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SageMath
E = EllipticCurve("fq1")
E.isogeny_class()
Elliptic curves in class 277200.fq
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
277200.fq1 | 277200fq4 | \([0, 0, 0, -7392675, -7736606750]\) | \(957681397954009/31185\) | \(1454967360000000\) | \([2]\) | \(4718592\) | \(2.4108\) | |
277200.fq2 | 277200fq3 | \([0, 0, 0, -732675, 36333250]\) | \(932288503609/527295615\) | \(24601504213440000000\) | \([2]\) | \(4718592\) | \(2.4108\) | |
277200.fq3 | 277200fq2 | \([0, 0, 0, -462675, -120536750]\) | \(234770924809/1334025\) | \(62240270400000000\) | \([2, 2]\) | \(2359296\) | \(2.0642\) | |
277200.fq4 | 277200fq1 | \([0, 0, 0, -12675, -3986750]\) | \(-4826809/144375\) | \(-6735960000000000\) | \([2]\) | \(1179648\) | \(1.7176\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 277200.fq have rank \(0\).
Complex multiplication
The elliptic curves in class 277200.fq do not have complex multiplication.Modular form 277200.2.a.fq
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.