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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 32175.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
32175.j1 | 32175p4 | \([1, -1, 1, -1544405, -738350778]\) | \(35765103905346817/1287\) | \(14659734375\) | \([2]\) | \(262144\) | \(1.8951\) | |
32175.j2 | 32175p6 | \([1, -1, 1, -677030, 207796722]\) | \(3013001140430737/108679952667\) | \(1237932585847546875\) | \([2]\) | \(524288\) | \(2.2417\) | |
32175.j3 | 32175p3 | \([1, -1, 1, -106655, -8945778]\) | \(11779205551777/3763454409\) | \(42868097877515625\) | \([2, 2]\) | \(262144\) | \(1.8951\) | |
32175.j4 | 32175p2 | \([1, -1, 1, -96530, -11517528]\) | \(8732907467857/1656369\) | \(18867078140625\) | \([2, 2]\) | \(131072\) | \(1.5485\) | |
32175.j5 | 32175p1 | \([1, -1, 1, -5405, -218028]\) | \(-1532808577/938223\) | \(-10686946359375\) | \([2]\) | \(65536\) | \(1.2020\) | \(\Gamma_0(N)\)-optimal |
32175.j6 | 32175p5 | \([1, -1, 1, 301720, -61217778]\) | \(266679605718863/296110251723\) | \(-3372880836032296875\) | \([2]\) | \(524288\) | \(2.2417\) |
Rank
sage: E.rank()
The elliptic curves in class 32175.j have rank \(1\).
Complex multiplication
The elliptic curves in class 32175.j do not have complex multiplication.Modular form 32175.2.a.j
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.