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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 165048.o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
165048.o1 | 165048bn3 | \([0, -1, 0, -160992, 24307452]\) | \(3044193988/85293\) | \(12929459282408448\) | \([2]\) | \(1441792\) | \(1.8706\) | |
165048.o2 | 165048bn2 | \([0, -1, 0, -23452, -834860]\) | \(37642192/13689\) | \(518774600837376\) | \([2, 2]\) | \(720896\) | \(1.5240\) | |
165048.o3 | 165048bn1 | \([0, -1, 0, -20807, -1148028]\) | \(420616192/117\) | \(277123184208\) | \([2]\) | \(360448\) | \(1.1774\) | \(\Gamma_0(N)\)-optimal |
165048.o4 | 165048bn4 | \([0, -1, 0, 71768, -5976740]\) | \(269676572/257049\) | \(-38965736685118464\) | \([2]\) | \(1441792\) | \(1.8706\) |
Rank
sage: E.rank()
The elliptic curves in class 165048.o have rank \(1\).
Complex multiplication
The elliptic curves in class 165048.o do not have complex multiplication.Modular form 165048.2.a.o
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.