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SageMath
E = EllipticCurve("ff1")
E.isogeny_class()
Elliptic curves in class 259920.ff
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
259920.ff1 | 259920ff3 | \([0, 0, 0, -32283147, 70498544186]\) | \(26487576322129/44531250\) | \(6255671978217600000000\) | \([4]\) | \(17694720\) | \(3.0776\) | |
259920.ff2 | 259920ff2 | \([0, 0, 0, -2652267, 350398874]\) | \(14688124849/8122500\) | \(1141034568826890240000\) | \([2, 2]\) | \(8847360\) | \(2.7310\) | |
259920.ff3 | 259920ff1 | \([0, 0, 0, -1612587, -783476134]\) | \(3301293169/22800\) | \(3202904052847411200\) | \([2]\) | \(4423680\) | \(2.3844\) | \(\Gamma_0(N)\)-optimal |
259920.ff4 | 259920ff4 | \([0, 0, 0, 10343733, 2770254074]\) | \(871257511151/527800050\) | \(-74144426282371327795200\) | \([2]\) | \(17694720\) | \(3.0776\) |
Rank
sage: E.rank()
The elliptic curves in class 259920.ff have rank \(1\).
Complex multiplication
The elliptic curves in class 259920.ff do not have complex multiplication.Modular form 259920.2.a.ff
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.