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SageMath
E = EllipticCurve("dt1")
E.isogeny_class()
Elliptic curves in class 466752.dt
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
466752.dt1 | 466752dt3 | \([0, 1, 0, -210894113, 1178543587935]\) | \(3957101249824708884951625/772310238681366528\) | \(202456495208888147116032\) | \([2]\) | \(87588864\) | \(3.4718\) | \(\Gamma_0(N)\)-optimal* |
466752.dt2 | 466752dt4 | \([0, 1, 0, -188611873, 1437325066847]\) | \(-2830680648734534916567625/1766676274677722124288\) | \(-463123585349116788549353472\) | \([2]\) | \(175177728\) | \(3.8184\) | |
466752.dt3 | 466752dt1 | \([0, 1, 0, -6420833, -4057229793]\) | \(111675519439697265625/37528570137307392\) | \(9837889490074308968448\) | \([2]\) | \(29196288\) | \(2.9225\) | \(\Gamma_0(N)\)-optimal* |
466752.dt4 | 466752dt2 | \([0, 1, 0, 18733727, -28019463649]\) | \(2773679829880629422375/2899504554614368272\) | \(-760087721964828956295168\) | \([2]\) | \(58392576\) | \(3.2691\) |
Rank
sage: E.rank()
The elliptic curves in class 466752.dt have rank \(1\).
Complex multiplication
The elliptic curves in class 466752.dt do not have complex multiplication.Modular form 466752.2.a.dt
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.