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SageMath
E = EllipticCurve("by1")
E.isogeny_class()
Elliptic curves in class 10800by
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
10800.bl3 | 10800by1 | \([0, 0, 0, 525, 3250]\) | \(9261/8\) | \(-13824000000\) | \([]\) | \(5184\) | \(0.63336\) | \(\Gamma_0(N)\)-optimal |
10800.bl2 | 10800by2 | \([0, 0, 0, -5475, -206750]\) | \(-1167051/512\) | \(-7962624000000\) | \([]\) | \(15552\) | \(1.1827\) | |
10800.bl1 | 10800by3 | \([0, 0, 0, -11475, 479250]\) | \(-132651/2\) | \(-2519424000000\) | \([]\) | \(15552\) | \(1.1827\) |
Rank
sage: E.rank()
The elliptic curves in class 10800by have rank \(0\).
Complex multiplication
The elliptic curves in class 10800by do not have complex multiplication.Modular form 10800.2.a.by
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.
\(\left(\begin{array}{rrr} 1 & 3 & 3 \\ 3 & 1 & 9 \\ 3 & 9 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.