Properties

Label 21.17.382...853.1
Degree $21$
Signature $[17, 2]$
Discriminant $3.829\times 10^{34}$
Root discriminant \(44.34\)
Ramified primes $7,13,97,8914877,5021940620101$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_7\wr C_3$ (as 21T159)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 21*x^19 - 3*x^18 + 186*x^17 + 55*x^16 - 897*x^15 - 410*x^14 + 2525*x^13 + 1584*x^12 - 4093*x^11 - 3339*x^10 + 3442*x^9 + 3694*x^8 - 1060*x^7 - 1902*x^6 - 106*x^5 + 410*x^4 + 98*x^3 - 25*x^2 - 11*x - 1)
 
gp: K = bnfinit(y^21 - 21*y^19 - 3*y^18 + 186*y^17 + 55*y^16 - 897*y^15 - 410*y^14 + 2525*y^13 + 1584*y^12 - 4093*y^11 - 3339*y^10 + 3442*y^9 + 3694*y^8 - 1060*y^7 - 1902*y^6 - 106*y^5 + 410*y^4 + 98*y^3 - 25*y^2 - 11*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 21*x^19 - 3*x^18 + 186*x^17 + 55*x^16 - 897*x^15 - 410*x^14 + 2525*x^13 + 1584*x^12 - 4093*x^11 - 3339*x^10 + 3442*x^9 + 3694*x^8 - 1060*x^7 - 1902*x^6 - 106*x^5 + 410*x^4 + 98*x^3 - 25*x^2 - 11*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 21*x^19 - 3*x^18 + 186*x^17 + 55*x^16 - 897*x^15 - 410*x^14 + 2525*x^13 + 1584*x^12 - 4093*x^11 - 3339*x^10 + 3442*x^9 + 3694*x^8 - 1060*x^7 - 1902*x^6 - 106*x^5 + 410*x^4 + 98*x^3 - 25*x^2 - 11*x - 1)
 

