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SageMath
E = EllipticCurve("on1")
E.isogeny_class()
Elliptic curves in class 100800.on
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
100800.on1 | 100800ia2 | \([0, 0, 0, -88500, 10145000]\) | \(-262885120/343\) | \(-100018800000000\) | \([]\) | \(414720\) | \(1.5934\) | |
100800.on2 | 100800ia1 | \([0, 0, 0, 1500, 65000]\) | \(1280/7\) | \(-2041200000000\) | \([]\) | \(138240\) | \(1.0441\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 100800.on have rank \(0\).
Complex multiplication
The elliptic curves in class 100800.on do not have complex multiplication.Modular form 100800.2.a.on
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.