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SageMath
E = EllipticCurve("dr1")
E.isogeny_class()
Elliptic curves in class 482790dr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
482790.dr4 | 482790dr1 | \([1, 0, 1, -1222708, -520495054]\) | \(114113060120923921/124104960\) | \(219859507042560\) | \([2]\) | \(9830400\) | \(2.0394\) | \(\Gamma_0(N)\)-optimal* |
482790.dr3 | 482790dr2 | \([1, 0, 1, -1232388, -511837262]\) | \(116844823575501841/3760263939600\) | \(6661536945101715600\) | \([2, 2]\) | \(19660800\) | \(2.3860\) | \(\Gamma_0(N)\)-optimal* |
482790.dr2 | 482790dr3 | \([1, 0, 1, -2996568, 1283392306]\) | \(1679731262160129361/570261564022500\) | \(1010253146621264122500\) | \([2, 2]\) | \(39321600\) | \(2.7326\) | \(\Gamma_0(N)\)-optimal* |
482790.dr5 | 482790dr4 | \([1, 0, 1, 376912, -1752929422]\) | \(3342636501165359/751262567039460\) | \(-1330907464526992797060\) | \([2]\) | \(39321600\) | \(2.7326\) | |
482790.dr1 | 482790dr5 | \([1, 0, 1, -43017318, 108571018906]\) | \(4969327007303723277361/1123462695162150\) | \(1990282695704153616150\) | \([2]\) | \(78643200\) | \(3.0791\) | \(\Gamma_0(N)\)-optimal* |
482790.dr6 | 482790dr6 | \([1, 0, 1, 8797302, 8897514778]\) | \(42502666283088696719/43898058864843750\) | \(-77768089060661458593750\) | \([2]\) | \(78643200\) | \(3.0791\) |
Rank
sage: E.rank()
The elliptic curves in class 482790dr have rank \(2\).
Complex multiplication
The elliptic curves in class 482790dr do not have complex multiplication.Modular form 482790.2.a.dr
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.