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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 2730.w
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2730.w1 | 2730v7 | \([1, 1, 1, -9218020, -10769952415]\) | \(86623684689189325642735681/56690726941459561860\) | \(56690726941459561860\) | \([2]\) | \(131072\) | \(2.7300\) | |
2730.w2 | 2730v4 | \([1, 1, 1, -5591040, 5086138017]\) | \(19328649688935739391016961/1048320\) | \(1048320\) | \([4]\) | \(32768\) | \(2.0368\) | |
2730.w3 | 2730v5 | \([1, 1, 1, -688720, -98092255]\) | \(36128658497509929012481/16775330746084419600\) | \(16775330746084419600\) | \([2, 2]\) | \(65536\) | \(2.3834\) | |
2730.w4 | 2730v3 | \([1, 1, 1, -350720, 78749345]\) | \(4770955732122964500481/71987251059360000\) | \(71987251059360000\) | \([2, 4]\) | \(32768\) | \(2.0368\) | |
2730.w5 | 2730v2 | \([1, 1, 1, -349440, 79361697]\) | \(4718909406724749250561/1098974822400\) | \(1098974822400\) | \([2, 4]\) | \(16384\) | \(1.6903\) | |
2730.w6 | 2730v6 | \([1, 1, 1, -33200, 216426017]\) | \(-4047051964543660801/20235220197806250000\) | \(-20235220197806250000\) | \([8]\) | \(65536\) | \(2.3834\) | |
2730.w7 | 2730v1 | \([1, 1, 1, -21760, 1242785]\) | \(-1139466686381936641/17587891077120\) | \(-17587891077120\) | \([4]\) | \(8192\) | \(1.3437\) | \(\Gamma_0(N)\)-optimal |
2730.w8 | 2730v8 | \([1, 1, 1, 2432580, -737334495]\) | \(1591934139020114746758719/1156766383092650262660\) | \(-1156766383092650262660\) | \([2]\) | \(131072\) | \(2.7300\) |
Rank
sage: E.rank()
The elliptic curves in class 2730.w have rank \(0\).
Complex multiplication
The elliptic curves in class 2730.w do not have complex multiplication.Modular form 2730.2.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.
\(\left(\begin{array}{rrrrrrrr} 1 & 16 & 2 & 4 & 8 & 8 & 16 & 4 \\ 16 & 1 & 8 & 4 & 2 & 8 & 4 & 16 \\ 2 & 8 & 1 & 2 & 4 & 4 & 8 & 2 \\ 4 & 4 & 2 & 1 & 2 & 2 & 4 & 4 \\ 8 & 2 & 4 & 2 & 1 & 4 & 2 & 8 \\ 8 & 8 & 4 & 2 & 4 & 1 & 8 & 8 \\ 16 & 4 & 8 & 4 & 2 & 8 & 1 & 16 \\ 4 & 16 & 2 & 4 & 8 & 8 & 16 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.