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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 117208.m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
117208.m1 | 117208d4 | \([0, 0, 0, -2187899, -1245629882]\) | \(9614292367656708/2093\) | \(252149101568\) | \([2]\) | \(786432\) | \(2.0121\) | |
117208.m2 | 117208d3 | \([0, 0, 0, -159299, -12610738]\) | \(3710860803108/1577224103\) | \(190012250617699328\) | \([2]\) | \(786432\) | \(2.0121\) | |
117208.m3 | 117208d2 | \([0, 0, 0, -136759, -19458390]\) | \(9392111857872/4380649\) | \(131937017395456\) | \([2, 2]\) | \(393216\) | \(1.6656\) | |
117208.m4 | 117208d1 | \([0, 0, 0, -7154, -406455]\) | \(-21511084032/25465531\) | \(-47935908105904\) | \([2]\) | \(196608\) | \(1.3190\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 117208.m have rank \(1\).
Complex multiplication
The elliptic curves in class 117208.m do not have complex multiplication.Modular form 117208.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.