Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - x^16 - 2*x^15 + 4*x^14 - x^13 + x^12 + x^11 + 4*x^10 - 8*x^9 + 2*x^8 - x^7 - 2*x^6 - x^5 + 8*x^4 - 5*x^3 + 3*x^2 - 2*x + 1)
gp: K = bnfinit(y^20 - y^19 - y^16 - 2*y^15 + 4*y^14 - y^13 + y^12 + y^11 + 4*y^10 - 8*y^9 + 2*y^8 - y^7 - 2*y^6 - y^5 + 8*y^4 - 5*y^3 + 3*y^2 - 2*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - x^19 - x^16 - 2*x^15 + 4*x^14 - x^13 + x^12 + x^11 + 4*x^10 - 8*x^9 + 2*x^8 - x^7 - 2*x^6 - x^5 + 8*x^4 - 5*x^3 + 3*x^2 - 2*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - x^19 - x^16 - 2*x^15 + 4*x^14 - x^13 + x^12 + x^11 + 4*x^10 - 8*x^9 + 2*x^8 - x^7 - 2*x^6 - x^5 + 8*x^4 - 5*x^3 + 3*x^2 - 2*x + 1)
\( x^{20} - x^{19} - x^{16} - 2 x^{15} + 4 x^{14} - x^{13} + x^{12} + x^{11} + 4 x^{10} - 8 x^{9} + \cdots + 1 \)
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: |
\(2462968747589111328125\)
\(\medspace = 5^{15}\cdot 13^{4}\cdot 41^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.74\) | 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: | $5^{3/4}13^{1/2}41^{1/2}\approx 77.19534416036332$ | ||
| Ramified primes: |
\(5\), \(13\), \(41\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5}a^{16}+\frac{2}{5}a^{15}+\frac{2}{5}a^{14}-\frac{1}{5}a^{12}+\frac{2}{5}a^{11}+\frac{2}{5}a^{10}-\frac{2}{5}a^{8}+\frac{2}{5}a^{7}+\frac{1}{5}a^{6}-\frac{2}{5}a^{5}-\frac{2}{5}a^{4}+\frac{1}{5}a^{3}-\frac{1}{5}a^{2}+\frac{1}{5}$, $\frac{1}{5}a^{17}-\frac{2}{5}a^{15}+\frac{1}{5}a^{14}-\frac{1}{5}a^{13}-\frac{1}{5}a^{12}-\frac{2}{5}a^{11}+\frac{1}{5}a^{10}-\frac{2}{5}a^{9}+\frac{1}{5}a^{8}+\frac{2}{5}a^{7}+\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{2}{5}a^{3}+\frac{2}{5}a^{2}+\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{18}-\frac{2}{5}a^{14}-\frac{1}{5}a^{13}+\frac{1}{5}a^{12}+\frac{2}{5}a^{10}+\frac{1}{5}a^{9}-\frac{2}{5}a^{8}-\frac{1}{5}a^{6}+\frac{1}{5}a^{5}-\frac{2}{5}a^{4}-\frac{1}{5}a^{3}-\frac{1}{5}a^{2}-\frac{2}{5}a+\frac{2}{5}$, $\frac{1}{25}a^{19}+\frac{1}{25}a^{18}+\frac{2}{25}a^{17}-\frac{1}{25}a^{16}-\frac{8}{25}a^{15}+\frac{7}{25}a^{14}+\frac{3}{25}a^{13}+\frac{2}{5}a^{12}+\frac{6}{25}a^{11}-\frac{12}{25}a^{10}-\frac{2}{5}a^{9}+\frac{7}{25}a^{8}-\frac{4}{25}a^{7}-\frac{4}{25}a^{6}+\frac{1}{5}a^{5}+\frac{9}{25}a^{4}+\frac{11}{25}a^{3}+\frac{2}{25}a^{2}+\frac{2}{25}a-\frac{3}{25}$
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 $1$
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: |
\( -\frac{1}{5} a^{19} - \frac{1}{5} a^{18} + \frac{4}{5} a^{17} + \frac{1}{5} a^{16} + \frac{1}{5} a^{15} + \frac{4}{5} a^{14} + \frac{1}{5} a^{13} - \frac{11}{5} a^{12} - \frac{3}{5} a^{11} - \frac{2}{5} a^{10} - \frac{7}{5} a^{9} - \frac{1}{5} a^{8} + \frac{21}{5} a^{7} + a^{6} + \frac{2}{5} a^{5} + \frac{6}{5} a^{4} - \frac{4}{5} a^{3} - 2 a^{2} - \frac{1}{5} a + \frac{1}{5} \)
