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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 19890d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
19890.k1 | 19890d1 | \([1, -1, 0, -5917050, -5538472524]\) | \(31427652507069423952801/654426190080\) | \(477076692568320\) | \([2]\) | \(389120\) | \(2.3459\) | \(\Gamma_0(N)\)-optimal |
19890.k2 | 19890d2 | \([1, -1, 0, -5910570, -5551213500]\) | \(-31324512477868037557921/143427974919699600\) | \(-104558993716461008400\) | \([2]\) | \(778240\) | \(2.6925\) |
Rank
sage: E.rank()
The elliptic curves in class 19890d have rank \(1\).
Complex multiplication
The elliptic curves in class 19890d do not have complex multiplication.Modular form 19890.2.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 Cremona 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 Cremona labels.