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SageMath
E = EllipticCurve("hr1")
E.isogeny_class()
Elliptic curves in class 486720.hr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
486720.hr1 | 486720hr4 | \([0, 0, 0, -389296908, -2956442825232]\) | \(261984288445803/42250\) | \(1052247836684648448000\) | \([2]\) | \(111476736\) | \(3.4360\) | |
486720.hr2 | 486720hr3 | \([0, 0, 0, -24256908, -46489961232]\) | \(-63378025803/812500\) | \(-20235535320858624000000\) | \([2]\) | \(55738368\) | \(3.0894\) | |
486720.hr3 | 486720hr2 | \([0, 0, 0, -5437068, -2922853648]\) | \(520300455507/193072360\) | \(6596049484455472005120\) | \([2]\) | \(37158912\) | \(2.8867\) | \(\Gamma_0(N)\)-optimal* |
486720.hr4 | 486720hr1 | \([0, 0, 0, 1052532, -324417808]\) | \(3774555693/3515200\) | \(-120091934173062758400\) | \([2]\) | \(18579456\) | \(2.5401\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 486720.hr have rank \(1\).
Complex multiplication
The elliptic curves in class 486720.hr do not have complex multiplication.Modular form 486720.2.a.hr
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.