Show commands:
SageMath
E = EllipticCurve("hf1")
E.isogeny_class()
Elliptic curves in class 221760hf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
221760.db1 | 221760hf1 | \([0, 0, 0, -1339308, -596504368]\) | \(37537160298467283/5519360000\) | \(39065411911680000\) | \([2]\) | \(2752512\) | \(2.1979\) | \(\Gamma_0(N)\)-optimal |
221760.db2 | 221760hf2 | \([0, 0, 0, -1216428, -710389552]\) | \(-28124139978713043/14526050000000\) | \(-102813754982400000000\) | \([2]\) | \(5505024\) | \(2.5445\) |
Rank
sage: E.rank()
The elliptic curves in class 221760hf have rank \(1\).
Complex multiplication
The elliptic curves in class 221760hf do not have complex multiplication.Modular form 221760.2.a.hf
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.