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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 340784h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
340784.h1 | 340784h1 | \([0, -1, 0, -144520, 21822704]\) | \(-1732323601/60416\) | \(-11642158884847616\) | \([]\) | \(1944000\) | \(1.8557\) | \(\Gamma_0(N)\)-optimal |
340784.h2 | 340784h2 | \([0, -1, 0, 664120, -1110273296]\) | \(168105213359/2859697196\) | \(-551063445418187472896\) | \([]\) | \(9720000\) | \(2.6604\) |
Rank
sage: E.rank()
The elliptic curves in class 340784h have rank \(1\).
Complex multiplication
The elliptic curves in class 340784h do not have complex multiplication.Modular form 340784.2.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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.