Properties

Label 2.2.21.1-175.1-b3
Base field \(\Q(\sqrt{21}) \)
Conductor \((5a+15)\)
Conductor norm \( 175 \)
CM no
Base change yes: 245.c3,315.b3
Q-curve yes
Torsion order \( 3 \)
Rank \( 1 \)

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Show commands: Magma / Pari/GP / SageMath

Base field \(\Q(\sqrt{21}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 5 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-5, -1, 1]))
 
gp: K = nfinit(Pol(Vecrev([-5, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, -1, 1]);
 

Weierstrass equation

\({y}^2+{y}={x}^{3}+a{x}^{2}+\left(-43a+123\right){x}-14a+40\)
sage: E = EllipticCurve([K([0,0]),K([0,1]),K([1,0]),K([123,-43]),K([40,-14])])
 
gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([0,1])),Pol(Vecrev([1,0])),Pol(Vecrev([123,-43])),Pol(Vecrev([40,-14]))], K);
 
magma: E := EllipticCurve([K![0,0],K![0,1],K![1,0],K![123,-43],K![40,-14]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((5a+15)\) = \((-a)\cdot(-a+1)\cdot(a+3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 175 \) = \(5\cdot5\cdot7\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-42875)\) = \((-a)^{3}\cdot(-a+1)^{3}\cdot(a+3)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1838265625 \) = \(5^{3}\cdot5^{3}\cdot7^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{71991296}{42875} \)
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 $\left(-5 a + 13 : -26 a + 72 : 1\right)$
Height \(0.122459267388325\)
Torsion structure: \(\Z/3\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-\frac{4}{3} a + 3 : -\frac{70}{9} a + \frac{188}{9} : 1\right)$
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: \( 0.122459267388325 \)
Period: \( 4.86222025956892 \)
Tamagawa product: \( 54 \)  =  \(3\cdot3\cdot( 2 \cdot 3 )\)
Torsion order: \(3\)
Leading coefficient: \( 1.55918584787907 \)
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\((j)_{-}\)
\((-a)\) \(5\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\((-a+1)\) \(5\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\((a+3)\) \(7\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3Cs.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 175.1-b consists of curves linked by isogenies of degrees dividing 9.

Base change

This curve is the base change of 245.c3, 315.b3, defined over \(\Q\), so it is also a \(\Q\)-curve.