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SageMath
E = EllipticCurve("hr1")
E.isogeny_class()
Elliptic curves in class 103488.hr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 103488.hr1 | 103488ht4 | \([0, 1, 0, -1103937, -446809665]\) | \(4824238966273/66\) | \(2035504644096\) | \([2]\) | \(884736\) | \(1.9177\) | |
| 103488.hr2 | 103488ht2 | \([0, 1, 0, -69057, -6985665]\) | \(1180932193/4356\) | \(134343306510336\) | \([2, 2]\) | \(442368\) | \(1.5711\) | |
| 103488.hr3 | 103488ht3 | \([0, 1, 0, -37697, -13326657]\) | \(-192100033/2371842\) | \(-73149930394877952\) | \([2]\) | \(884736\) | \(1.9177\) | |
| 103488.hr4 | 103488ht1 | \([0, 1, 0, -6337, 1343]\) | \(912673/528\) | \(16284037152768\) | \([2]\) | \(221184\) | \(1.2246\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 103488.hr have rank \(0\).
Complex multiplication
The elliptic curves in class 103488.hr do not have complex multiplication.Modular form 103488.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
