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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 139650.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
139650.j1 | 139650ip5 | \([1, 1, 0, -4245421275, 106468732963125]\) | \(4603390551972799451373601/3745967689800\) | \(6886083636520003125000\) | \([2]\) | \(84934656\) | \(3.9280\) | |
139650.j2 | 139650ip3 | \([1, 1, 0, -265396275, 1662734638125]\) | \(1124604760397601117601/1013798336040000\) | \(1863630631824530625000000\) | \([2, 2]\) | \(42467328\) | \(3.5814\) | |
139650.j3 | 139650ip6 | \([1, 1, 0, -205371275, 2435076313125]\) | \(-521116167586355661601/1092005739697609800\) | \(-2007396613588813990003125000\) | \([2]\) | \(84934656\) | \(3.9280\) | |
139650.j4 | 139650ip4 | \([1, 1, 0, -176804275, -895737249875]\) | \(332501596620668284321/3896484375000000\) | \(7162773284912109375000000\) | \([2]\) | \(42467328\) | \(3.5814\) | |
139650.j5 | 139650ip2 | \([1, 1, 0, -20396275, 13149638125]\) | \(510467451652317601/254721600000000\) | \(468245961225000000000000\) | \([2, 2]\) | \(21233664\) | \(3.2348\) | |
139650.j6 | 139650ip1 | \([1, 1, 0, 4691725, 1584070125]\) | \(6213165856218719/4183818240000\) | \(-7690969251840000000000\) | \([2]\) | \(10616832\) | \(2.8882\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 139650.j have rank \(1\).
Complex multiplication
The elliptic curves in class 139650.j do not have complex multiplication.Modular form 139650.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.