Show commands:
SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 41405f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
41405.m1 | 41405f1 | \([1, 1, 0, -8453, -67192]\) | \(117649/65\) | \(36911501382665\) | \([2]\) | \(96768\) | \(1.2938\) | \(\Gamma_0(N)\)-optimal |
41405.m2 | 41405f2 | \([1, 1, 0, 32952, -489523]\) | \(6967871/4225\) | \(-2399247589873225\) | \([2]\) | \(193536\) | \(1.6404\) |
Rank
sage: E.rank()
The elliptic curves in class 41405f have rank \(0\).
Complex multiplication
The elliptic curves in class 41405f do not have complex multiplication.Modular form 41405.2.a.f
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.