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SageMath
E = EllipticCurve("du1")
E.isogeny_class()
Elliptic curves in class 193600du
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
193600.gr2 | 193600du1 | \([0, 1, 0, -733, -8087]\) | \(-32768\) | \(-1331000000\) | \([]\) | \(62208\) | \(0.52822\) | \(\Gamma_0(N)\)-optimal | \(-11\) |
193600.gr1 | 193600du2 | \([0, 1, 0, -88733, 10408913]\) | \(-32768\) | \(-2357947691000000\) | \([]\) | \(684288\) | \(1.7272\) | \(-11\) |
Rank
sage: E.rank()
The elliptic curves in class 193600du have rank \(0\).
Complex multiplication
Each elliptic curve in class 193600du has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-11}) \).Modular form 193600.2.a.du
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.
\(\left(\begin{array}{rr} 1 & 11 \\ 11 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.