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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 8280.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8280.r1 | 8280k4 | \([0, 0, 0, -19441587, 32994813166]\) | \(544328872410114151778/14166950625\) | \(21151143947520000\) | \([2]\) | \(196608\) | \(2.6479\) | |
8280.r2 | 8280k3 | \([0, 0, 0, -1887267, -115818626]\) | \(497927680189263938/284271240234375\) | \(424414687500000000000\) | \([2]\) | \(196608\) | \(2.6479\) | |
8280.r3 | 8280k2 | \([0, 0, 0, -1216587, 514218166]\) | \(266763091319403556/1355769140625\) | \(1012076240400000000\) | \([2, 2]\) | \(98304\) | \(2.3013\) | |
8280.r4 | 8280k1 | \([0, 0, 0, -35607, 16553194]\) | \(-26752376766544/618796614375\) | \(-115482299361120000\) | \([2]\) | \(49152\) | \(1.9547\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 8280.r have rank \(0\).
Complex multiplication
The elliptic curves in class 8280.r do not have complex multiplication.Modular form 8280.2.a.r
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.