Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 6*x^18 + 8*x^17 - x^16 + 7*x^15 + 6*x^14 - 10*x^13 + 11*x^12 + 15*x^11 - x^10 + 15*x^9 + 11*x^8 - 10*x^7 + 6*x^6 + 7*x^5 - x^4 + 8*x^3 + 6*x^2 - 2*x + 1)
gp: K = bnfinit(y^20 - 2*y^19 + 6*y^18 + 8*y^17 - y^16 + 7*y^15 + 6*y^14 - 10*y^13 + 11*y^12 + 15*y^11 - y^10 + 15*y^9 + 11*y^8 - 10*y^7 + 6*y^6 + 7*y^5 - y^4 + 8*y^3 + 6*y^2 - 2*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 2*x^19 + 6*x^18 + 8*x^17 - x^16 + 7*x^15 + 6*x^14 - 10*x^13 + 11*x^12 + 15*x^11 - x^10 + 15*x^9 + 11*x^8 - 10*x^7 + 6*x^6 + 7*x^5 - x^4 + 8*x^3 + 6*x^2 - 2*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 2*x^19 + 6*x^18 + 8*x^17 - x^16 + 7*x^15 + 6*x^14 - 10*x^13 + 11*x^12 + 15*x^11 - x^10 + 15*x^9 + 11*x^8 - 10*x^7 + 6*x^6 + 7*x^5 - x^4 + 8*x^3 + 6*x^2 - 2*x + 1)
\( x^{20} - 2 x^{19} + 6 x^{18} + 8 x^{17} - x^{16} + 7 x^{15} + 6 x^{14} - 10 x^{13} + 11 x^{12} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(39479125871598264344535849\)
\(\medspace = 3^{10}\cdot 401^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(19.05\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $3^{1/2}401^{1/2}\approx 34.68429039204925$ | ||
| Ramified primes: |
\(3\), \(401\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{10}+\frac{1}{3}a^{8}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{14}+\frac{1}{3}a^{11}+\frac{1}{3}a^{9}+\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a$, $\frac{1}{9}a^{15}+\frac{1}{9}a^{14}+\frac{1}{9}a^{13}+\frac{1}{9}a^{12}+\frac{1}{9}a^{11}-\frac{4}{9}a^{10}+\frac{4}{9}a^{9}+\frac{1}{9}a^{8}+\frac{1}{9}a^{7}-\frac{2}{9}a^{6}-\frac{4}{9}a^{5}-\frac{2}{9}a^{4}+\frac{1}{9}a^{3}+\frac{1}{9}a^{2}+\frac{4}{9}a+\frac{4}{9}$, $\frac{1}{9}a^{16}+\frac{4}{9}a^{11}-\frac{1}{9}a^{10}-\frac{1}{3}a^{9}-\frac{1}{3}a^{7}-\frac{2}{9}a^{6}+\frac{2}{9}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}-\frac{4}{9}$, $\frac{1}{9}a^{17}+\frac{1}{9}a^{12}+\frac{2}{9}a^{11}+\frac{1}{3}a^{10}-\frac{1}{3}a^{8}+\frac{4}{9}a^{7}-\frac{4}{9}a^{6}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{9}a-\frac{1}{3}$, $\frac{1}{27}a^{18}-\frac{1}{27}a^{16}+\frac{1}{9}a^{14}+\frac{4}{27}a^{13}+\frac{2}{27}a^{12}+\frac{11}{27}a^{11}+\frac{4}{27}a^{10}+\frac{4}{9}a^{9}-\frac{11}{27}a^{8}+\frac{8}{27}a^{7}+\frac{5}{27}a^{6}+\frac{1}{27}a^{5}+\frac{4}{9}a^{4}-\frac{4}{27}a^{2}+\frac{7}{27}$, $\frac{1}{27}a^{19}-\frac{1}{27}a^{17}+\frac{1}{27}a^{14}-\frac{1}{27}a^{13}-\frac{1}{27}a^{12}+\frac{10}{27}a^{11}-\frac{4}{9}a^{10}+\frac{4}{27}a^{9}+\frac{5}{27}a^{8}-\frac{7}{27}a^{7}-\frac{11}{27}a^{6}-\frac{4}{9}a^{5}+\frac{2}{9}a^{4}-\frac{7}{27}a^{3}-\frac{4}{9}a^{2}+\frac{4}{27}a+\frac{2}{9}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{22}{27} a^{19} + \frac{7}{27} a^{18} - \frac{95}{27} a^{17} - \frac{364}{27} a^{16} - \frac{148}{9} a^{15} - \frac{616}{27} a^{14} - \frac{799}{27} a^{13} - \frac{181}{9} a^{12} - \frac{485}{27} a^{11} - \frac{617}{27} a^{10} - \frac{475}{27} a^{9} - \frac{631}{27} a^{8} - 30 a^{7} - \frac{614}{27} a^{6} - \frac{521}{27} a^{5} - \frac{143}{9} a^{4} - \frac{182}{27} a^{3} - \frac{109}{27} a^{2} - \frac{64}{27} a + \frac{10}{27} \)
