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SageMath
E = EllipticCurve("gk1")
E.isogeny_class()
Elliptic curves in class 478800gk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
478800.gk4 | 478800gk1 | \([0, 0, 0, -36378075, 84451482250]\) | \(114113060120923921/124104960\) | \(5790241013760000000\) | \([2]\) | \(35389440\) | \(2.8876\) | \(\Gamma_0(N)\)-optimal |
478800.gk3 | 478800gk2 | \([0, 0, 0, -36666075, 83046330250]\) | \(116844823575501841/3760263939600\) | \(175438874365977600000000\) | \([2, 2]\) | \(70778880\) | \(3.2342\) | |
478800.gk5 | 478800gk3 | \([0, 0, 0, 11213925, 284477490250]\) | \(3342636501165359/751262567039460\) | \(-35050906327793045760000000\) | \([2]\) | \(141557760\) | \(3.5808\) | |
478800.gk2 | 478800gk4 | \([0, 0, 0, -89154075, -208314557750]\) | \(1679731262160129361/570261564022500\) | \(26606123531033760000000000\) | \([2, 2]\) | \(141557760\) | \(3.5808\) | |
478800.gk6 | 478800gk5 | \([0, 0, 0, 261737925, -1443805289750]\) | \(42502666283088696719/43898058864843750\) | \(-2048107834398150000000000000\) | \([2]\) | \(283115520\) | \(3.9274\) | |
478800.gk1 | 478800gk6 | \([0, 0, 0, -1279854075, -17619920657750]\) | \(4969327007303723277361/1123462695162150\) | \(52416275505485270400000000\) | \([2]\) | \(283115520\) | \(3.9274\) |
Rank
sage: E.rank()
The elliptic curves in class 478800gk have rank \(0\).
Complex multiplication
The elliptic curves in class 478800gk do not have complex multiplication.Modular form 478800.2.a.gk
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.