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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 10080i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
10080.j3 | 10080i1 | \([0, 0, 0, -8013, 267388]\) | \(1219555693504/43758225\) | \(2041583745600\) | \([2, 2]\) | \(12288\) | \(1.1323\) | \(\Gamma_0(N)\)-optimal |
10080.j2 | 10080i2 | \([0, 0, 0, -20163, -736202]\) | \(2428799546888/778248135\) | \(290479559892480\) | \([2]\) | \(24576\) | \(1.4789\) | |
10080.j1 | 10080i3 | \([0, 0, 0, -127083, 17437282]\) | \(608119035935048/826875\) | \(308629440000\) | \([2]\) | \(24576\) | \(1.4789\) | |
10080.j4 | 10080i4 | \([0, 0, 0, 3012, 946528]\) | \(1012048064/130203045\) | \(-388784209121280\) | \([2]\) | \(24576\) | \(1.4789\) |
Rank
sage: E.rank()
The elliptic curves in class 10080i have rank \(0\).
Complex multiplication
The elliptic curves in class 10080i do not have complex multiplication.Modular form 10080.2.a.i
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.