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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 7920bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 7920.m6 | 7920bc1 | \([0, 0, 0, 36717, 257362]\) | \(1833318007919/1070530560\) | \(-3196587123671040\) | \([2]\) | \(36864\) | \(1.6645\) | \(\Gamma_0(N)\)-optimal |
| 7920.m5 | 7920bc2 | \([0, 0, 0, -147603, 2063698]\) | \(119102750067601/68309049600\) | \(203969729160806400\) | \([2, 2]\) | \(73728\) | \(2.0111\) | |
| 7920.m3 | 7920bc3 | \([0, 0, 0, -1541523, -733647278]\) | \(135670761487282321/643043610000\) | \(1920117930762240000\) | \([2, 2]\) | \(147456\) | \(2.3576\) | |
| 7920.m2 | 7920bc4 | \([0, 0, 0, -1702803, 853380178]\) | \(182864522286982801/463015182960\) | \(1382555928075632640\) | \([2]\) | \(147456\) | \(2.3576\) | |
| 7920.m1 | 7920bc5 | \([0, 0, 0, -24636243, -47066274542]\) | \(553808571467029327441/12529687500\) | \(37413446400000000\) | \([2]\) | \(294912\) | \(2.7042\) | |
| 7920.m4 | 7920bc6 | \([0, 0, 0, -749523, -1486522478]\) | \(-15595206456730321/310672490129100\) | \(-927663084765650534400\) | \([2]\) | \(294912\) | \(2.7042\) |
Rank
sage: E.rank()
The elliptic curves in class 7920bc have rank \(0\).
Complex multiplication
The elliptic curves in class 7920bc do not have complex multiplication.Modular form 7920.2.a.bc
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