\( x^{21} - 21 x^{19} - 3 x^{18} + 186 x^{17} + 55 x^{16} - 897 x^{15} - 410 x^{14} + 2525 x^{13} + \cdots - 1 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $21$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[17, 2]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(38289048631639269472806869479351853\) \(\medspace = 7^{14}\cdot 13\cdot 97\cdot 8914877\cdot 5021940620101\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(44.34\)
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:  $7^{2/3}13^{1/2}97^{1/2}8914877^{1/2}5021940620101^{1/2}\approx 869460188568.4381$
Ramified primes:   \(7\), \(13\), \(97\), \(8914877\), \(5021940620101\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  $\Q(\sqrt{56454\!\cdots\!89597}$)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Yes
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:  $18$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
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:   $a^{19}-a^{18}-20a^{17}+17a^{16}+169a^{15}-114a^{14}-783a^{13}+373a^{12}+2152a^{11}-568a^{10}-3525a^{9}+186a^{8}+3256a^{7}+438a^{6}-1499a^{5}-403a^{4}+304a^{3}+107a^{2}-22a-7$, $a^{20}+3a^{19}-24a^{18}-64a^{17}+239a^{16}+580a^{15}-1274a^{14}-2893a^{13}+3892a^{12}+8575a^{11}-6706a^{10}-15156a^{9}+5823a^{8}+15153a^{7}-1687a^{6}-7670a^{5}-335a^{4}+1763a^{3}+226a^{2}-142a-26$, $a$, $a+1$, $14a^{20}-5a^{19}-292a^{18}+62a^{17}+2578a^{16}-146a^{15}-12477a^{14}-1315a^{13}+35703a^{12}+9529a^{11}-60428a^{10}-25351a^{9}+56848a^{8}+31585a^{7}-25797a^{6}-17485a^{5}+4620a^{4}+4098a^{3}-63a^{2}-326a-40$, $a^{20}-21a^{18}-3a^{17}+186a^{16}+55a^{15}-897a^{14}-410a^{13}+2525a^{12}+1584a^{11}-4093a^{10}-3339a^{9}+3442a^{8}+3694a^{7}-1060a^{6}-1902a^{5}-106a^{4}+411a^{3}+98a^{2}-29a-11$, $a^{20}-21a^{18}-3a^{17}+186a^{16}+55a^{15}-897a^{14}-410a^{13}+2525a^{12}+1584a^{11}-4093a^{10}-3339a^{9}+3442a^{8}+3694a^{7}-1060a^{6}-1902a^{5}-106a^{4}+410a^{3}+98a^{2}-26a-10$, $a^{20}+2a^{19}-23a^{18}-44a^{17}+222a^{16}+411a^{15}-1160a^{14}-2110a^{13}+3519a^{12}+6423a^{11}-6138a^{10}-11631a^{9}+5637a^{8}+11897a^{7}-2125a^{6}-6171a^{5}+68a^{4}+1459a^{3}+119a^{2}-121a-18$, $2a^{20}-2a^{19}-41a^{18}+36a^{17}+356a^{16}-263a^{15}-1700a^{14}+994a^{13}+4839a^{12}-2045a^{11}-8292a^{10}+2195a^{9}+8203a^{8}-1065a^{7}-4269a^{6}+174a^{5}+1049a^{4}+21a^{3}-95a^{2}-8a+2$, $a^{20}-21a^{18}-3a^{17}+186a^{16}+55a^{15}-897a^{14}-410a^{13}+2525a^{12}+1584a^{11}-4093a^{10}-3339a^{9}+3442a^{8}+3694a^{7}-1060a^{6}-1902a^{5}-106a^{4}+411a^{3}+97a^{2}-28a-10$, $8a^{20}-2a^{19}-167a^{18}+17a^{17}+1474a^{16}+85a^{15}-7117a^{14}-1600a^{13}+20237a^{12}+7995a^{11}-33760a^{10}-19094a^{9}+30689a^{8}+22854a^{7}-12613a^{6}-12656a^{5}+1464a^{4}+3076a^{3}+254a^{2}-268a-50$, $a^{20}-a^{19}-20a^{18}+17a^{17}+169a^{16}-114a^{15}-783a^{14}+373a^{13}+2152a^{12}-568a^{11}-3525a^{10}+186a^{9}+3256a^{8}+438a^{7}-1499a^{6}-403a^{5}+304a^{4}+107a^{3}-22a^{2}-7a+1$, $3a^{20}-6a^{19}-58a^{18}+114a^{17}+472a^{16}-902a^{15}-2106a^{14}+3850a^{13}+5624a^{12}-9604a^{11}-9204a^{10}+14198a^{9}+9078a^{8}-12086a^{7}-5141a^{6}+5538a^{5}+1583a^{4}-1219a^{3}-250a^{2}+94a+18$, $55a^{20}-22a^{19}-1143a^{18}+289a^{17}+10050a^{16}-940a^{15}-48411a^{14}-3561a^{13}+137741a^{12}+33294a^{11}-231345a^{10}-93219a^{9}+214899a^{8}+118449a^{7}-94808a^{6}-66090a^{5}+15595a^{4}+15562a^{3}+168a^{2}-1251a-169$, $a^{19}-a^{18}-20a^{17}+17a^{16}+169a^{15}-114a^{14}-783a^{13}+373a^{12}+2152a^{11}-568a^{10}-3525a^{9}+186a^{8}+3256a^{7}+438a^{6}-1499a^{5}-403a^{4}+303a^{3}+107a^{2}-19a-7$, $2a^{19}-2a^{18}-41a^{17}+36a^{16}+356a^{15}-263a^{14}-1700a^{13}+994a^{12}+4839a^{11}-2045a^{10}-8292a^{9}+2195a^{8}+8203a^{7}-1065a^{6}-4269a^{5}+174a^{4}+1049a^{3}+21a^{2}-94a-8$, $33a^{20}-20a^{19}-680a^{18}+312a^{17}+5932a^{16}-1761a^{15}-28397a^{14}+3550a^{13}+80570a^{12}+3886a^{11}-135849a^{10}-28627a^{9}+128492a^{8}+44610a^{7}-59893a^{6}-26502a^{5}+11665a^{4}+6348a^{3}-454a^{2}-517a-58$, $45a^{20}-25a^{19}-929a^{18}+379a^{17}+8117a^{16}-1998a^{15}-38894a^{14}+2907a^{13}+110323a^{12}+10852a^{11}-185520a^{10}-48678a^{9}+174094a^{8}+70520a^{7}-79384a^{6}-41251a^{5}+14461a^{4}+9920a^{3}-273a^{2}-816a-107$ Copy content Toggle raw display (assuming GRH)
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:  \( 14921510518.5 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{17}\cdot(2\pi)^{2}\cdot 14921510518.5 \cdot 1}{2\cdot\sqrt{38289048631639269472806869479351853}}\cr\approx \mathstrut & 0.197294586501 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^21 - 21*x^19 - 3*x^18 + 186*x^17 + 55*x^16 - 897*x^15 - 410*x^14 + 2525*x^13 + 1584*x^12 - 4093*x^11 - 3339*x^10 + 3442*x^9 + 3694*x^8 - 1060*x^7 - 1902*x^6 - 106*x^5 + 410*x^4 + 98*x^3 - 25*x^2 - 11*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^21 - 21*x^19 - 3*x^18 + 186*x^17 + 55*x^16 - 897*x^15 - 410*x^14 + 2525*x^13 + 1584*x^12 - 4093*x^11 - 3339*x^10 + 3442*x^9 + 3694*x^8 - 1060*x^7 - 1902*x^6 - 106*x^5 + 410*x^4 + 98*x^3 - 25*x^2 - 11*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^21 - 21*x^19 - 3*x^18 + 186*x^17 + 55*x^16 - 897*x^15 - 410*x^14 + 2525*x^13 + 1584*x^12 - 4093*x^11 - 3339*x^10 + 3442*x^9 + 3694*x^8 - 1060*x^7 - 1902*x^6 - 106*x^5 + 410*x^4 + 98*x^3 - 25*x^2 - 11*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^21 - 21*x^19 - 3*x^18 + 186*x^17 + 55*x^16 - 897*x^15 - 410*x^14 + 2525*x^13 + 1584*x^12 - 4093*x^11 - 3339*x^10 + 3442*x^9 + 3694*x^8 - 1060*x^7 - 1902*x^6 - 106*x^5 + 410*x^4 + 98*x^3 - 25*x^2 - 11*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

