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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 32368.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 32368.f1 | 32368bd6 | \([0, 1, 0, -12625928, 17263833076]\) | \(2251439055699625/25088\) | \(2480387404070912\) | \([2]\) | \(663552\) | \(2.5229\) | |
| 32368.f2 | 32368bd5 | \([0, 1, 0, -788488, 270004212]\) | \(-548347731625/1835008\) | \(-181422621554900992\) | \([2]\) | \(331776\) | \(2.1763\) | |
| 32368.f3 | 32368bd4 | \([0, 1, 0, -164248, 20946324]\) | \(4956477625/941192\) | \(93053283705847808\) | \([2]\) | \(221184\) | \(1.9735\) | |
| 32368.f4 | 32368bd2 | \([0, 1, 0, -48648, -4143500]\) | \(128787625/98\) | \(9689013297152\) | \([2]\) | \(73728\) | \(1.4242\) | |
| 32368.f5 | 32368bd1 | \([0, 1, 0, -2408, -92876]\) | \(-15625/28\) | \(-2768289513472\) | \([2]\) | \(36864\) | \(1.0777\) | \(\Gamma_0(N)\)-optimal |
| 32368.f6 | 32368bd3 | \([0, 1, 0, 20712, 1932436]\) | \(9938375/21952\) | \(-2170338978562048\) | \([2]\) | \(110592\) | \(1.6270\) |
Rank
sage: E.rank()
The elliptic curves in class 32368.f have rank \(1\).
Complex multiplication
The elliptic curves in class 32368.f do not have complex multiplication.Modular form 32368.2.a.f
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
