Show commands:
SageMath
E = EllipticCurve("co1")
E.isogeny_class()
Elliptic curves in class 47040.co
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
47040.co1 | 47040bi1 | \([0, -1, 0, -47105, 1519617]\) | \(1092727/540\) | \(5712366214840320\) | \([2]\) | \(258048\) | \(1.7165\) | \(\Gamma_0(N)\)-optimal |
47040.co2 | 47040bi2 | \([0, -1, 0, 172415, 11485825]\) | \(53582633/36450\) | \(-385584719501721600\) | \([2]\) | \(516096\) | \(2.0631\) |
Rank
sage: E.rank()
The elliptic curves in class 47040.co have rank \(1\).
Complex multiplication
The elliptic curves in class 47040.co do not have complex multiplication.Modular form 47040.2.a.co
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.