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SageMath
E = EllipticCurve("ef1")
E.isogeny_class()
Elliptic curves in class 100800.ef
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
100800.ef1 | 100800c3 | \([0, 0, 0, -1514700, -716634000]\) | \(4767078987/6860\) | \(553063956480000000\) | \([2]\) | \(1327104\) | \(2.3069\) | |
100800.ef2 | 100800c4 | \([0, 0, 0, -1082700, -1133946000]\) | \(-1740992427/5882450\) | \(-474252342681600000000\) | \([2]\) | \(2654208\) | \(2.6535\) | |
100800.ef3 | 100800c1 | \([0, 0, 0, -74700, 6886000]\) | \(416832723/56000\) | \(6193152000000000\) | \([2]\) | \(442368\) | \(1.7576\) | \(\Gamma_0(N)\)-optimal |
100800.ef4 | 100800c2 | \([0, 0, 0, 117300, 36454000]\) | \(1613964717/6125000\) | \(-677376000000000000\) | \([2]\) | \(884736\) | \(2.1042\) |
Rank
sage: E.rank()
The elliptic curves in class 100800.ef have rank \(1\).
Complex multiplication
The elliptic curves in class 100800.ef do not have complex multiplication.Modular form 100800.2.a.ef
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.