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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 11913.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
11913.d1 | 11913i3 | \([1, 0, 0, -52894, -4682095]\) | \(347873904937/395307\) | \(18597566080467\) | \([2]\) | \(43200\) | \(1.4596\) | |
11913.d2 | 11913i2 | \([1, 0, 0, -4159, -32776]\) | \(169112377/88209\) | \(4149870117129\) | \([2, 2]\) | \(21600\) | \(1.1130\) | |
11913.d3 | 11913i1 | \([1, 0, 0, -2354, 43395]\) | \(30664297/297\) | \(13972626657\) | \([2]\) | \(10800\) | \(0.76644\) | \(\Gamma_0(N)\)-optimal |
11913.d4 | 11913i4 | \([1, 0, 0, 15696, -251181]\) | \(9090072503/5845851\) | \(-275023210489731\) | \([2]\) | \(43200\) | \(1.4596\) |
Rank
sage: E.rank()
The elliptic curves in class 11913.d have rank \(0\).
Complex multiplication
The elliptic curves in class 11913.d do not have complex multiplication.Modular form 11913.2.a.d
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.