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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 104742.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
104742.p1 | 104742a4 | \([1, -1, 0, -4878537, 4148676557]\) | \(4406910829875/7744\) | \(22564392882280128\) | \([2]\) | \(2433024\) | \(2.3971\) | |
104742.p2 | 104742a3 | \([1, -1, 0, -307977, 63510029]\) | \(1108717875/45056\) | \(131283740405993472\) | \([2]\) | \(1216512\) | \(2.0505\) | |
104742.p3 | 104742a2 | \([1, -1, 0, -77862, 2109528]\) | \(13060888875/7086244\) | \(28323497615694732\) | \([2]\) | \(811008\) | \(1.8478\) | |
104742.p4 | 104742a1 | \([1, -1, 0, -46122, -3775068]\) | \(2714704875/21296\) | \(85119451887888\) | \([2]\) | \(405504\) | \(1.5012\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 104742.p have rank \(2\).
Complex multiplication
The elliptic curves in class 104742.p do not have complex multiplication.Modular form 104742.2.a.p
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.