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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 6720.x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6720.x1 | 6720o3 | \([0, -1, 0, -1205, -14475]\) | \(189123395584/16078125\) | \(16464000000\) | \([2]\) | \(6912\) | \(0.70178\) | |
6720.x2 | 6720o1 | \([0, -1, 0, -245, 1557]\) | \(1594753024/4725\) | \(4838400\) | \([2]\) | \(2304\) | \(0.15247\) | \(\Gamma_0(N)\)-optimal |
6720.x3 | 6720o2 | \([0, -1, 0, -145, 2737]\) | \(-20720464/178605\) | \(-2926264320\) | \([2]\) | \(4608\) | \(0.49904\) | |
6720.x4 | 6720o4 | \([0, -1, 0, 1295, -68975]\) | \(14647977776/132355125\) | \(-2168506368000\) | \([2]\) | \(13824\) | \(1.0483\) |
Rank
sage: E.rank()
The elliptic curves in class 6720.x have rank \(1\).
Complex multiplication
The elliptic curves in class 6720.x do not have complex multiplication.Modular form 6720.2.a.x
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.