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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 71478.s
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
71478.s1 | 71478c4 | \([1, -1, 0, -478212, -31693132]\) | \(13060888875/7086244\) | \(6561890825567654412\) | \([2]\) | \(1368576\) | \(2.3016\) | |
71478.s2 | 71478c2 | \([1, -1, 0, -369912, -86503040]\) | \(4406910829875/7744\) | \(9836729166528\) | \([2]\) | \(456192\) | \(1.7523\) | |
71478.s3 | 71478c3 | \([1, -1, 0, -283272, 57706352]\) | \(2714704875/21296\) | \(19720182796597008\) | \([2]\) | \(684288\) | \(1.9550\) | |
71478.s4 | 71478c1 | \([1, -1, 0, -23352, -1318592]\) | \(1108717875/45056\) | \(57231878787072\) | \([2]\) | \(228096\) | \(1.4057\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 71478.s have rank \(0\).
Complex multiplication
The elliptic curves in class 71478.s do not have complex multiplication.Modular form 71478.2.a.s
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.