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SageMath
E = EllipticCurve("oz1")
E.isogeny_class()
Elliptic curves in class 187200.oz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
187200.oz1 | 187200fz3 | \([0, 0, 0, -7488300, 7887202000]\) | \(31103978031362/195\) | \(291133440000000\) | \([2]\) | \(4718592\) | \(2.3810\) | |
187200.oz2 | 187200fz4 | \([0, 0, 0, -648300, 19762000]\) | \(20183398562/11567205\) | \(17269744527360000000\) | \([2]\) | \(4718592\) | \(2.3810\) | |
187200.oz3 | 187200fz2 | \([0, 0, 0, -468300, 123082000]\) | \(15214885924/38025\) | \(28385510400000000\) | \([2, 2]\) | \(2359296\) | \(2.0344\) | |
187200.oz4 | 187200fz1 | \([0, 0, 0, -18300, 3382000]\) | \(-3631696/24375\) | \(-4548960000000000\) | \([2]\) | \(1179648\) | \(1.6878\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 187200.oz have rank \(0\).
Complex multiplication
The elliptic curves in class 187200.oz do not have complex multiplication.Modular form 187200.2.a.oz
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.