Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 3*x^18 - 6*x^17 + 8*x^16 - 9*x^15 + 11*x^14 - 10*x^13 + 6*x^12 - 5*x^11 + x^10 + 4*x^9 - 4*x^8 + 5*x^7 - 5*x^6 + 3*x^5 - x^4 + x^2 - x + 1)
gp: K = bnfinit(y^20 - 2*y^19 + 3*y^18 - 6*y^17 + 8*y^16 - 9*y^15 + 11*y^14 - 10*y^13 + 6*y^12 - 5*y^11 + y^10 + 4*y^9 - 4*y^8 + 5*y^7 - 5*y^6 + 3*y^5 - y^4 + y^2 - y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 2*x^19 + 3*x^18 - 6*x^17 + 8*x^16 - 9*x^15 + 11*x^14 - 10*x^13 + 6*x^12 - 5*x^11 + x^10 + 4*x^9 - 4*x^8 + 5*x^7 - 5*x^6 + 3*x^5 - x^4 + x^2 - x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 2*x^19 + 3*x^18 - 6*x^17 + 8*x^16 - 9*x^15 + 11*x^14 - 10*x^13 + 6*x^12 - 5*x^11 + x^10 + 4*x^9 - 4*x^8 + 5*x^7 - 5*x^6 + 3*x^5 - x^4 + x^2 - x + 1)
\( x^{20} - 2 x^{19} + 3 x^{18} - 6 x^{17} + 8 x^{16} - 9 x^{15} + 11 x^{14} - 10 x^{13} + 6 x^{12} + \cdots + 1 \)
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: |
\(3301013298634667867241\)
\(\medspace = 3^{10}\cdot 236438047^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.91\) | 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: | $3^{1/2}236438047^{1/2}\approx 26632.95216456486$ | ||
| Ramified primes: |
\(3\), \(236438047\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{59}a^{19}-\frac{26}{59}a^{18}-\frac{22}{59}a^{17}-\frac{9}{59}a^{16}-\frac{12}{59}a^{15}-\frac{16}{59}a^{14}-\frac{18}{59}a^{13}+\frac{9}{59}a^{12}+\frac{26}{59}a^{11}+\frac{20}{59}a^{10}-\frac{7}{59}a^{9}-\frac{5}{59}a^{8}-\frac{2}{59}a^{7}-\frac{6}{59}a^{6}+\frac{21}{59}a^{5}-\frac{29}{59}a^{4}-\frac{13}{59}a^{3}+\frac{17}{59}a^{2}+\frac{6}{59}a-\frac{27}{59}$
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
Trivial group, which has order $1$ (assuming GRH)
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: |
\( \frac{13}{59} a^{19} - \frac{43}{59} a^{18} + \frac{68}{59} a^{17} - \frac{117}{59} a^{16} + \frac{198}{59} a^{15} - \frac{267}{59} a^{14} + \frac{297}{59} a^{13} - \frac{355}{59} a^{12} + \frac{338}{59} a^{11} - \frac{271}{59} a^{10} + \frac{263}{59} a^{9} - \frac{183}{59} a^{8} + \frac{33}{59} a^{7} - \frac{19}{59} a^{6} + \frac{37}{59} a^{5} + \frac{95}{59} a^{4} - \frac{51}{59} a^{3} + \frac{44}{59} a^{2} - \frac{40}{59} a + \frac{62}{59} \)
(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: |
