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SageMath
E = EllipticCurve("ex1")
E.isogeny_class()
Elliptic curves in class 159600ex
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 159600.bn4 | 159600ex1 | \([0, -1, 0, -18691008, 36642880512]\) | \(-11283450590382195961/2530373271552000\) | \(-161943889379328000000000\) | \([2]\) | \(15482880\) | \(3.1737\) | \(\Gamma_0(N)\)-optimal |
| 159600.bn3 | 159600ex2 | \([0, -1, 0, -313603008, 2137595968512]\) | \(53294746224000958661881/1997017344000000\) | \(127809110016000000000000\) | \([2, 2]\) | \(30965760\) | \(3.5202\) | |
| 159600.bn1 | 159600ex3 | \([0, -1, 0, -5017603008, 136803707968512]\) | \(218289391029690300712901881/306514992000\) | \(19616959488000000000\) | \([4]\) | \(61931520\) | \(3.8668\) | |
| 159600.bn2 | 159600ex4 | \([0, -1, 0, -328195008, 1927763008512]\) | \(61085713691774408830201/10268551781250000000\) | \(657187314000000000000000000\) | \([2]\) | \(61931520\) | \(3.8668\) |
Rank
sage: E.rank()
The elliptic curves in class 159600ex have rank \(1\).
Complex multiplication
The elliptic curves in class 159600ex do not have complex multiplication.Modular form 159600.2.a.ex
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.
