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SageMath
E = EllipticCurve("dd1")
E.isogeny_class()
Elliptic curves in class 38720.dd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
38720.dd1 | 38720bm3 | \([0, -1, 0, -20005, 1095525]\) | \(488095744/125\) | \(226759808000\) | \([2]\) | \(69120\) | \(1.1649\) | |
38720.dd2 | 38720bm4 | \([0, -1, 0, -17585, 1368017]\) | \(-20720464/15625\) | \(-453519616000000\) | \([2]\) | \(138240\) | \(1.5115\) | |
38720.dd3 | 38720bm1 | \([0, -1, 0, -645, -4123]\) | \(16384/5\) | \(9070392320\) | \([2]\) | \(23040\) | \(0.61557\) | \(\Gamma_0(N)\)-optimal |
38720.dd4 | 38720bm2 | \([0, -1, 0, 1775, -29775]\) | \(21296/25\) | \(-725631385600\) | \([2]\) | \(46080\) | \(0.96214\) |
Rank
sage: E.rank()
The elliptic curves in class 38720.dd have rank \(1\).
Complex multiplication
The elliptic curves in class 38720.dd do not have complex multiplication.Modular form 38720.2.a.dd
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.