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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 6800.n
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6800.n1 | 6800h3 | \([0, 0, 0, -36275, 2659250]\) | \(82483294977/17\) | \(1088000000\) | \([2]\) | \(8192\) | \(1.1212\) | |
6800.n2 | 6800h2 | \([0, 0, 0, -2275, 41250]\) | \(20346417/289\) | \(18496000000\) | \([2, 2]\) | \(4096\) | \(0.77466\) | |
6800.n3 | 6800h1 | \([0, 0, 0, -275, -750]\) | \(35937/17\) | \(1088000000\) | \([2]\) | \(2048\) | \(0.42808\) | \(\Gamma_0(N)\)-optimal |
6800.n4 | 6800h4 | \([0, 0, 0, -275, 111250]\) | \(-35937/83521\) | \(-5345344000000\) | \([2]\) | \(8192\) | \(1.1212\) |
Rank
sage: E.rank()
The elliptic curves in class 6800.n have rank \(0\).
Complex multiplication
The elliptic curves in class 6800.n do not have complex multiplication.Modular form 6800.2.a.n
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.