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SageMath
E = EllipticCurve("ee1")
E.isogeny_class()
Elliptic curves in class 100800.ee
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
100800.ee1 | 100800d3 | \([0, 0, 0, -672300, -185922000]\) | \(416832723/56000\) | \(4514807808000000000\) | \([2]\) | \(1327104\) | \(2.3069\) | |
100800.ee2 | 100800d1 | \([0, 0, 0, -168300, 26542000]\) | \(4767078987/6860\) | \(758661120000000\) | \([2]\) | \(442368\) | \(1.7576\) | \(\Gamma_0(N)\)-optimal |
100800.ee3 | 100800d2 | \([0, 0, 0, -120300, 41998000]\) | \(-1740992427/5882450\) | \(-650551910400000000\) | \([2]\) | \(884736\) | \(2.1042\) | |
100800.ee4 | 100800d4 | \([0, 0, 0, 1055700, -984258000]\) | \(1613964717/6125000\) | \(-493807104000000000000\) | \([2]\) | \(2654208\) | \(2.6535\) |
Rank
sage: E.rank()
The elliptic curves in class 100800.ee have rank \(1\).
Complex multiplication
The elliptic curves in class 100800.ee do not have complex multiplication.Modular form 100800.2.a.ee
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.