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Properties

Label 21.17.380...957.1
Degree $21$
Signature $[17, 2]$
Discriminant $3.803\times 10^{35}$
Root discriminant \(49.46\)
Ramified primes see page
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 + 49*x^16 - 903*x^15 - 326*x^14 + 2621*x^13 + 1146*x^12 - 4667*x^11 - 2321*x^10 + 5032*x^9 + 2776*x^8 - 3086*x^7 - 1910*x^6 + 894*x^5 + 670*x^4 - 46*x^3 - 87*x^2 - 17*x - 1)
 
gp: K = bnfinit(y^21 - 21*y^19 - 3*y^18 + 186*y^17 + 49*y^16 - 903*y^15 - 326*y^14 + 2621*y^13 + 1146*y^12 - 4667*y^11 - 2321*y^10 + 5032*y^9 + 2776*y^8 - 3086*y^7 - 1910*y^6 + 894*y^5 + 670*y^4 - 46*y^3 - 87*y^2 - 17*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 21*x^19 - 3*x^18 + 186*x^17 + 49*x^16 - 903*x^15 - 326*x^14 + 2621*x^13 + 1146*x^12 - 4667*x^11 - 2321*x^10 + 5032*x^9 + 2776*x^8 - 3086*x^7 - 1910*x^6 + 894*x^5 + 670*x^4 - 46*x^3 - 87*x^2 - 17*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 21*x^19 - 3*x^18 + 186*x^17 + 49*x^16 - 903*x^15 - 326*x^14 + 2621*x^13 + 1146*x^12 - 4667*x^11 - 2321*x^10 + 5032*x^9 + 2776*x^8 - 3086*x^7 - 1910*x^6 + 894*x^5 + 670*x^4 - 46*x^3 - 87*x^2 - 17*x - 1)
 

