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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 259920.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
259920.k1 | 259920k3 | \([0, 0, 0, -312294963, 2124189650738]\) | \(23977812996389881/146611125\) | \(20595673967325368832000\) | \([2]\) | \(53084160\) | \(3.4688\) | |
259920.k2 | 259920k4 | \([0, 0, 0, -64331283, -160831217518]\) | \(209595169258201/41748046875\) | \(5864692479579000000000000\) | \([2]\) | \(53084160\) | \(3.4688\) | |
259920.k3 | 259920k2 | \([0, 0, 0, -19884963, 31879136738]\) | \(6189976379881/456890625\) | \(64183194496512576000000\) | \([2, 2]\) | \(26542080\) | \(3.1222\) | |
259920.k4 | 259920k1 | \([0, 0, 0, 1168557, 2197884242]\) | \(1256216039/15582375\) | \(-2188984738617902592000\) | \([2]\) | \(13271040\) | \(2.7757\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 259920.k have rank \(2\).
Complex multiplication
The elliptic curves in class 259920.k do not have complex multiplication.Modular form 259920.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.