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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 8712.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8712.r1 | 8712j4 | \([0, 0, 0, -767019, 258557398]\) | \(37736227588/33\) | \(43641285608448\) | \([2]\) | \(92160\) | \(1.9176\) | |
8712.r2 | 8712j3 | \([0, 0, 0, -113619, -9101378]\) | \(122657188/43923\) | \(58086551144844288\) | \([2]\) | \(92160\) | \(1.9176\) | |
8712.r3 | 8712j2 | \([0, 0, 0, -48279, 3979690]\) | \(37642192/1089\) | \(360040606269696\) | \([2, 2]\) | \(46080\) | \(1.5710\) | |
8712.r4 | 8712j1 | \([0, 0, 0, 726, 206305]\) | \(2048/891\) | \(-18411167366064\) | \([2]\) | \(23040\) | \(1.2244\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 8712.r have rank \(1\).
Complex multiplication
The elliptic curves in class 8712.r do not have complex multiplication.Modular form 8712.2.a.r
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.