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SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 16830.bk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
16830.bk1 | 16830cd8 | \([1, -1, 1, -1811128343, 29667338187311]\) | \(901247067798311192691198986281/552431869440\) | \(402722832821760\) | \([6]\) | \(5308416\) | \(3.5101\) | |
16830.bk2 | 16830cd7 | \([1, -1, 1, -113955863, 457030494767]\) | \(224494757451893010998773801/6152490825146276160000\) | \(4485165811531635320640000\) | \([6]\) | \(5308416\) | \(3.5101\) | |
16830.bk3 | 16830cd6 | \([1, -1, 1, -113195543, 463573200431]\) | \(220031146443748723000125481/172266701724057600\) | \(125582425556837990400\) | \([2, 6]\) | \(2654208\) | \(3.1635\) | |
16830.bk4 | 16830cd5 | \([1, -1, 1, -22364168, 40683483731]\) | \(1696892787277117093383481/1440538624914939000\) | \(1050152657562990531000\) | \([2]\) | \(1769472\) | \(2.9608\) | |
16830.bk5 | 16830cd4 | \([1, -1, 1, -14646488, -21340722733]\) | \(476646772170172569823801/5862293314453125000\) | \(4273611826236328125000\) | \([2]\) | \(1769472\) | \(2.9608\) | |
16830.bk6 | 16830cd3 | \([1, -1, 1, -7027223, 7346695727]\) | \(-52643812360427830814761/1504091705903677440\) | \(-1096482853603780853760\) | \([6]\) | \(1327104\) | \(2.8169\) | |
16830.bk7 | 16830cd2 | \([1, -1, 1, -1709168, 331875731]\) | \(757443433548897303481/373234243041000000\) | \(272087763176889000000\) | \([2, 2]\) | \(884736\) | \(2.6142\) | |
16830.bk8 | 16830cd1 | \([1, -1, 1, 390352, 39622547]\) | \(9023321954633914439/6156756739584000\) | \(-4488275663156736000\) | \([2]\) | \(442368\) | \(2.2676\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 16830.bk have rank \(0\).
Complex multiplication
The elliptic curves in class 16830.bk do not have complex multiplication.Modular form 16830.2.a.bk
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 3 & 12 & 4 & 6 & 12 \\ 4 & 1 & 2 & 12 & 3 & 4 & 6 & 12 \\ 2 & 2 & 1 & 6 & 6 & 2 & 3 & 6 \\ 3 & 12 & 6 & 1 & 4 & 12 & 2 & 4 \\ 12 & 3 & 6 & 4 & 1 & 12 & 2 & 4 \\ 4 & 4 & 2 & 12 & 12 & 1 & 6 & 3 \\ 6 & 6 & 3 & 2 & 2 & 6 & 1 & 2 \\ 12 & 12 & 6 & 4 & 4 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.