Show commands:
SageMath
E = EllipticCurve("hw1")
E.isogeny_class()
Elliptic curves in class 478800.hw
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
478800.hw1 | 478800hw3 | \([0, 0, 0, -502891875, 4340702513250]\) | \(11165451838341046875/572244736\) | \(720863560876032000000\) | \([2]\) | \(95551488\) | \(3.4752\) | \(\Gamma_0(N)\)-optimal* |
478800.hw2 | 478800hw4 | \([0, 0, 0, -502027875, 4356360785250]\) | \(-11108001800138902875/79947274872976\) | \(-100710541524786342912000000\) | \([2]\) | \(191102976\) | \(3.8218\) | |
478800.hw3 | 478800hw1 | \([0, 0, 0, -6763875, 4825905250]\) | \(19804628171203875/5638671302656\) | \(9743624010989568000000\) | \([2]\) | \(31850496\) | \(2.9259\) | \(\Gamma_0(N)\)-optimal* |
478800.hw4 | 478800hw2 | \([0, 0, 0, 17812125, 31834929250]\) | \(361682234074684125/462672528510976\) | \(-799498129266966528000000\) | \([2]\) | \(63700992\) | \(3.2725\) |
Rank
sage: E.rank()
The elliptic curves in class 478800.hw have rank \(1\).
Complex multiplication
The elliptic curves in class 478800.hw do not have complex multiplication.Modular form 478800.2.a.hw
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.