Show commands:
SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 346710.cm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
346710.cm1 | 346710cm2 | \([1, 0, 0, -850830470, -9551623712610]\) | \(68116577000481061136745854064481/7021805693801784439900710\) | \(7021805693801784439900710\) | \([]\) | \(171028032\) | \(3.8003\) | |
346710.cm2 | 346710cm1 | \([1, 0, 0, -19796420, 33899552400]\) | \(857992894283086952856137281/29735971811910000000\) | \(29735971811910000000\) | \([7]\) | \(24432576\) | \(2.8273\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 346710.cm have rank \(0\).
Complex multiplication
The elliptic curves in class 346710.cm do not have complex multiplication.Modular form 346710.2.a.cm
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.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.