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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 690k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
690.k5 | 690k1 | \([1, 0, 0, -420, 3600]\) | \(-8194759433281/965779200\) | \(-965779200\) | \([8]\) | \(384\) | \(0.45987\) | \(\Gamma_0(N)\)-optimal |
690.k4 | 690k2 | \([1, 0, 0, -6900, 220032]\) | \(36330796409313601/428490000\) | \(428490000\) | \([2, 4]\) | \(768\) | \(0.80644\) | |
690.k3 | 690k3 | \([1, 0, 0, -7080, 207900]\) | \(39248884582600321/3935264062500\) | \(3935264062500\) | \([2, 2]\) | \(1536\) | \(1.1530\) | |
690.k1 | 690k4 | \([1, 0, 0, -110400, 14109732]\) | \(148809678420065817601/20700\) | \(20700\) | \([4]\) | \(1536\) | \(1.1530\) | |
690.k2 | 690k5 | \([1, 0, 0, -25830, -1370850]\) | \(1905890658841300321/293666194803750\) | \(293666194803750\) | \([2]\) | \(3072\) | \(1.4996\) | |
690.k6 | 690k6 | \([1, 0, 0, 8790, 1010922]\) | \(75108181893694559/484313964843750\) | \(-484313964843750\) | \([2]\) | \(3072\) | \(1.4996\) |
Rank
sage: E.rank()
The elliptic curves in class 690k have rank \(0\).
Complex multiplication
The elliptic curves in class 690k do not have complex multiplication.Modular form 690.2.a.k
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.