(order $6$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: |
$\frac{37}{27}a^{19}-\frac{37}{27}a^{18}+\frac{188}{27}a^{17}+\frac{466}{27}a^{16}+\frac{154}{9}a^{15}+\frac{667}{27}a^{14}+\frac{763}{27}a^{13}+9a^{12}+\frac{287}{27}a^{11}+\frac{497}{27}a^{10}+\frac{301}{27}a^{9}+\frac{568}{27}a^{8}+\frac{278}{9}a^{7}+\frac{389}{27}a^{6}+\frac{275}{27}a^{5}+\frac{68}{9}a^{4}-\frac{67}{27}a^{3}-\frac{68}{27}a^{2}+\frac{61}{27}a-\frac{22}{27}$, $\frac{25}{27}a^{19}-\frac{7}{9}a^{18}+\frac{122}{27}a^{17}+\frac{115}{9}a^{16}+\frac{115}{9}a^{15}+\frac{550}{27}a^{14}+\frac{650}{27}a^{13}+\frac{362}{27}a^{12}+\frac{478}{27}a^{11}+22a^{10}+\frac{337}{27}a^{9}+\frac{620}{27}a^{8}+\frac{716}{27}a^{7}+\frac{376}{27}a^{6}+17a^{5}+\frac{44}{3}a^{4}+\frac{107}{27}a^{3}+\frac{46}{9}a^{2}+\frac{100}{27}a-\frac{10}{9}$, $\frac{10}{27}a^{19}+\frac{4}{27}a^{18}+\frac{11}{27}a^{17}+\frac{212}{27}a^{16}+\frac{58}{9}a^{15}-\frac{11}{27}a^{14}-\frac{4}{3}a^{13}-\frac{320}{27}a^{12}-\frac{55}{3}a^{11}-\frac{170}{27}a^{10}-\frac{98}{27}a^{9}-\frac{8}{3}a^{8}+\frac{58}{27}a^{7}-8a^{6}-\frac{461}{27}a^{5}-\frac{122}{9}a^{4}-\frac{346}{27}a^{3}-\frac{187}{27}a^{2}-\frac{32}{27}a-\frac{8}{27}$, $\frac{10}{27}a^{19}+\frac{2}{3}a^{18}+\frac{23}{27}a^{17}+\frac{29}{3}a^{16}+\frac{164}{9}a^{15}+\frac{466}{27}a^{14}+\frac{707}{27}a^{13}+\frac{479}{27}a^{12}+\frac{64}{27}a^{11}+\frac{29}{3}a^{10}+\frac{271}{27}a^{9}+\frac{236}{27}a^{8}+\frac{707}{27}a^{7}+\frac{556}{27}a^{6}+7a^{5}+\frac{58}{9}a^{4}-\frac{82}{27}a^{3}-\frac{65}{9}a^{2}-\frac{14}{27}a-\frac{11}{9}$, $\frac{1}{3}a^{19}-\frac{7}{9}a^{18}+\frac{23}{9}a^{17}+\frac{10}{9}a^{16}+\frac{10}{9}a^{15}+\frac{34}{9}a^{14}-\frac{4}{3}a^{13}-\frac{29}{9}a^{12}+2a^{11}-\frac{23}{9}a^{10}+\frac{1}{9}a^{9}+\frac{17}{3}a^{8}-\frac{2}{9}a^{7}-2a^{6}+\frac{10}{9}a^{5}-\frac{35}{9}a^{4}-\frac{14}{9}a^{3}+\frac{2}{9}a^{2}-\frac{1}{9}a-\frac{1}{3}$, $\frac{13}{27}a^{19}+\frac{1}{9}a^{18}+\frac{74}{27}a^{17}+\frac{68}{9}a^{16}+\frac{166}{9}a^{15}+\frac{646}{27}a^{14}+\frac{641}{27}a^{13}+\frac{524}{27}a^{12}+\frac{304}{27}a^{11}+\frac{56}{9}a^{10}+\frac{316}{27}a^{9}+\frac{494}{27}a^{8}+\frac{617}{27}a^{7}+\frac{616}{27}a^{6}+\frac{137}{9}a^{5}+4a^{4}-\frac{25}{27}a^{3}-\frac{31}{9}a^{2}-\frac{23}{27}a-\frac{5}{9}$, $\frac{28}{27}a^{19}-\frac{16}{27}a^{18}+\frac{131}{27}a^{17}+\frac{409}{27}a^{16}+\frac{58}{3}a^{15}+\frac{628}{27}a^{14}+\frac{826}{27}a^{13}+\frac{61}{3}a^{12}+\frac{392}{27}a^{11}+\frac{593}{27}a^{10}+\frac{523}{27}a^{9}+\frac{568}{27}a^{8}+\frac{284}{9}a^{7}+\frac{611}{27}a^{6}+\frac{452}{27}a^{5}+\frac{124}{9}a^{4}+\frac{119}{27}a^{3}+\frac{43}{27