Show commands:
SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 9408cm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
9408.cv2 | 9408cm1 | \([0, 1, 0, -457, -4243]\) | \(-28672/3\) | \(-1106841792\) | \([]\) | \(3360\) | \(0.47444\) | \(\Gamma_0(N)\)-optimal |
9408.cv1 | 9408cm2 | \([0, 1, 0, -178817, 29068437]\) | \(-1713910976512/1594323\) | \(-588221108782272\) | \([]\) | \(43680\) | \(1.7569\) |
Rank
sage: E.rank()
The elliptic curves in class 9408cm have rank \(1\).
Complex multiplication
The elliptic curves in class 9408cm do not have complex multiplication.Modular form 9408.2.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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 13 \\ 13 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.