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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 81120.x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
81120.x1 | 81120bi4 | \([0, -1, 0, -13745, -615135]\) | \(14526784/15\) | \(296559144960\) | \([2]\) | \(147456\) | \(1.1192\) | |
81120.x2 | 81120bi3 | \([0, -1, 0, -9520, 357460]\) | \(38614472/405\) | \(1000887114240\) | \([2]\) | \(147456\) | \(1.1192\) | |
81120.x3 | 81120bi1 | \([0, -1, 0, -1070, -4200]\) | \(438976/225\) | \(69506049600\) | \([2, 2]\) | \(73728\) | \(0.77263\) | \(\Gamma_0(N)\)-optimal |
81120.x4 | 81120bi2 | \([0, -1, 0, 4000, -36648]\) | \(2863288/1875\) | \(-4633736640000\) | \([2]\) | \(147456\) | \(1.1192\) |
Rank
sage: E.rank()
The elliptic curves in class 81120.x have rank \(1\).
Complex multiplication
The elliptic curves in class 81120.x do not have complex multiplication.Modular form 81120.2.a.x
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.