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SageMath
E = EllipticCurve("fk1")
E.isogeny_class()
Elliptic curves in class 129360.fk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
129360.fk1 | 129360cb6 | \([0, 1, 0, -82559136, -288751969836]\) | \(258286045443018193442/8440380939375\) | \(2033668868375612160000\) | \([2]\) | \(12582912\) | \(3.1826\) | |
129360.fk2 | 129360cb4 | \([0, 1, 0, -23324016, 43348202820]\) | \(11647843478225136004/128410942275\) | \(15469997002456550400\) | \([2]\) | \(6291456\) | \(2.8360\) | |
129360.fk3 | 129360cb3 | \([0, 1, 0, -5384136, -4099699836]\) | \(143279368983686884/22699269140625\) | \(2734639426688400000000\) | \([2, 2]\) | \(6291456\) | \(2.8360\) | |
129360.fk4 | 129360cb2 | \([0, 1, 0, -1494516, 640969020]\) | \(12257375872392016/1191317675625\) | \(35880277304219040000\) | \([2, 2]\) | \(3145728\) | \(2.4895\) | |
129360.fk5 | 129360cb1 | \([0, 1, 0, 112929, 48143304]\) | \(84611246065664/580054565475\) | \(-1091885433177092400\) | \([2]\) | \(1572864\) | \(2.1429\) | \(\Gamma_0(N)\)-optimal |
129360.fk6 | 129360cb5 | \([0, 1, 0, 9556944, -22788002700]\) | \(400647648358480318/1163177490234375\) | \(-280261977187500000000000\) | \([2]\) | \(12582912\) | \(3.1826\) |
Rank
sage: E.rank()
The elliptic curves in class 129360.fk have rank \(0\).
Complex multiplication
The elliptic curves in class 129360.fk do not have complex multiplication.Modular form 129360.2.a.fk
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.