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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 21904.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
21904.d1 | 21904h3 | \([0, -1, 0, -41033493, 101184633181]\) | \(727057727488000/37\) | \(388840968736768\) | \([]\) | \(590976\) | \(2.7207\) | |
21904.d2 | 21904h2 | \([0, -1, 0, -511093, 136355645]\) | \(1404928000/50653\) | \(532323286200635392\) | \([]\) | \(196992\) | \(2.1714\) | |
21904.d3 | 21904h1 | \([0, -1, 0, -73013, -7509827]\) | \(4096000/37\) | \(388840968736768\) | \([]\) | \(65664\) | \(1.6221\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 21904.d have rank \(0\).
Complex multiplication
The elliptic curves in class 21904.d do not have complex multiplication.Modular form 21904.2.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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.