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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 27200.h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
27200.h1 | 27200o3 | \([0, 1, 0, -6669633, -1306311137]\) | \(8010684753304969/4456448000000\) | \(18253611008000000000000\) | \([2]\) | \(2211840\) | \(2.9619\) | |
27200.h2 | 27200o1 | \([0, 1, 0, -4085633, 3177200863]\) | \(1841373668746009/31443200\) | \(128791347200000000\) | \([2]\) | \(737280\) | \(2.4126\) | \(\Gamma_0(N)\)-optimal |
27200.h3 | 27200o2 | \([0, 1, 0, -3957633, 3385712863]\) | \(-1673672305534489/241375690000\) | \(-988674826240000000000\) | \([2]\) | \(1474560\) | \(2.7592\) | |
27200.h4 | 27200o4 | \([0, 1, 0, 26098367, -10317511137]\) | \(479958568556831351/289000000000000\) | \(-1183744000000000000000000\) | \([2]\) | \(4423680\) | \(3.3085\) |
Rank
sage: E.rank()
The elliptic curves in class 27200.h have rank \(1\).
Complex multiplication
The elliptic curves in class 27200.h do not have complex multiplication.Modular form 27200.2.a.h
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.