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SageMath
E = EllipticCurve("by1")
E.isogeny_class()
Elliptic curves in class 7056by
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
7056.k5 | 7056by1 | \([0, 0, 0, -28371, -4061806]\) | \(-7189057/16128\) | \(-5665734653902848\) | \([2]\) | \(36864\) | \(1.7114\) | \(\Gamma_0(N)\)-optimal |
7056.k4 | 7056by2 | \([0, 0, 0, -592851, -175550830]\) | \(65597103937/63504\) | \(22308830199742464\) | \([2, 2]\) | \(73728\) | \(2.0579\) | |
7056.k1 | 7056by3 | \([0, 0, 0, -9483411, -11240741806]\) | \(268498407453697/252\) | \(88527103967232\) | \([2]\) | \(147456\) | \(2.4045\) | |
7056.k3 | 7056by4 | \([0, 0, 0, -733971, -85657390]\) | \(124475734657/63011844\) | \(22135936765694459904\) | \([2, 2]\) | \(147456\) | \(2.4045\) | |
7056.k2 | 7056by5 | \([0, 0, 0, -6449331, 6243532274]\) | \(84448510979617/933897762\) | \(328076445521187643392\) | \([2]\) | \(294912\) | \(2.7511\) | |
7056.k6 | 7056by6 | \([0, 0, 0, 2723469, -661666894]\) | \(6359387729183/4218578658\) | \(-1481978378772666851328\) | \([2]\) | \(294912\) | \(2.7511\) |
Rank
sage: E.rank()
The elliptic curves in class 7056by have rank \(0\).
Complex multiplication
The elliptic curves in class 7056by do not have complex multiplication.Modular form 7056.2.a.by
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 Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.