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SageMath
E = EllipticCurve("y1")
E.isogeny_class()
Elliptic curves in class 204624y
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
204624.bu5 | 204624y1 | \([0, 0, 0, -5532051, -5362105966]\) | \(-53297461115137/4513839183\) | \(-1585702820019073609728\) | \([2]\) | \(9437184\) | \(2.8129\) | \(\Gamma_0(N)\)-optimal |
204624.bu4 | 204624y2 | \([0, 0, 0, -90239331, -329943461470]\) | \(231331938231569617/1472026689\) | \(517120078331917799424\) | \([2, 2]\) | \(18874368\) | \(3.1595\) | |
204624.bu3 | 204624y3 | \([0, 0, 0, -91968051, -316644418510]\) | \(244883173420511137/18418027974129\) | \(6470216973559941872062464\) | \([2, 2]\) | \(37748736\) | \(3.5061\) | |
204624.bu1 | 204624y4 | \([0, 0, 0, -1443827091, -21116449256686]\) | \(947531277805646290177/38367\) | \(13478251579011072\) | \([2]\) | \(37748736\) | \(3.5061\) | |
204624.bu2 | 204624y5 | \([0, 0, 0, -299661411, 1623086947874]\) | \(8471112631466271697/1662662681263647\) | \(584090127169299994637463552\) | \([2]\) | \(75497472\) | \(3.8526\) | |
204624.bu6 | 204624y6 | \([0, 0, 0, 88065789, -1405237035454]\) | \(215015459663151503/2552757445339983\) | \(-896778665741024739802902528\) | \([2]\) | \(75497472\) | \(3.8526\) |
Rank
sage: E.rank()
The elliptic curves in class 204624y have rank \(0\).
Complex multiplication
The elliptic curves in class 204624y do not have complex multiplication.Modular form 204624.2.a.y
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.