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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 570.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
570.k1 | 570k4 | \([1, 0, 0, -463231, 77449961]\) | \(10993009831928446009969/3767761230468750000\) | \(3767761230468750000\) | \([2]\) | \(17280\) | \(2.2664\) | |
570.k2 | 570k2 | \([1, 0, 0, -414991, 102863225]\) | \(7903870428425797297009/886464000000\) | \(886464000000\) | \([6]\) | \(5760\) | \(1.7171\) | |
570.k3 | 570k1 | \([1, 0, 0, -25871, 1614201]\) | \(-1914980734749238129/20440940544000\) | \(-20440940544000\) | \([6]\) | \(2880\) | \(1.3705\) | \(\Gamma_0(N)\)-optimal |
570.k4 | 570k3 | \([1, 0, 0, 85489, 8420985]\) | \(69096190760262356111/70568821500000000\) | \(-70568821500000000\) | \([2]\) | \(8640\) | \(1.9198\) |
Rank
sage: E.rank()
The elliptic curves in class 570.k have rank \(0\).
Complex multiplication
The elliptic curves in class 570.k do not have complex multiplication.Modular form 570.2.a.k
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.