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SageMath
E = EllipticCurve("kd1")
E.isogeny_class()
Elliptic curves in class 379050kd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
379050.kd4 | 379050kd1 | \([1, 0, 0, -91197813, -335223292383]\) | \(114113060120923921/124104960\) | \(91228549682340000000\) | \([2]\) | \(66355200\) | \(3.1174\) | \(\Gamma_0(N)\)-optimal |
379050.kd3 | 379050kd2 | \([1, 0, 0, -91919813, -329645842383]\) | \(116844823575501841/3760263939600\) | \(2764139528609574806250000\) | \([2, 2]\) | \(132710400\) | \(3.4640\) | |
379050.kd2 | 379050kd3 | \([1, 0, 0, -223504313, 826850328117]\) | \(1679731262160129361/570261564022500\) | \(419194651248069005039062500\) | \([2, 2]\) | \(265420800\) | \(3.8105\) | |
379050.kd5 | 379050kd4 | \([1, 0, 0, 28112687, -1129182324883]\) | \(3342636501165359/751262567039460\) | \(-552247020760827460379062500\) | \([2]\) | \(265420800\) | \(3.8105\) | |
379050.kd1 | 379050kd5 | \([1, 0, 0, -3208523063, 69938989446867]\) | \(4969327007303723277361/1123462695162150\) | \(825848316633402884439843750\) | \([2]\) | \(530841600\) | \(4.1571\) | |
379050.kd6 | 379050kd6 | \([1, 0, 0, 656162437, 5730992459367]\) | \(42502666283088696719/43898058864843750\) | \(-32269107085725533532714843750\) | \([2]\) | \(530841600\) | \(4.1571\) |
Rank
sage: E.rank()
The elliptic curves in class 379050kd have rank \(1\).
Complex multiplication
The elliptic curves in class 379050kd do not have complex multiplication.Modular form 379050.2.a.kd
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.