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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 162450.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
162450.d1 | 162450dd3 | \([1, -1, 0, -246925692, 1493535273966]\) | \(3107086841064961/570\) | \(305452733311406250\) | \([2]\) | \(26542080\) | \(3.1908\) | |
162450.d2 | 162450dd4 | \([1, -1, 0, -17871192, 15475826466]\) | \(1177918188481/488703750\) | \(261887537222866933593750\) | \([2]\) | \(26542080\) | \(3.1908\) | |
162450.d3 | 162450dd2 | \([1, -1, 0, -15434442, 23334345216]\) | \(758800078561/324900\) | \(174108057987501562500\) | \([2, 2]\) | \(13271040\) | \(2.8442\) | |
162450.d4 | 162450dd1 | \([1, -1, 0, -813942, 482503716]\) | \(-111284641/123120\) | \(-65977790395263750000\) | \([2]\) | \(6635520\) | \(2.4976\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 162450.d have rank \(1\).
Complex multiplication
The elliptic curves in class 162450.d do not have complex multiplication.Modular form 162450.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.