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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 127296.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
127296.d1 | 127296bi2 | \([0, 0, 0, -697932, -219419280]\) | \(196741326427281/5020614352\) | \(959454344015511552\) | \([2]\) | \(2359296\) | \(2.2329\) | |
127296.d2 | 127296bi1 | \([0, 0, 0, -98892, 7017840]\) | \(559679941521/212556032\) | \(40620090281951232\) | \([2]\) | \(1179648\) | \(1.8863\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 127296.d have rank \(1\).
Complex multiplication
The elliptic curves in class 127296.d do not have complex multiplication.Modular form 127296.2.a.d
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.