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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 35490.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
35490.f1 | 35490k8 | \([1, 1, 0, -1557845383, -23653796228447]\) | \(86623684689189325642735681/56690726941459561860\) | \(273635311017579486321904740\) | \([2]\) | \(22020096\) | \(4.0125\) | |
35490.f2 | 35490k4 | \([1, 1, 0, -944885763, 11178969652557]\) | \(19328649688935739391016961/1048320\) | \(5060040410880\) | \([2]\) | \(5505024\) | \(3.3193\) | |
35490.f3 | 35490k6 | \([1, 1, 0, -116393683, -214926715427]\) | \(36128658497509929012481/16775330746084419600\) | \(80971317423176991285056400\) | \([2, 2]\) | \(11010048\) | \(3.6659\) | |
35490.f4 | 35490k3 | \([1, 1, 0, -59271683, 173308669773]\) | \(4770955732122964500481/71987251059360000\) | \(347468711298578382240000\) | \([2, 2]\) | \(5505024\) | \(3.3193\) | |
35490.f5 | 35490k2 | \([1, 1, 0, -59055363, 174652925517]\) | \(4718909406724749250561/1098974822400\) | \(5304541563533721600\) | \([2, 2]\) | \(2752512\) | \(2.9728\) | |
35490.f6 | 35490k5 | \([1, 1, 0, -5610803, 475516013757]\) | \(-4047051964543660801/20235220197806250000\) | \(-97671542967752987756250000\) | \([2]\) | \(11010048\) | \(3.6659\) | |
35490.f7 | 35490k1 | \([1, 1, 0, -3677443, 2748786253]\) | \(-1139466686381936641/17587891077120\) | \(-84893390942062510080\) | \([2]\) | \(1376256\) | \(2.6262\) | \(\Gamma_0(N)\)-optimal |
35490.f8 | 35490k7 | \([1, 1, 0, 411106017, -1621979415207]\) | \(1591934139020114746758719/1156766383092650262660\) | \(-5583490388809052121659651940\) | \([2]\) | \(22020096\) | \(4.0125\) |
Rank
sage: E.rank()
The elliptic curves in class 35490.f have rank \(0\).
Complex multiplication
The elliptic curves in class 35490.f do not have complex multiplication.Modular form 35490.2.a.f
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.