$\frac{17}{59}a^{19}-\frac{29}{59}a^{18}+\frac{39}{59}a^{17}-\frac{94}{59}a^{16}+\frac{150}{59}a^{15}-\frac{154}{59}a^{14}+\frac{225}{59}a^{13}-\frac{260}{59}a^{12}+\frac{206}{59}a^{11}-\frac{250}{59}a^{10}+\frac{235}{59}a^{9}-\frac{85}{59}a^{8}+\frac{84}{59}a^{7}-\frac{102}{59}a^{6}-\frac{56}{59}a^{5}+\frac{38}{59}a^{4}-\frac{44}{59}a^{3}+\frac{53}{59}a^{2}-\frac{75}{59}a+\frac{13}{59}$, $\frac{1}{59}a^{19}+\frac{33}{59}a^{18}-\frac{81}{59}a^{17}+\frac{109}{59}a^{16}-\frac{189}{59}a^{15}+\frac{279}{59}a^{14}-\frac{313}{59}a^{13}+\frac{363}{59}a^{12}-\frac{387}{59}a^{11}+\frac{256}{59}a^{10}-\frac{243}{59}a^{9}+\frac{231}{59}a^{8}-\frac{61}{59}a^{7}-\frac{6}{59}a^{6}+\frac{21}{59}a^{5}-\frac{88}{59}a^{4}+\frac{105}{59}a^{3}-\frac{42}{59}a^{2}+\frac{65}{59}a-\frac{27}{59}$, $\frac{39}{59}a^{19}-\frac{129}{59}a^{18}+\frac{204}{59}a^{17}-\frac{351}{59}a^{16}+\frac{535}{59}a^{15}-\frac{624}{59}a^{14}+\frac{714}{59}a^{13}-\frac{711}{59}a^{12}+\frac{483}{59}a^{11}-\frac{282}{59}a^{10}+\frac{140}{59}a^{9}+\frac{159}{59}a^{8}-\frac{255}{59}a^{7}+\frac{238}{59}a^{6}-\frac{243}{59}a^{5}+\frac{108}{59}a^{4}-\frac{35}{59}a^{3}-\frac{45}{59}a^{2}+\frac{57}{59}a-\frac{50}{59}$, $\frac{13}{59}a^{19}-\frac{43}{59}a^{18}+\frac{68}{59}a^{17}-\frac{117}{59}a^{16}+\frac{139}{59}a^{15}-\frac{149}{59}a^{14}+\frac{179}{59}a^{13}-\frac{119}{59}a^{12}+\frac{43}{59}a^{11}-\frac{35}{59}a^{10}-\frac{32}{59}a^{9}+\frac{53}{59}a^{8}+\frac{33}{59}a^{7}+\frac{40}{59}a^{6}-\frac{22}{59}a^{5}-\frac{82}{59}a^{4}+\frac{8}{59}a^{3}-\frac{15}{59}a^{2}+\frac{19}{59}a+\frac{3}{59}$, $\frac{53}{59}a^{19}-\frac{80}{59}a^{18}+\frac{73}{59}a^{17}-\frac{182}{59}a^{16}+\frac{190}{59}a^{15}-\frac{81}{59}a^{14}+\frac{108}{59}a^{13}+\frac{5}{59}a^{12}-\frac{274}{59}a^{11}+\frac{175}{59}a^{10}-\frac{194}{59}a^{9}+\frac{384}{59}a^{8}-\frac{106}{59}a^{7}-\frac{23}{59}a^{6}-\frac{67}{59}a^{5}-\frac{62}{59}a^{4}+\frac{137}{59}a^{3}-\frac{43}{59}a^{2}+\frac{23}{59}a-\frac{15}{59}$, $\frac{36}{59}a^{19}-\frac{51}{59}a^{18}+\frac{34}{59}a^{17}-\frac{147}{59}a^{16}+\frac{158}{59}a^{15}-\frac{104}{59}a^{14}+\frac{237}{59}a^{13}-\frac{207}{59}a^{12}+\frac{51}{59}a^{11}-\frac{224}{59}a^{10}+\frac{161}{59}a^{9}+\frac{115}{59}a^{8}+\frac{105}{59}a^{7}-\frac{39}{59}a^{6}-\frac{188}{59}a^{5}+\frac{18}{59}a^{4}+\frac{4}{59}a^{3}+\frac{81}{59}a^{2}-\frac{20}{59}a-\frac{28}{59}$, $\frac{14}{59}a^{19}-\frac{69}{59}a^{18}+\frac{105}{59}a^{17}-\frac{126}{59}a^{16}+\frac{186}{59}a^{15}-\frac{224}{59}a^{14}+\frac{161}{59}a^{13}-\frac{110}{59}a^{12}+\frac{10}{59}a^{11}+\frac{162}{59}a^{10}-\frac{157}{59}a^{9}+\frac{166}{59}a^{8}-\frac{264}{59}a^{7}+\frac{211}{59}a^{6}-\frac{60}{59}a^{5}+\frac{66}{59}a^{4}-\frac{5}{59}a^{3}-\frac{57}{59}a^{2}+\frac{25}{59}a-\frac{24}{59}$, $\frac{29}{59}a^{19}-\frac{46}{59}a^{18}+\frac{70}{59}a^{17}-\frac{143}{59}a^{16}+\frac{183}{59}a^{15}-\frac{228}{59}a^{14}+\frac{245}{59}a^{13}-\frac{211}{59}a^{12}+\frac{164}{59}a^{11}-\frac{128}{59}a^{10}+\frac{33}{59}a^{9}+\frac{32}{59}a^{8}-\frac{58}{59}a^{7}+\frac{62}{59}a^{6}+\frac{19}{59}a^{5}+\frac{44}{59}a^