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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 300042h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
300042.h2 | 300042h1 | \([1, -1, 0, -105363, -13014243]\) | \(177444640175483953/1913455446264\) | \(1394909020326456\) | \([]\) | \(3237120\) | \(1.7214\) | \(\Gamma_0(N)\)-optimal |
300042.h1 | 300042h2 | \([1, -1, 0, -783333, 259831503]\) | \(72918170522696196433/2250945132306174\) | \(1640939001451200846\) | \([3]\) | \(9711360\) | \(2.2707\) |
Rank
sage: E.rank()
The elliptic curves in class 300042h have rank \(0\).
Complex multiplication
The elliptic curves in class 300042h do not have complex multiplication.Modular form 300042.2.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 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.