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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 53361bp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
53361.q4 | 53361bp1 | \([1, -1, 1, -1815386, -940703200]\) | \(4354703137/1617\) | \(245686842692252577\) | \([2]\) | \(921600\) | \(2.3043\) | \(\Gamma_0(N)\)-optimal |
53361.q3 | 53361bp2 | \([1, -1, 1, -2082191, -645830314]\) | \(6570725617/2614689\) | \(397275624633372417009\) | \([2, 2]\) | \(1843200\) | \(2.6509\) | |
53361.q6 | 53361bp3 | \([1, -1, 1, 6722374, -4681842910]\) | \(221115865823/190238433\) | \(-28904811355900823431473\) | \([2]\) | \(3686400\) | \(2.9974\) | |
53361.q2 | 53361bp4 | \([1, -1, 1, -15155636, 22258845326]\) | \(2533811507137/58110129\) | \(8829248065831480859649\) | \([2, 2]\) | \(3686400\) | \(2.9974\) | |
53361.q5 | 53361bp5 | \([1, -1, 1, 1653079, 68913114680]\) | \(3288008303/13504609503\) | \(-2051889221140297004224143\) | \([2]\) | \(7372800\) | \(3.3440\) | |
53361.q1 | 53361bp6 | \([1, -1, 1, -241139471, 1441346935592]\) | \(10206027697760497/5557167\) | \(844355482092502892127\) | \([2]\) | \(7372800\) | \(3.3440\) |
Rank
sage: E.rank()
The elliptic curves in class 53361bp have rank \(0\).
Complex multiplication
The elliptic curves in class 53361bp do not have complex multiplication.Modular form 53361.2.a.bp
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.