\( x^{21} - 21 x^{19} - 3 x^{18} + 186 x^{17} + 49 x^{16} - 903 x^{15} - 326 x^{14} + 2621 x^{13} + \cdots - 1 \) 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:   \(380273806833409289291563492222717957\) \(\medspace = 7^{14}\cdot 757\cdot 1373\cdot 537599\cdot 1003457499787\) 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:  \(49.46\)
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:  $7^{2/3}757^{1/2}1373^{1/2}537599^{1/2}1003457499787^{1/2}\approx 2740063448627.555$
Ramified primes:   \(7\), \(757\), \(1373\), \(537599\), \(1003457499787\) 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:  $\Q(\sqrt{56069\!\cdots\!07093}$)
$\card{ \Aut(K/\Q) }$:  $1$
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:   $a^{20}+4a^{19}-25a^{18}-84a^{17}+256a^{16}+743a^{15}-1418a^{14}-3592a^{13}+4703a^{12}+10331a^{11}-9726a^{10}-18093a^{9}+12551a^{8}+19023a^{7}-9621a^{6}-11377a^{5}+3796a^{4}+3411a^{3}-513a^{2}-370a-34$, $a^{20}+4a^{19}-25a^{18}-84a^{17}+256a^{16}+743a^{15}-1418a^{14}-3592a^{13}+4703a^{12}+10331a^{11}-9726a^{10}-18093a^{9}+12551a^{8}+19023a^{7}-9621a^{6}-11377a^{5}+3796a^{4}+3411a^{3}-513a^{2}-370a-35$, $a$, $a+1$, $a^{20}+5a^{19}-26a^{18}-104a^{17}+273a^{16}+912a^{15}-1538a^{14}-4375a^{13}+5160a^{12}+12495a^{11}-10744a^{10}-21742a^{9}+13879a^{8}+22727a^{7}-10549a^{6}-13536a^{5}+4045a^{4}+4063a^{3}-494a^{2}-449a-46$, $a^{2}+a-1$, $25a^{20}-9a^{19}-519a^{18}+108a^{17}+4558a^{16}-350a^{15}-22017a^{14}-694a^{13}+63848a^{12}+7477a^{11}-114238a^{10}-20999a^{9}+125022a^{8}+29906a^{7}-79735a^{6}-23266a^{5}+26101a^{4}+8965a^{3}-3038a^{2}-1286a-107$, $6a^{20}+a^{19}-128a^{18}-37a^{17}+1153a^{16}+446a^{15}-5707a^{14}-2619a^{13}+16967a^{12}+8581a^{11}-31196a^{10}-16533a^{9}+35222a^{8}+18923a^{7}-23250a^{6}-12459a^{5}+7843a^{4}+4194a^{3}-901a^{2}-533a-50$, $10a^{20}-4a^{19}-208a^{18}+53a^{17}+1830a^{16}-238a^{15}-8854a^{14}+248a^{13}+25708a^{12}+1333a^{11}-46016a^{10}-5244a^{9}+50306a^{8}+8398a^{7}-32001a^{6}-7056a^{5}+10474a^{4}+2883a^{3}-1252a^{2}-435a-30$, $a^{19}-a^{18}-20a^{17}+17a^{16}+169a^{15}-120a^{14}-783a^{13}+457a^{12}+2164a^{11}-1018a^{10}-3649a^{9}+1328a^{8}+3704a^{7}-928a^{6}-2159a^{5}+249a^{4}+652a^{3}+18a^{2}-78a-9$, $14a^{20}-6a^{19}-290a^{18}+81a^{17}+2540a^{16}-380a^{15}-12226a^{14}+510a^{13}+35278a^{12}+1589a^{11}-62647a^{10}-7232a^{9}+67770a^{8}+12112a^{7}-42472a^{6}-10435a^{5}+13556a^{4}+4333a^{3}-1511a^{2}-660a-58$, $22a^{20}-11a^{19}-455a^{18}+159a^{17}+3983a^{16}-868a^{15}-19185a^{14}+2072a^{13}+55480a^{12}-1067a^{11}-98924a^{10}-5246a^{9}+107717a^{8}+12711a^{7}-68212a^{6}-12729a^{5}+22176a^{4}+5819a^{3}-2587a^{2}-962a-77$, $42a^{20}-29a^{19}-860a^{18}+466a^{17}+7450a^{16}-3054a^{15}-35472a^{14}+10562a^{13}+101168a^{12}-20746a^{11}-177158a^{10}+22442a^{9}+188147a^{8}-9722a^{7}-115090a^{6}-3883a^{5}+35793a^{4}+4800a^{3}-4020a^{2}-1093a-73$, $37a^{20}-26a^{19}-757a^{18}+419a^{17}+6553a^{16}-2758a^{15}-31180a^{14}+9602a^{13}+88862a^{12}-19061a^{11}-155453a^{10}+21032a^{9}+164806a^{8}-9751a^{7}-100435a^{6}-2869a^{5}+30937a^{4}+4191a^{3}-3359a^{2}-999a-76$, $15a^{20}-2a^{19}-316a^{18}+2812a^{16}+311a^{15}-13745a^{14}-2710a^{13}+40289a^{12}+10504a^{11}-72781a^{10}-22244a^{9}+80285a^{8}+27308a^{7}-51425a^{6}-19250a^{5}+16755a^{4}+6935a^{3}-1873a^{2}-949a-87$, $38a^{20}-44a^{19}-760a^{18}+779a^{17}+6427a^{16}-5805a^{15}-29818a^{14}+23772a^{13}+82494a^{12}-58452a^{11}-138824a^{10}+87899a^{9}+139271a^{8}-77852a^{7}-78289a^{6}+36421a^{5}+21629a^{4}-7157a^{3}-2164a^{2}+196a+45$, $4a^{20}-84a^{18}-12a^{17}+744a^{16}+196a^{15}-3613a^{14}-1303a^{13}+10499a^{12}+4571a^{11}-18758a^{10}-9217a^{9}+20404a^{8}+10930a^{7}-12805a^{6}-7404a^{5}+3992a^{4}+2529a^{3}-372a^{2}-320a-36$, $3a^{20}-5a^{19}-58a^{18}+91a^{17}+473a^{16}-699a^{15}-2107a^{14}+2953a^{13}+5547a^{12}-7484a^{11}-8719a^{10}+11607a^{9}+7853a^{8}-10718a^{7}-3618a^{6}+5439a^{5}+628a^{4}-1297a^{3}-2a^{2}+94a+9$ 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:  \( 56379445831.8 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{17}\cdot(2\pi)^{2}\cdot 56379445831.8 \cdot 1}{2\cdot\sqrt{380273806833409289291563492222717957}}\cr\approx \mathstrut & 0.236544178322 \end{aligned}\] (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 + 49*x^16 - 903*x^15 - 326*x^14 + 2621*x^13 + 1146*x^12 - 4667*x^11 - 2321*x^10 + 5032*x^9 + 2776*x^8 - 3086*x^7 - 1910*x^6 + 894*x^5 + 670*x^4 - 46*x^3 - 87*x^2 - 17*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 + 49*x^16 - 903*x^15 - 326*x^14 + 2621*x^13 + 1146*x^12 - 4667*x^11 - 2321*x^10 + 5032*x^9 + 2776*x^8 - 3086*x^7 - 1910*x^6 + 894*x^5 + 670*x^4 - 46*x^3 - 87*x^2 - 17*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 + 49*x^16 - 903*x^15 - 326*x^14 + 2621*x^13 + 1146*x^12 - 4667*x^11 - 2321*x^10 + 5032*x^9 + 2776*x^8 - 3086*x^7 - 1910*x^6 + 894*x^5 + 670*x^4 - 46*x^3 - 87*x^2 - 17*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 + 49*x^16 - 903*x^15 - 326*x^14 + 2621*x^13 + 1146*x^12 - 4667*x^11 - 2321*x^10 + 5032*x^9 + 2776*x^8 - 3086*x^7 - 1910*x^6 + 894*x^5 + 670*x^4 - 46*x^3 - 87*x^2 - 17*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 $18{,}\,{\href{/padicField/2.3.0.1}{3} }$ ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ ${\href{/padicField/5.9.0.1}{9} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ R $21$ ${\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{3}$ $18{,}\,{\href{/padicField/19.3.0.1}{3} }$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ $18{,}\,{\href{/padicField/31.3.0.1}{3} }$ $21$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ $15{,}\,{\href{/padicField/53.6.0.1}{6} }$ $15{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(7\) Copy content Toggle raw display 7.9.6.1$x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.12.8.1$x^{12} + 15 x^{10} + 40 x^{9} + 84 x^{8} + 120 x^{7} + 53 x^{6} + 414 x^{5} - 1293 x^{4} - 1830 x^{3} + 10968 x^{2} - 13836 x + 12004$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
\(757\) Copy content Toggle raw display $\Q_{757}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{757}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
Deg $7$$1$$7$$0$$C_7$$[\ ]^{7}$
\(1373\) Copy content Toggle raw display $\Q_{1373}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{1373}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{1373}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
\(537599\) Copy content Toggle raw display $\Q_{537599}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{537599}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{537599}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{537599}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{537599}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{537599}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
\(1003457499787\) Copy content Toggle raw display $\Q_{1003457499787}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{1003457499787}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{1003457499787}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{1003457499787}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{1003457499787}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{1003457499787}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$