Show commands:
SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 312325o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
312325.o2 | 312325o1 | \([0, -1, 1, -80083, -8076182]\) | \(163840/13\) | \(4506854630078125\) | \([]\) | \(1814400\) | \(1.7473\) | \(\Gamma_0(N)\)-optimal |
312325.o1 | 312325o2 | \([0, -1, 1, -1281333, 557111943]\) | \(671088640/2197\) | \(761658432483203125\) | \([]\) | \(5443200\) | \(2.2966\) |
Rank
sage: E.rank()
The elliptic curves in class 312325o have rank \(1\).
Complex multiplication
The elliptic curves in class 312325o do not have complex multiplication.Modular form 312325.2.a.o
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.