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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 127050.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
127050.d1 | 127050k2 | \([1, 1, 0, -49687200, -134853838500]\) | \(-6480608299825/1411788\) | \(-2955364217668242187500\) | \([]\) | \(17107200\) | \(3.1140\) | |
127050.d2 | 127050k1 | \([1, 1, 0, 225300, -639126000]\) | \(604175/84672\) | \(-177247999726875000000\) | \([]\) | \(5702400\) | \(2.5647\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 127050.d have rank \(2\).
Complex multiplication
The elliptic curves in class 127050.d do not have complex multiplication.Modular form 127050.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.