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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 108900p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
108900.d1 | 108900p1 | \([0, 0, 0, 0, -503284375]\) | \(0\) | \(-109423510035468750000\) | \([]\) | \(3706560\) | \(2.5240\) | \(\Gamma_0(N)\)-optimal | \(-3\) |
108900.d2 | 108900p2 | \([0, 0, 0, 0, 13588678125]\) | \(0\) | \(-79769738815856718750000\) | \([]\) | \(11119680\) | \(3.0733\) | \(-3\) |
Rank
sage: E.rank()
The elliptic curves in class 108900p have rank \(1\).
Complex multiplication
Each elliptic curve in class 108900p has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).Modular form 108900.2.a.p
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.