Show commands:
SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 481650x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
481650.x3 | 481650x1 | \([1, 1, 0, -21203250, -37588393500]\) | \(13978188933715369/3853200\) | \(290604069356250000\) | \([2]\) | \(24772608\) | \(2.7205\) | \(\Gamma_0(N)\)-optimal* |
481650.x2 | 481650x2 | \([1, 1, 0, -21287750, -37273800000]\) | \(14145975058083049/231986722500\) | \(17496181250679726562500\) | \([2, 2]\) | \(49545216\) | \(3.0671\) | \(\Gamma_0(N)\)-optimal* |
481650.x1 | 481650x3 | \([1, 1, 0, -42708500, 50272805250]\) | \(114231674639984329/51619333593750\) | \(3893072874442419433593750\) | \([2]\) | \(99090432\) | \(3.4136\) | \(\Gamma_0(N)\)-optimal* |
481650.x4 | 481650x4 | \([1, 1, 0, -1219000, -104684731250]\) | \(-2656166199049/62770478980950\) | \(-4734079888743129508593750\) | \([2]\) | \(99090432\) | \(3.4136\) |
Rank
sage: E.rank()
The elliptic curves in class 481650x have rank \(0\).
Complex multiplication
The elliptic curves in class 481650x do not have complex multiplication.Modular form 481650.2.a.x
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.