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SageMath
E = EllipticCurve("ce1")
E.isogeny_class()
Elliptic curves in class 209814ce
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
209814.bu5 | 209814ce1 | \([1, 0, 1, -9931925, -9126414112]\) | \(2533811507137/625016832\) | \(26726454678117956517888\) | \([2]\) | \(17694720\) | \(3.0138\) | \(\Gamma_0(N)\)-optimal |
209814.bu4 | 209814ce2 | \([1, 0, 1, -54692245, 148089733856]\) | \(423108074414017/23284318464\) | \(995664836973480278558976\) | \([2, 2]\) | \(35389440\) | \(3.3603\) | |
209814.bu2 | 209814ce3 | \([1, 0, 1, -863175525, 9760955933056]\) | \(1663303207415737537/5483698704\) | \(234489404728399659029136\) | \([2, 2]\) | \(70778880\) | \(3.7069\) | |
209814.bu6 | 209814ce4 | \([1, 0, 1, 37625915, 597236045888]\) | \(137763859017023/3683199928848\) | \(-157497959941026602927129232\) | \([2]\) | \(70778880\) | \(3.7069\) | |
209814.bu1 | 209814ce5 | \([1, 0, 1, -13810797465, 624705670448968]\) | \(6812873765474836663297/74052\) | \(3166550595910976868\) | \([2]\) | \(141557760\) | \(4.0535\) | |
209814.bu3 | 209814ce6 | \([1, 0, 1, -851286065, 10042912099064]\) | \(-1595514095015181697/95635786040388\) | \(-4089498666836889000521941092\) | \([2]\) | \(141557760\) | \(4.0535\) |
Rank
sage: E.rank()
The elliptic curves in class 209814ce have rank \(1\).
Complex multiplication
The elliptic curves in class 209814ce do not have complex multiplication.Modular form 209814.2.a.ce
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.