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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 1682a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1682.d2 | 1682a1 | \([1, 0, 1, 4187, 173584]\) | \(13651919/29696\) | \(-17663873340416\) | \([]\) | \(3360\) | \(1.2261\) | \(\Gamma_0(N)\)-optimal |
1682.d1 | 1682a2 | \([1, 0, 1, -382673, -91764536]\) | \(-10418796526321/82044596\) | \(-48802039062823316\) | \([]\) | \(16800\) | \(2.0308\) |
Rank
sage: E.rank()
The elliptic curves in class 1682a have rank \(1\).
Complex multiplication
The elliptic curves in class 1682a do not have complex multiplication.Modular form 1682.2.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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.