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SageMath
E = EllipticCurve("dw1")
E.isogeny_class()
Elliptic curves in class 159936dw
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
159936.cs2 | 159936dw1 | \([0, -1, 0, -801313, 276255169]\) | \(1845026709625/793152\) | \(24461584537485312\) | \([2]\) | \(1658880\) | \(2.1048\) | \(\Gamma_0(N)\)-optimal |
159936.cs3 | 159936dw2 | \([0, -1, 0, -675873, 365543361]\) | \(-1107111813625/1228691592\) | \(-37894052146631933952\) | \([2]\) | \(3317760\) | \(2.4514\) | |
159936.cs1 | 159936dw3 | \([0, -1, 0, -2353633, -1049962367]\) | \(46753267515625/11591221248\) | \(357484616379518681088\) | \([2]\) | \(4976640\) | \(2.6541\) | |
159936.cs4 | 159936dw4 | \([0, -1, 0, 5674527, -6664857471]\) | \(655215969476375/1001033261568\) | \(-30872846254791362347008\) | \([2]\) | \(9953280\) | \(3.0007\) |
Rank
sage: E.rank()
The elliptic curves in class 159936dw have rank \(0\).
Complex multiplication
The elliptic curves in class 159936dw do not have complex multiplication.Modular form 159936.2.a.dw
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.