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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 159936.z
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
159936.z1 | 159936cy6 | \([0, -1, 0, -87005249, -312338826015]\) | \(2361739090258884097/5202\) | \(160434775130112\) | \([2]\) | \(9437184\) | \(2.8598\) | |
159936.z2 | 159936cy4 | \([0, -1, 0, -5437889, -4878819231]\) | \(576615941610337/27060804\) | \(834581700226842624\) | \([2, 2]\) | \(4718592\) | \(2.5132\) | |
159936.z3 | 159936cy5 | \([0, -1, 0, -5155649, -5408132127]\) | \(-491411892194497/125563633938\) | \(-3872505454702562377728\) | \([2]\) | \(9437184\) | \(2.8598\) | |
159936.z4 | 159936cy2 | \([0, -1, 0, -357569, -67756191]\) | \(163936758817/30338064\) | \(935655608558813184\) | \([2, 2]\) | \(2359296\) | \(2.1666\) | |
159936.z5 | 159936cy1 | \([0, -1, 0, -106689, 12475233]\) | \(4354703137/352512\) | \(10871815349993472\) | \([2]\) | \(1179648\) | \(1.8200\) | \(\Gamma_0(N)\)-optimal |
159936.z6 | 159936cy3 | \([0, -1, 0, 708671, -395518367]\) | \(1276229915423/2927177028\) | \(-90277006584623136768\) | \([2]\) | \(4718592\) | \(2.5132\) |
Rank
sage: E.rank()
The elliptic curves in class 159936.z have rank \(1\).
Complex multiplication
The elliptic curves in class 159936.z do not have complex multiplication.Modular form 159936.2.a.z
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.