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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 16758r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 16758.bh3 | 16758r1 | \([1, -1, 1, -828575, 207117831]\) | \(19804628171203875/5638671302656\) | \(17911369082326745088\) | \([2]\) | \(442368\) | \(2.4010\) | \(\Gamma_0(N)\)-optimal |
| 16758.bh4 | 16758r2 | \([1, -1, 1, 2181985, 1364377095]\) | \(361682234074684125/462672528510976\) | \(-1469689928283271016448\) | \([2]\) | \(884736\) | \(2.7476\) | |
| 16758.bh1 | 16758r3 | \([1, -1, 1, -61604255, 186123021319]\) | \(11165451838341046875/572244736\) | \(1325138704273504512\) | \([2]\) | \(1327104\) | \(2.9503\) | |
| 16758.bh2 | 16758r4 | \([1, -1, 1, -61498415, 186794343271]\) | \(-11108001800138902875/79947274872976\) | \(-185132726560149819644592\) | \([2]\) | \(2654208\) | \(3.2969\) |
Rank
sage: E.rank()
The elliptic curves in class 16758r have rank \(1\).
Complex multiplication
The elliptic curves in class 16758r do not have complex multiplication.Modular form 16758.2.a.r
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
