4} a^{3} + 42719 a^{2} - \frac{897099}{4} a - \frac{1238855}{8} : 1\right)$ $\left(-10 a^{3} - 10 a^{2} + 60 a + 59 : 98 a^{3} + 147 a^{2} - 637 a - 564 : 1\right)$
Heights \(8.5640330262502875900809453554505545886\) \(0.77696068475826206156102008416506032866\)
Torsion structure: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 2 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(2\)
Regulator: \( 6.6539169643677947389472076178142794581 \)
Period: \( 1.0929619239614507396819991504797728308 \)
Tamagawa product: \( 2 \)
Torsion order: \(1\)
Leading coefficient: \( 2.97357804265446 \)
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\((j)_{-}\)
\((-a^3-2a^2+5a+12)\) \(49\) \(2\) \(I_{10}\) Non-split multiplicative \(1\) \(1\) \(10\) \(10\)

Galois Representations

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

prime Image of Galois Representation
\(5\) 5B.1.4[2]

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5.
Its isogeny class 49.1-b consists of curves linked by isogenies of degree 5.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following 4 elliptic curves:

Base field Curve
\(\Q\) 1225.i1
\(\Q\) 1225.a1
\(\Q(\sqrt{5}) \) a curve with conductor norm 60025 (not in the database)
\(\Q(\sqrt{5}) \) 2.2.5.1-49.1-a1