}a^{2}+\frac{52}{27}a-\frac{40}{27}$, $\frac{5}{27}a^{19}-\frac{2}{9}a^{18}+\frac{28}{27}a^{17}+\frac{14}{9}a^{16}+\frac{10}{3}a^{15}-\frac{4}{27}a^{14}+\frac{25}{27}a^{13}+\frac{34}{27}a^{12}-\frac{130}{27}a^{11}-\frac{10}{9}a^{10}+\frac{92}{27}a^{9}-\frac{116}{27}a^{8}+\frac{40}{27}a^{7}+\frac{62}{27}a^{6}-\frac{46}{9}a^{5}-\frac{20}{9}a^{4}+\frac{10}{27}a^{3}-\frac{16}{3}a^{2}-\frac{13}{27}a-\frac{1}{9}$, $\frac{32}{27}a^{19}-\frac{14}{9}a^{18}+\frac{172}{27}a^{17}+13a^{16}+\frac{92}{9}a^{15}+\frac{434}{27}a^{14}+\frac{463}{27}a^{13}+\frac{22}{27}a^{12}+\frac{212}{27}a^{11}+\frac{113}{9}a^{10}+\frac{107}{27}a^{9}+\frac{430}{27}a^{8}+\frac{532}{27}a^{7}+\frac{71}{27}a^{6}+\frac{53}{9}a^{5}+\frac{11}{3}a^{4}-\frac{137}{27}a^{3}-\frac{7}{9}a^{2}+\frac{56}{27}a-\frac{8}{3}$
| sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
| Regulator: | \( 74630.3756382 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 74630.3756382 \cdot 2}{6\cdot\sqrt{39479125871598264344535849}}\cr\approx \mathstrut & 0.379672631472 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 6*x^18 + 8*x^17 - x^16 + 7*x^15 + 6*x^14 - 10*x^13 + 11*x^12 + 15*x^11 - x^10 + 15*x^9 + 11*x^8 - 10*x^7 + 6*x^6 + 7*x^5 - x^4 + 8*x^3 + 6*x^2 - 2*x + 1)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^20 - 2*x^19 + 6*x^18 + 8*x^17 - x^16 + 7*x^15 + 6*x^14 - 10*x^13 + 11*x^12 + 15*x^11 - x^10 + 15*x^9 + 11*x^8 - 10*x^7 + 6*x^6 + 7*x^5 - x^4 + 8*x^3 + 6*x^2 - 2*x + 1, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^20 - 2*x^19 + 6*x^18 + 8*x^17 - x^16 + 7*x^15 + 6*x^14 - 10*x^13 + 11*x^12 + 15*x^11 - x^10 + 15*x^9 + 11*x^8 - 10*x^7 + 6*x^6 + 7*x^5 - x^4 + 8*x^3 + 6*x^2 - 2*x + 1);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 2*x^19 + 6*x^18 + 8*x^17 - x^16 + 7*x^15 + 6*x^14 - 10*x^13 + 11*x^12 + 15*x^11 - x^10 + 15*x^9 + 11*x^8 - 10*x^7 + 6*x^6 + 7*x^5 - x^4 + 8*x^3 + 6*x^2 - 2*x + 1);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
$C_2\wr D_5$ (as 20T73):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A solvable group of order 320 |
| The 20 conjugacy class representatives for $C_2\wr D_5$ |
| Character table for $C_2\wr D_5$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 5.5.160801.1, 10.8.698137963227.1, 10.0.6283241669043.1, 10.2.232712654409.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Sibling fields
| Degree 10 siblings: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 10.4.77570884803.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }^{4}$ | ${\href{/padicField/11.10.0.1}{10} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{10}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.5.0.1}{5} }^{4}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{10}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{6}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(401\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |