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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 405720.m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
405720.m1 | 405720m3 | \([0, 0, 0, -465843, 120429358]\) | \(63649751618/1164375\) | \(204521322781440000\) | \([2]\) | \(4718592\) | \(2.1165\) | \(\Gamma_0(N)\)-optimal* |
405720.m2 | 405720m2 | \([0, 0, 0, -60123, -2828378]\) | \(273671716/119025\) | \(10453312053273600\) | \([2, 2]\) | \(2359296\) | \(1.7699\) | \(\Gamma_0(N)\)-optimal* |
405720.m3 | 405720m1 | \([0, 0, 0, -51303, -4470662]\) | \(680136784/345\) | \(7574863806720\) | \([2]\) | \(1179648\) | \(1.4233\) | \(\Gamma_0(N)\)-optimal* |
405720.m4 | 405720m4 | \([0, 0, 0, 204477, -20979938]\) | \(5382838942/4197615\) | \(-737306943490897920\) | \([2]\) | \(4718592\) | \(2.1165\) |
Rank
sage: E.rank()
The elliptic curves in class 405720.m have rank \(0\).
Complex multiplication
The elliptic curves in class 405720.m do not have complex multiplication.Modular form 405720.2.a.m
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.