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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 882a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
882.a1 | 882a1 | \([1, -1, 0, -4566, 119916]\) | \(-67645179/8\) | \(-1245197016\) | \([3]\) | \(1008\) | \(0.77073\) | \(\Gamma_0(N)\)-optimal |
882.a2 | 882a2 | \([1, -1, 0, 579, 366533]\) | \(189/512\) | \(-58095911978496\) | \([]\) | \(3024\) | \(1.3200\) |
Rank
sage: E.rank()
The elliptic curves in class 882a have rank \(1\).
Complex multiplication
The elliptic curves in class 882a do not have complex multiplication.Modular form 882.2.a.a
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.