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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 1440.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1440.f1 | 1440d3 | \([0, 0, 0, -4323, 109402]\) | \(23937672968/45\) | \(16796160\) | \([2]\) | \(1024\) | \(0.64163\) | |
1440.f2 | 1440d2 | \([0, 0, 0, -723, -5258]\) | \(111980168/32805\) | \(12244400640\) | \([2]\) | \(1024\) | \(0.64163\) | |
1440.f3 | 1440d1 | \([0, 0, 0, -273, 1672]\) | \(48228544/2025\) | \(94478400\) | \([2, 2]\) | \(512\) | \(0.29505\) | \(\Gamma_0(N)\)-optimal |
1440.f4 | 1440d4 | \([0, 0, 0, 132, 6208]\) | \(85184/5625\) | \(-16796160000\) | \([2]\) | \(1024\) | \(0.64163\) |
Rank
sage: E.rank()
The elliptic curves in class 1440.f have rank \(0\).
Complex multiplication
The elliptic curves in class 1440.f do not have complex multiplication.Modular form 1440.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.