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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 177744.x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
177744.x1 | 177744br4 | \([0, -1, 0, -967188, -365789844]\) | \(2640279346000/3087\) | \(116988618071808\) | \([2]\) | \(1710720\) | \(1.9826\) | |
177744.x2 | 177744br3 | \([0, -1, 0, -59953, -5798996]\) | \(-10061824000/352947\) | \(-835981166638128\) | \([2]\) | \(855360\) | \(1.6360\) | |
177744.x3 | 177744br2 | \([0, -1, 0, -14988, -221220]\) | \(9826000/5103\) | \(193389348241152\) | \([2]\) | \(570240\) | \(1.4333\) | |
177744.x4 | 177744br1 | \([0, -1, 0, 3527, -28664]\) | \(2048000/1323\) | \(-3133623698352\) | \([2]\) | \(285120\) | \(1.0867\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 177744.x have rank \(0\).
Complex multiplication
The elliptic curves in class 177744.x do not have complex multiplication.Modular form 177744.2.a.x
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.