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SageMath
E = EllipticCurve("ke1")
E.isogeny_class()
Elliptic curves in class 463680ke
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
463680.ke4 | 463680ke1 | \([0, 0, 0, -1963674732, 29202036672944]\) | \(4381924769947287308715481/608122186185572352000\) | \(116213959551688964730519552000\) | \([2]\) | \(495452160\) | \(4.3036\) | \(\Gamma_0(N)\)-optimal* |
463680.ke2 | 463680ke2 | \([0, 0, 0, -30287023212, 2028728475306416]\) | \(16077778198622525072705635801/388799208512064000000\) | \(74300685813099962302464000000\) | \([2, 2]\) | \(990904320\) | \(4.6502\) | \(\Gamma_0(N)\)-optimal* |
463680.ke1 | 463680ke3 | \([0, 0, 0, -484589558892, 129840204932360624]\) | \(65853432878493908038433301506521/38511703125000000\) | \(7359701078016000000000000\) | \([2]\) | \(1981808640\) | \(4.9968\) | \(\Gamma_0(N)\)-optimal* |
463680.ke3 | 463680ke4 | \([0, 0, 0, -29158063212, 2186944090794416]\) | \(-14346048055032350809895395801/2509530875136386550792000\) | \(-479578818602447875742726356992000\) | \([2]\) | \(1981808640\) | \(4.9968\) |
Rank
sage: E.rank()
The elliptic curves in class 463680ke have rank \(1\).
Complex multiplication
The elliptic curves in class 463680ke do not have complex multiplication.Modular form 463680.2.a.ke
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 Cremona 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 Cremona labels.