(order $10$)
| 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: |
$a$, $\frac{2}{5}a^{19}+\frac{2}{5}a^{17}+\frac{2}{5}a^{16}-\frac{4}{5}a^{15}-\frac{6}{5}a^{14}-\frac{4}{5}a^{12}-\frac{1}{5}a^{11}+\frac{8}{5}a^{10}+\frac{12}{5}a^{9}-\frac{2}{5}a^{8}+\frac{6}{5}a^{7}+\frac{1}{5}a^{6}-\frac{14}{5}a^{5}-\frac{6}{5}a^{4}+\frac{4}{5}a^{3}-\frac{2}{5}a^{2}+\frac{11}{5}a-\frac{2}{5}$, $\frac{8}{25}a^{19}-\frac{7}{25}a^{18}-\frac{4}{25}a^{17}-\frac{3}{25}a^{16}-\frac{14}{25}a^{15}-\frac{24}{25}a^{14}+\frac{34}{25}a^{13}+\frac{1}{5}a^{12}+\frac{23}{25}a^{11}+\frac{14}{25}a^{10}+\frac{9}{5}a^{9}-\frac{69}{25}a^{8}-\frac{12}{25}a^{7}-\frac{32}{25}a^{6}-a^{5}-\frac{33}{25}a^{4}+\frac{93}{25}a^{3}-\frac{14}{25}a^{2}+\frac{51}{25}a-\frac{34}{25}$, $\frac{31}{25}a^{19}-\frac{9}{25}a^{18}-\frac{8}{25}a^{17}-\frac{1}{25}a^{16}-\frac{23}{25}a^{15}-\frac{88}{25}a^{14}+\frac{53}{25}a^{13}+\frac{2}{5}a^{12}+\frac{11}{25}a^{11}+\frac{38}{25}a^{10}+\frac{33}{5}a^{9}-\frac{108}{25}a^{8}-\frac{29}{25}a^{7}+\frac{1}{25}a^{6}-\frac{12}{5}a^{5}-\frac{101}{25}a^{4}+\frac{146}{25}a^{3}-\frac{43}{25}a^{2}-\frac{3}{25}a-\frac{3}{25}$, $\frac{23}{25}a^{19}-\frac{12}{25}a^{18}-\frac{4}{25}a^{17}+\frac{7}{25}a^{16}+\frac{1}{25}a^{15}-\frac{59}{25}a^{14}+\frac{54}{25}a^{13}-\frac{2}{5}a^{12}-\frac{2}{25}a^{11}-\frac{11}{25}a^{10}+\frac{22}{5}a^{9}-\frac{104}{25}a^{8}+\frac{18}{25}a^{7}+\frac{23}{25}a^{6}+\frac{4}{5}a^{5}-\frac{58}{25}a^{4}+\frac{118}{25}a^{3}-\frac{74}{25}a^{2}+\frac{16}{25}a-\frac{9}{25}$, $\frac{3}{25}a^{19}+\frac{3}{25}a^{18}-\frac{9}{25}a^{17}+\frac{12}{25}a^{16}-\frac{14}{25}a^{15}-\frac{14}{25}a^{14}-\frac{1}{25}a^{13}+\frac{1}{5}a^{12}-\frac{22}{25}a^{11}+\frac{29}{25}a^{10}+a^{9}+\frac{1}{25}a^{8}-\frac{12}{25}a^{7}+\frac{38}{25}a^{6}-\frac{9}{5}a^{5}-\frac{28}{25}a^{4}+\frac{18}{25}a^{3}-\frac{39}{25}a^{2}-\frac{9}{25}a+\frac{11}{25}$, $\frac{6}{25}a^{19}-\frac{14}{25}a^{18}-\frac{3}{25}a^{17}-\frac{1}{25}a^{16}-\frac{8}{25}a^{15}+\frac{2}{25}a^{14}+\frac{53}{25}a^{13}+\frac{1}{25}a^{11}+\frac{8}{25}a^{10}-\frac{93}{25}a^{8}+\frac{6}{25}a^{7}+\frac{11}{25}a^{6}-\frac{6}{5}a^{5}+\frac{34}{25}a^{4}+\frac{86}{25}a^{3}-\frac{28}{25}a^{2}+\frac{12}{25}a+\frac{2}{25}$, $\frac{3}{25}a^{19}+\frac{3}{25}a^{18}-\frac{4}{25}a^{17}+\frac{7}{25}a^{16}-\frac{9}{25}a^{15}-\frac{19}{25}a^{14}-\frac{6}{25}a^{13}+\frac{1}{5}a^{12}-\frac{17}{25}a^{11}+\frac{24}{25}a^{10}+\frac{8}{5}a^{9}+\frac{16}{25}a^{8}-\frac{12}{25}a^{7}+\frac{38}{25}a^{6}-2a^{5}-\frac{43}{25}a^{4}-\frac{2}{25}a^{3}+\frac{1}{25}a^{2}-\frac{29}{25}a+\frac{21}{25}$, $\frac{1}{25}a^{19}+\frac{16}{25}a^{18}+\frac{7}{25}a^{17}-\frac{11}{25}a^{16}-\frac{13}{25}a^{15}-\frac{13}{25}a^{14}-\frac{42}{25}a^{13}+\frac{1}{5}a^{12}+\frac{26}{25}a^{11}+\frac{28}{25}a^{10}+\frac{4}{5}a^{9}+\frac{77}{25}a^{8}-\frac{14}{25}a^{7}-\frac{49}{25}a^{6}-2a^{5}-\frac{26}{25}a^{4}-\frac{29}{25}a^{3}+\frac{57}{25}a^{2}+\frac{2}{25}a+\frac{7}{25}$
