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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 169050.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
169050.i1 | 169050ic2 | \([1, 1, 0, -171700, 27313000]\) | \(104453838382375/14904\) | \(79876125000\) | \([2]\) | \(663552\) | \(1.5017\) | |
169050.i2 | 169050ic1 | \([1, 1, 0, -10700, 426000]\) | \(-25282750375/304704\) | \(-1633023000000\) | \([2]\) | \(331776\) | \(1.1551\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 169050.i have rank \(2\).
Complex multiplication
The elliptic curves in class 169050.i do not have complex multiplication.Modular form 169050.2.a.i
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.