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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 3570n
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3570.p3 | 3570n1 | \([1, 0, 1, -17998, 928256]\) | \(-644706081631626841/347004000000\) | \(-347004000000\) | \([6]\) | \(9216\) | \(1.1636\) | \(\Gamma_0(N)\)-optimal |
3570.p2 | 3570n2 | \([1, 0, 1, -287998, 59464256]\) | \(2641739317048851306841/764694000\) | \(764694000\) | \([6]\) | \(18432\) | \(1.5101\) | |
3570.p4 | 3570n3 | \([1, 0, 1, 14627, 3777656]\) | \(346124368852751159/6361262220902400\) | \(-6361262220902400\) | \([2]\) | \(27648\) | \(1.7129\) | |
3570.p1 | 3570n4 | \([1, 0, 1, -292573, 57476216]\) | \(2769646315294225853641/174474906948464640\) | \(174474906948464640\) | \([2]\) | \(55296\) | \(2.0594\) |
Rank
sage: E.rank()
The elliptic curves in class 3570n have rank \(0\).
Complex multiplication
The elliptic curves in class 3570n do not have complex multiplication.Modular form 3570.2.a.n
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.