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SageMath
E = EllipticCurve("ef1")
E.isogeny_class()
Elliptic curves in class 333270.ef
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
333270.ef1 | 333270ef2 | \([1, -1, 1, -3511337, -3646389]\) | \(44365623586201/25674468750\) | \(2770741505580538218750\) | \([2]\) | \(19464192\) | \(2.8039\) | |
333270.ef2 | 333270ef1 | \([1, -1, 1, -2416307, -1440763761]\) | \(14457238157881/49990500\) | \(5394882931500730500\) | \([2]\) | \(9732096\) | \(2.4573\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 333270.ef have rank \(1\).
Complex multiplication
The elliptic curves in class 333270.ef do not have complex multiplication.Modular form 333270.2.a.ef
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.