$S_7\wr C_3$ (as 21T159):

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 384072192000
The 1165 conjugacy class representatives for $S_7\wr C_3$
Character table for $S_7\wr C_3$

Intermediate fields

\(\Q(\zeta_{7})^+\)

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 42 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 $18{,}\,{\href{/padicField/2.3.0.1}{3} }$ $18{,}\,{\href{/padicField/3.3.0.1}{3} }$ ${\href{/padicField/5.9.0.1}{9} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ R $15{,}\,{\href{/padicField/11.6.0.1}{6} }$ R ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ $15{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ $18{,}\,{\href{/padicField/31.3.0.1}{3} }$ $21$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ $18{,}\,{\href{/padicField/53.3.0.1}{3} }$ ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(7\) Copy content Toggle raw display Deg $21$$3$$7$$14$
\(13\) Copy content Toggle raw display 13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.3.0.1$x^{3} + 2 x + 11$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.0.1$x^{3} + 2 x + 11$$1$$3$$0$$C_3$$[\ ]^{3}$
13.7.0.1$x^{7} + 3 x + 11$$1$$7$$0$$C_7$$[\ ]^{7}$
\(97\) Copy content Toggle raw display $\Q_{97}$$x + 92$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 92$$1$$1$$0$Trivial$[\ ]$
97.2.0.1$x^{2} + 96 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} + 96 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.4.0.1$x^{4} + 6 x^{2} + 80 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} + 6 x^{2} + 80 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
97.5.0.1$x^{5} + 3 x + 92$$1$$5$$0$$C_5$$[\ ]^{5}$
\(8914877\) Copy content Toggle raw display $\Q_{8914877}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{8914877}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{8914877}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
\(5021940620101\) Copy content Toggle raw display $\Q_{5021940620101}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{5021940620101}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{5021940620101}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{5021940620101}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{5021940620101}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{5021940620101}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{5021940620101}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$