Properties

Base field 3.3.733.1
Label 3.3.733.1-10.1-a6
Conductor \((10,-a - 2)\)
Conductor norm \( 10 \)
CM no
base-change no
Q-curve no
Torsion order \( 2 \)
Rank not available

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Base field 3.3.733.1

Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 7 x + 8 \); class number \(1\).

sage: x = polygen(QQ); K.<a> = NumberField(x^3 - x^2 - 7*x + 8)
 
gp: K = nfinit(a^3 - a^2 - 7*a + 8);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, -7, -1, 1]);
 

Weierstrass equation

\( y^2 + \left(a + 1\right) x y = x^{3} + \left(225468796079696669707481 a^{2} + 342306432946871581914275 a - 716285958340025181167230\right) x - 102770037091536501869102455441620105963 a^{2} - 156025336642090576588977058556709893561 a + 326487459846688471086063418466138671196 \)
sage: E = EllipticCurve(K, [a + 1, 0, 0, 225468796079696669707481*a^2 + 342306432946871581914275*a - 716285958340025181167230, -102770037091536501869102455441620105963*a^2 - 156025336642090576588977058556709893561*a + 326487459846688471086063418466138671196])
 
gp: E = ellinit([a + 1, 0, 0, 225468796079696669707481*a^2 + 342306432946871581914275*a - 716285958340025181167230, -102770037091536501869102455441620105963*a^2 - 156025336642090576588977058556709893561*a + 326487459846688471086063418466138671196],K)
 
magma: E := ChangeRing(EllipticCurve([a + 1, 0, 0, 225468796079696669707481*a^2 + 342306432946871581914275*a - 716285958340025181167230, -102770037091536501869102455441620105963*a^2 - 156025336642090576588977058556709893561*a + 326487459846688471086063418466138671196]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((10,-a - 2)\) = \( \left(a^{2} - 6\right) \cdot \left(-a + 3\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
\(N(\mathfrak{N}) \) = \( 10 \) = \( 2 \cdot 5 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
\(\mathfrak{D}\) = \((1280,a + 1032,a^{2} + 1216)\) = \( \left(a^{2} - 6\right)^{8} \cdot \left(-a + 3\right) \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\mathfrak{D})\) = \( 1280 \) = \( 2^{8} \cdot 5 \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(j\) = \( -\frac{3912802297187}{1280} a^{2} - \frac{697237971589}{1280} a + \frac{26568133564637}{1280} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank not available.

sage: E.rank()
 
magma: Rank(E);
 

Regulator: not available

sage: gens = E.gens(); gens
 
magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;
 
sage: E.regulator_of_points(gens)
 
magma: Regulator(gens);
 

Torsion subgroup

Structure: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Generator: $\left(1273164004570 a^{2} + \frac{7731663743069}{4} a - 4044681636634 : -\frac{17916975779629}{8} a^{2} - \frac{27201529324493}{8} a + 7114996836597 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

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\((j)_{-}\)
\( \left(a^{2} - 6\right) \) \(2\) \(8\) \(I_{8}\) Split multiplicative \(-1\) \(1\) \(8\) \(8\)
\( \left(-a + 3\right) \) \(5\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

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 10.1-a consists of curves linked by isogenies of degrees dividing 8.

Base change

This curve is not the base-change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.