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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 1005a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1005.b1 | 1005a1 | \([1, 1, 0, -297, -1944]\) | \(2912566550041/254390625\) | \(254390625\) | \([2]\) | \(360\) | \(0.35342\) | \(\Gamma_0(N)\)-optimal |
1005.b2 | 1005a2 | \([1, 1, 0, 328, -8319]\) | \(3883959939959/33133870125\) | \(-33133870125\) | \([2]\) | \(720\) | \(0.69999\) |
Rank
sage: E.rank()
The elliptic curves in class 1005a have rank \(0\).
Complex multiplication
The elliptic curves in class 1005a do not have complex multiplication.Modular form 1005.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.