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SageMath
E = EllipticCurve("cs1")
E.isogeny_class()
Elliptic curves in class 357390.cs
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
357390.cs1 | 357390cs6 | \([1, -1, 1, -555855233, -5044041928123]\) | \(553808571467029327441/12529687500\) | \(429723766390204687500\) | \([2]\) | \(84934656\) | \(3.4833\) | |
357390.cs2 | 357390cs3 | \([1, -1, 1, -38419493, 91467958637]\) | \(182864522286982801/463015182960\) | \(15879775797873723677040\) | \([2]\) | \(42467328\) | \(3.1367\) | |
357390.cs3 | 357390cs4 | \([1, -1, 1, -34780613, -78617659219]\) | \(135670761487282321/643043610000\) | \(22054111249171528890000\) | \([2, 2]\) | \(42467328\) | \(3.1367\) | |
357390.cs4 | 357390cs5 | \([1, -1, 1, -16911113, -159309173419]\) | \(-15595206456730321/310672490129100\) | \(-10654962669428151349845900\) | \([2]\) | \(84934656\) | \(3.4833\) | |
357390.cs5 | 357390cs2 | \([1, -1, 1, -3330293, 222002957]\) | \(119102750067601/68309049600\) | \(2342757716235724550400\) | \([2, 2]\) | \(21233664\) | \(2.7901\) | |
357390.cs6 | 357390cs1 | \([1, -1, 1, 828427, 27374861]\) | \(1833318007919/1070530560\) | \(-36715394879482429440\) | \([2]\) | \(10616832\) | \(2.4436\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 357390.cs have rank \(2\).
Complex multiplication
The elliptic curves in class 357390.cs do not have complex multiplication.Modular form 357390.2.a.cs
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.