Show commands:
SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 678.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
678.c1 | 678c1 | \([1, 1, 1, -148, -427]\) | \(358667682625/149962752\) | \(149962752\) | \([2]\) | \(224\) | \(0.26588\) | \(\Gamma_0(N)\)-optimal |
678.c2 | 678c2 | \([1, 1, 1, 492, -2475]\) | \(13169335133375/10723508352\) | \(-10723508352\) | \([2]\) | \(448\) | \(0.61245\) |
Rank
sage: E.rank()
The elliptic curves in class 678.c have rank \(1\).
Complex multiplication
The elliptic curves in class 678.c do not have complex multiplication.Modular form 678.2.a.c
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.