{4}-\frac{23}{59}a^{3}-\frac{38}{59}a^{2}+\frac{56}{59}a-\frac{16}{59}$, $\frac{26}{59}a^{19}-\frac{27}{59}a^{18}+\frac{77}{59}a^{17}-\frac{116}{59}a^{16}+\frac{101}{59}a^{15}-\frac{180}{59}a^{14}+\frac{181}{59}a^{13}-\frac{120}{59}a^{12}+\frac{145}{59}a^{11}-\frac{129}{59}a^{10}-\frac{5}{59}a^{9}-\frac{130}{59}a^{8}+\frac{66}{59}a^{7}+\frac{80}{59}a^{6}+\frac{74}{59}a^{5}+\frac{13}{59}a^{4}-\frac{43}{59}a^{3}-\frac{30}{59}a^{2}-\frac{21}{59}a+\frac{65}{59}$
| 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: | \( 622.109043514 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 622.109043514 \cdot 1}{6\cdot\sqrt{3301013298634667867241}}\cr\approx \mathstrut & 0.173057454298 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 3*x^18 - 6*x^17 + 8*x^16 - 9*x^15 + 11*x^14 - 10*x^13 + 6*x^12 - 5*x^11 + x^10 + 4*x^9 - 4*x^8 + 5*x^7 - 5*x^6 + 3*x^5 - x^4 + x^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 + 3*x^18 - 6*x^17 + 8*x^16 - 9*x^15 + 11*x^14 - 10*x^13 + 6*x^12 - 5*x^11 + x^10 + 4*x^9 - 4*x^8 + 5*x^7 - 5*x^6 + 3*x^5 - x^4 + x^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 + 3*x^18 - 6*x^17 + 8*x^16 - 9*x^15 + 11*x^14 - 10*x^13 + 6*x^12 - 5*x^11 + x^10 + 4*x^9 - 4*x^8 + 5*x^7 - 5*x^6 + 3*x^5 - x^4 + x^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 + 3*x^18 - 6*x^17 + 8*x^16 - 9*x^15 + 11*x^14 - 10*x^13 + 6*x^12 - 5*x^11 + x^10 + 4*x^9 - 4*x^8 + 5*x^7 - 5*x^6 + 3*x^5 - x^4 + x^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
$C_2\times S_{10}$ (as 20T1021):
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 non-solvable group of order 7257600 |
| The 84 conjugacy class representatives for $C_2\times S_{10}$ |
| Character table for $C_2\times S_{10}$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 10.0.236438047.1 |
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 20 sibling: | data not computed |
| Degree 40 sibling: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
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 | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.9.0.1}{9} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.10.0.1}{10} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.9.0.1}{9} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.6.0.1}{6} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.20.10.1 | $x^{20} + 30 x^{18} + 409 x^{16} + 4 x^{15} + 3244 x^{14} - 60 x^{13} + 16162 x^{12} - 1250 x^{11} + 53008 x^{10} - 7102 x^{9} + 121150 x^{8} - 12140 x^{7} + 219264 x^{6} + 5736 x^{5} + 257465 x^{4} + 35250 x^{3} + 250183 x^{2} + 51502 x + 77812$ | $2$ | $10$ | $10$ | 20T3 | $[\ ]_{2}^{10}$ |
|
\(236438047\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |