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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 486720.s
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
486720.s1 | 486720s3 | \([0, 0, 0, -1384233708, 19816551254032]\) | \(2543984126301795848/909361981125\) | \(104851432982911172972544000\) | \([2]\) | \(264241152\) | \(3.9621\) | \(\Gamma_0(N)\)-optimal* |
486720.s2 | 486720s4 | \([0, 0, 0, -714993708, -7207366029968]\) | \(350584567631475848/8259273550125\) | \(952312374063748366995456000\) | \([2]\) | \(264241152\) | \(3.9621\) | |
486720.s3 | 486720s2 | \([0, 0, 0, -98988708, 214508612032]\) | \(7442744143086784/2927948765625\) | \(42199865145633259584000000\) | \([2, 2]\) | \(132120576\) | \(3.6156\) | \(\Gamma_0(N)\)-optimal* |
486720.s4 | 486720s1 | \([0, 0, 0, 19839417, 24193487032]\) | \(3834800837445824/3342041015625\) | \(-752626302255140625000000\) | \([2]\) | \(66060288\) | \(3.2690\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 486720.s have rank \(1\).
Complex multiplication
The elliptic curves in class 486720.s do not have complex multiplication.Modular form 486720.2.a.s
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.