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SageMath
sage: E = EllipticCurve("265200.dz1")
sage: E.isogeny_class()
Elliptic curves in class 265200.dz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
265200.dz1 | 265200dz7 | [0, 1, 0, -2636400002408, 1647649523114107188] | [2] | 1911029760 | |
265200.dz2 | 265200dz6 | [0, 1, 0, -164775002408, 25744481864107188] | [2, 2] | 955514880 | |
265200.dz3 | 265200dz8 | [0, 1, 0, -164476210408, 25842499384539188] | [2] | 1911029760 | |
265200.dz4 | 265200dz4 | [0, 1, 0, -32549306408, 2259971694187188] | [2] | 637009920 | |
265200.dz5 | 265200dz3 | [0, 1, 0, -10317114408, 400722685291188] | [2] | 477757440 | |
265200.dz6 | 265200dz2 | [0, 1, 0, -2222906408, 28373223787188] | [2, 2] | 318504960 | |
265200.dz7 | 265200dz1 | [0, 1, 0, -838458408, -8998565524812] | [2] | 159252480 | \(\Gamma_0(N)\)-optimal |
265200.dz8 | 265200dz5 | [0, 1, 0, 5952325592, 188591420523188] | [2] | 637009920 |
Rank
sage: E.rank()
The elliptic curves in class 265200.dz have rank \(1\).
Modular form 265200.2.a.dz
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.