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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 1254.g
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1254.g1 | 1254h5 | \([1, 1, 1, -42359, -3373225]\) | \(8405459297332260337/52107462\) | \(52107462\) | \([2]\) | \(2048\) | \(1.0870\) | |
| 1254.g2 | 1254h3 | \([1, 1, 1, -2649, -53469]\) | \(2055795133410577/5109104484\) | \(5109104484\) | \([2, 2]\) | \(1024\) | \(0.74046\) | |
| 1254.g3 | 1254h6 | \([1, 1, 1, -1659, -92673]\) | \(-504985875929137/3362745482118\) | \(-3362745482118\) | \([2]\) | \(2048\) | \(1.0870\) | |
| 1254.g4 | 1254h2 | \([1, 1, 1, -229, -229]\) | \(1328460616657/761097744\) | \(761097744\) | \([2, 4]\) | \(512\) | \(0.39388\) | |
| 1254.g5 | 1254h1 | \([1, 1, 1, -149, 635]\) | \(365986170577/1765632\) | \(1765632\) | \([4]\) | \(256\) | \(0.047310\) | \(\Gamma_0(N)\)-optimal |
| 1254.g6 | 1254h4 | \([1, 1, 1, 911, -685]\) | \(83608233481583/48873824868\) | \(-48873824868\) | \([4]\) | \(1024\) | \(0.74046\) |
Rank
sage: E.rank()
The elliptic curves in class 1254.g have rank \(1\).
Complex multiplication
The elliptic curves in class 1254.g do not have complex multiplication.Modular form 1254.2.a.g
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
