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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 1650q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1650.r3 | 1650q1 | \([1, 0, 0, -34838, -2505708]\) | \(299270638153369/1069200\) | \(16706250000\) | \([2]\) | \(3840\) | \(1.1796\) | \(\Gamma_0(N)\)-optimal |
1650.r2 | 1650q2 | \([1, 0, 0, -35338, -2430208]\) | \(312341975961049/17862322500\) | \(279098789062500\) | \([2, 2]\) | \(7680\) | \(1.5261\) | |
1650.r1 | 1650q3 | \([1, 0, 0, -104088, 9876042]\) | \(7981893677157049/1917731420550\) | \(29964553446093750\) | \([2]\) | \(15360\) | \(1.8727\) | |
1650.r4 | 1650q4 | \([1, 0, 0, 25412, -9902458]\) | \(116149984977671/2779502343750\) | \(-43429724121093750\) | \([2]\) | \(15360\) | \(1.8727\) |
Rank
sage: E.rank()
The elliptic curves in class 1650q have rank \(0\).
Complex multiplication
The elliptic curves in class 1650q do not have complex multiplication.Modular form 1650.2.a.q
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.