Show commands:
SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 47040.cm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
47040.cm1 | 47040ff1 | \([0, -1, 0, -5945, 161337]\) | \(140608/15\) | \(2479325614080\) | \([2]\) | \(71680\) | \(1.1126\) | \(\Gamma_0(N)\)-optimal |
47040.cm2 | 47040ff2 | \([0, -1, 0, 7775, 784225]\) | \(39304/225\) | \(-297519073689600\) | \([2]\) | \(143360\) | \(1.4592\) |
Rank
sage: E.rank()
The elliptic curves in class 47040.cm have rank \(0\).
Complex multiplication
The elliptic curves in class 47040.cm do not have complex multiplication.Modular form 47040.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.