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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 5040.w
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5040.w1 | 5040z4 | \([0, 0, 0, -3807, 36234]\) | \(1210991472/588245\) | \(2964077141760\) | \([2]\) | \(6912\) | \(1.0866\) | |
5040.w2 | 5040z3 | \([0, 0, 0, -3132, 67419]\) | \(10788913152/8575\) | \(2700507600\) | \([2]\) | \(3456\) | \(0.74005\) | |
5040.w3 | 5040z2 | \([0, 0, 0, -2007, -34606]\) | \(129348709488/6125\) | \(42336000\) | \([2]\) | \(2304\) | \(0.53732\) | |
5040.w4 | 5040z1 | \([0, 0, 0, -132, -481]\) | \(588791808/109375\) | \(47250000\) | \([2]\) | \(1152\) | \(0.19075\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 5040.w have rank \(1\).
Complex multiplication
The elliptic curves in class 5040.w do not have complex multiplication.Modular form 5040.2.a.w
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.