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SageMath
E = EllipticCurve("er1")
E.isogeny_class()
Elliptic curves in class 28224.er
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
28224.er1 | 28224cc2 | \([0, 0, 0, -2604, 50960]\) | \(238328\) | \(8193540096\) | \([2]\) | \(24576\) | \(0.75684\) | |
28224.er2 | 28224cc1 | \([0, 0, 0, -84, 1568]\) | \(-64\) | \(-1024192512\) | \([2]\) | \(12288\) | \(0.41027\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 28224.er have rank \(1\).
Complex multiplication
The elliptic curves in class 28224.er do not have complex multiplication.Modular form 28224.2.a.er
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.