| 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: | \( 887.733037213 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 887.733037213 \cdot 1}{10\cdot\sqrt{2462968747589111328125}}\cr\approx \mathstrut & 0.171534553169 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - x^16 - 2*x^15 + 4*x^14 - x^13 + x^12 + x^11 + 4*x^10 - 8*x^9 + 2*x^8 - x^7 - 2*x^6 - x^5 + 8*x^4 - 5*x^3 + 3*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 - x^19 - x^16 - 2*x^15 + 4*x^14 - x^13 + x^12 + x^11 + 4*x^10 - 8*x^9 + 2*x^8 - x^7 - 2*x^6 - x^5 + 8*x^4 - 5*x^3 + 3*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 - x^19 - x^16 - 2*x^15 + 4*x^14 - x^13 + x^12 + x^11 + 4*x^10 - 8*x^9 + 2*x^8 - x^7 - 2*x^6 - x^5 + 8*x^4 - 5*x^3 + 3*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 - x^19 - x^16 - 2*x^15 + 4*x^14 - x^13 + x^12 + x^11 + 4*x^10 - 8*x^9 + 2*x^8 - x^7 - 2*x^6 - x^5 + 8*x^4 - 5*x^3 + 3*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
$C_4\times S_5$ (as 20T123):
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 480 |
| The 28 conjugacy class representatives for $C_4\times S_5$ |
| Character table for $C_4\times S_5$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.1.2665.1, 10.2.887778125.1 |
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 24 siblings: | data not computed |
| Degree 40 siblings: | 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 | $20$ | $20$ | R | ${\href{/padicField/7.4.0.1}{4} }^{5}$ | ${\href{/padicField/11.5.0.1}{5} }^{4}$ | R | $20$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{5}$ | ${\href{/padicField/53.4.0.1}{4} }^{5}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.12.9.2 | $x^{12} + 12 x^{10} + 12 x^{9} + 69 x^{8} + 108 x^{7} + 42 x^{6} - 396 x^{5} + 840 x^{4} + 252 x^{3} + 1476 x^{2} + 684 x + 1601$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ | |
|
\(13\)
| 13.8.4.1 | $x^{8} + 520 x^{7} + 101458 x^{6} + 8810644 x^{5} + 288610205 x^{4} + 142111548 x^{3} + 982314112 x^{2} + 3617879976 x + 920156436$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 13.12.0.1 | $x^{12} + x^{8} + 5 x^{7} + 8 x^{6} + 11 x^{5} + 3 x^{4} + x^{3} + x^{2} + 4 x + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
|
\(41\)
| $\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.1.1 | $x^{2} + 41$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 41.2.1.1 | $x^{2} + 41$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 41.2.1.1 | $x^{2} + 41$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.1.1 | $x^{2} + 41$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |