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Properties

Label 21.1.168...000.1
Degree $21$
Signature $[1, 10]$
Discriminant $1.686\times 10^{70}$
Root discriminant \(2208.68\)
Ramified primes $2,5,7,97$
Class number not computed
Class group not computed
Galois group $S_3\times F_7$ (as 21T15)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 + 77*x^19 - 14*x^18 + 2541*x^17 - 924*x^16 + 46669*x^15 - 24570*x^14 + 517055*x^13 - 480760*x^12 + 3613071*x^11 - 3850*x^10 + 11794727*x^9 - 4705764*x^8 + 41542631*x^7 + 118482154*x^6 + 57561084*x^5 + 954860816*x^4 + 1014152048*x^3 + 3747573928*x^2 - 813508752*x + 5460680552)
 
gp: K = bnfinit(y^21 + 77*y^19 - 14*y^18 + 2541*y^17 - 924*y^16 + 46669*y^15 - 24570*y^14 + 517055*y^13 - 480760*y^12 + 3613071*y^11 - 3850*y^10 + 11794727*y^9 - 4705764*y^8 + 41542631*y^7 + 118482154*y^6 + 57561084*y^5 + 954860816*y^4 + 1014152048*y^3 + 3747573928*y^2 - 813508752*y + 5460680552, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 + 77*x^19 - 14*x^18 + 2541*x^17 - 924*x^16 + 46669*x^15 - 24570*x^14 + 517055*x^13 - 480760*x^12 + 3613071*x^11 - 3850*x^10 + 11794727*x^9 - 4705764*x^8 + 41542631*x^7 + 118482154*x^6 + 57561084*x^5 + 954860816*x^4 + 1014152048*x^3 + 3747573928*x^2 - 813508752*x + 5460680552);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 + 77*x^19 - 14*x^18 + 2541*x^17 - 924*x^16 + 46669*x^15 - 24570*x^14 + 517055*x^13 - 480760*x^12 + 3613071*x^11 - 3850*x^10 + 11794727*x^9 - 4705764*x^8 + 41542631*x^7 + 118482154*x^6 + 57561084*x^5 + 954860816*x^4 + 1014152048*x^3 + 3747573928*x^2 - 813508752*x + 5460680552)
 

\( x^{21} + 77 x^{19} - 14 x^{18} + 2541 x^{17} - 924 x^{16} + 46669 x^{15} - 24570 x^{14} + \cdots + 5460680552 \) 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:  $[1, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(16856743381788797964169462556045972312361821474422784000000000000000000\) \(\medspace = 2^{33}\cdot 5^{18}\cdot 7^{40}\cdot 97^{7}\) 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:  \(2208.68\)
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:  $2^{27/14}5^{6/7}7^{83/42}97^{1/2}\approx 6968.439929406937$
Ramified primes:   \(2\), \(5\), \(7\), \(97\) 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:  \(\Q(\sqrt{194}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}+\frac{1}{4}a^{5}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}+\frac{1}{4}a^{5}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{5}$, $\frac{1}{392}a^{14}-\frac{1}{56}a^{13}+\frac{1}{56}a^{12}-\frac{1}{8}a^{11}+\frac{1}{56}a^{10}+\frac{5}{56}a^{9}-\frac{5}{56}a^{8}+\frac{41}{392}a^{7}-\frac{1}{14}a^{6}-\frac{1}{7}a^{5}-\frac{1}{7}a^{4}-\frac{1}{2}a^{3}+\frac{3}{7}a^{2}-\frac{1}{7}a-\frac{2}{49}$, $\frac{1}{392}a^{15}-\frac{3}{28}a^{13}-\frac{3}{28}a^{11}-\frac{1}{28}a^{10}+\frac{1}{28}a^{9}-\frac{1}{49}a^{8}-\frac{5}{56}a^{7}+\frac{3}{28}a^{6}-\frac{1}{7}a^{5}-\frac{1}{2}a^{4}+\frac{3}{7}a^{3}-\frac{1}{7}a^{2}-\frac{2}{49}a-\frac{2}{7}$, $\frac{1}{392}a^{16}-\frac{3}{28}a^{12}-\frac{1}{28}a^{11}+\frac{1}{28}a^{10}-\frac{1}{49}a^{9}-\frac{5}{56}a^{8}-\frac{1}{4}a^{7}+\frac{3}{28}a^{6}-\frac{1}{14}a^{4}-\frac{1}{7}a^{3}-\frac{2}{49}a^{2}-\frac{2}{7}a+\frac{2}{7}$, $\frac{1}{784}a^{17}-\frac{1}{784}a^{15}-\frac{1}{56}a^{12}+\frac{1}{14}a^{11}+\frac{3}{392}a^{10}-\frac{1}{16}a^{9}-\frac{45}{392}a^{8}-\frac{17}{112}a^{7}+\frac{11}{56}a^{6}-\frac{3}{14}a^{5}-\frac{1}{14}a^{4}-\frac{23}{98}a^{3}-\frac{1}{14}a^{2}+\frac{8}{49}a-\frac{5}{14}$, $\frac{1}{833392}a^{18}-\frac{171}{416696}a^{17}+\frac{443}{833392}a^{16}-\frac{73}{208348}a^{15}+\frac{185}{416696}a^{14}-\frac{538}{7441}a^{13}+\frac{2531}{59528}a^{12}-\frac{74}{52087}a^{11}+\frac{48901}{833392}a^{10}+\frac{57595}{416696}a^{9}-\frac{186013}{833392}a^{8}+\frac{19749}{208348}a^{7}-\frac{1901}{14882}a^{6}+\frac{6693}{14882}a^{5}-\frac{35583}{104174}a^{4}-\frac{7687}{52087}a^{3}+\frac{15097}{52087}a^{2}-\frac{50579}{104174}a-\frac{7615}{52087}$, $\frac{1}{11667488}a^{19}-\frac{1}{5833744}a^{18}+\frac{801}{1458436}a^{17}+\frac{377}{2916872}a^{16}-\frac{51}{11667488}a^{15}+\frac{4071}{5833744}a^{14}+\frac{46107}{416696}a^{13}+\frac{121771}{1458436}a^{12}-\frac{606653}{11667488}a^{11}-\frac{94967}{5833744}a^{10}+\frac{32883}{1458436}a^{9}+\frac{477859}{2916872}a^{8}+\frac{1071167}{11667488}a^{7}-\frac{138137}{833392}a^{6}+\frac{1078507}{2916872}a^{5}-\frac{267699}{729218}a^{4}-\frac{94055}{729218}a^{3}-\frac{97011}{364609}a^{2}+\frac{82384}{364609}a-\frac{353243}{1458436}$, $\frac{1}{62\!\cdots\!16}a^{20}+\frac{28\!\cdots\!69}{31\!\cdots\!08}a^{19}-\frac{12\!\cdots\!11}{31\!\cdots\!08}a^{18}-\frac{65\!\cdots\!87}{11\!\cdots\!36}a^{17}+\frac{82\!\cdots\!87}{62\!\cdots\!16}a^{16}-\frac{19\!\cdots\!49}{31\!\cdots\!08}a^{15}-\frac{47\!\cdots\!49}{39\!\cdots\!26}a^{14}+\frac{20\!\cdots\!26}{19\!\cdots\!13}a^{13}+\frac{35\!\cdots\!83}{62\!\cdots\!16}a^{12}-\frac{24\!\cdots\!17}{31\!\cdots\!08}a^{11}-\frac{40\!\cdots\!89}{44\!\cdots\!44}a^{10}+\frac{18\!\cdots\!41}{78\!\cdots\!52}a^{9}+\frac{36\!\cdots\!49}{62\!\cdots\!16}a^{8}-\frac{68\!\cdots\!23}{31\!\cdots\!08}a^{7}+\frac{32\!\cdots\!95}{15\!\cdots\!04}a^{6}+\frac{20\!\cdots\!55}{78\!\cdots\!52}a^{5}-\frac{32\!\cdots\!69}{19\!\cdots\!13}a^{4}+\frac{10\!\cdots\!01}{27\!\cdots\!59}a^{3}+\frac{72\!\cdots\!12}{19\!\cdots\!13}a^{2}+\frac{35\!\cdots\!95}{78\!\cdots\!52}a+\frac{39\!\cdots\!94}{19\!\cdots\!13}$ Copy content Toggle raw display

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

not computed

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:  $10$
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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^21 + 77*x^19 - 14*x^18 + 2541*x^17 - 924*x^16 + 46669*x^15 - 24570*x^14 + 517055*x^13 - 480760*x^12 + 3613071*x^11 - 3850*x^10 + 11794727*x^9 - 4705764*x^8 + 41542631*x^7 + 118482154*x^6 + 57561084*x^5 + 954860816*x^4 + 1014152048*x^3 + 3747573928*x^2 - 813508752*x + 5460680552)
 
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 + 77*x^19 - 14*x^18 + 2541*x^17 - 924*x^16 + 46669*x^15 - 24570*x^14 + 517055*x^13 - 480760*x^12 + 3613071*x^11 - 3850*x^10 + 11794727*x^9 - 4705764*x^8 + 41542631*x^7 + 118482154*x^6 + 57561084*x^5 + 954860816*x^4 + 1014152048*x^3 + 3747573928*x^2 - 813508752*x + 5460680552, 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 + 77*x^19 - 14*x^18 + 2541*x^17 - 924*x^16 + 46669*x^15 - 24570*x^14 + 517055*x^13 - 480760*x^12 + 3613071*x^11 - 3850*x^10 + 11794727*x^9 - 4705764*x^8 + 41542631*x^7 + 118482154*x^6 + 57561084*x^5 + 954860816*x^4 + 1014152048*x^3 + 3747573928*x^2 - 813508752*x + 5460680552);
 
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 + 77*x^19 - 14*x^18 + 2541*x^17 - 924*x^16 + 46669*x^15 - 24570*x^14 + 517055*x^13 - 480760*x^12 + 3613071*x^11 - 3850*x^10 + 11794727*x^9 - 4705764*x^8 + 41542631*x^7 + 118482154*x^6 + 57561084*x^5 + 954860816*x^4 + 1014152048*x^3 + 3747573928*x^2 - 813508752*x + 5460680552);
 
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

$S_3\times F_7$ (as 21T15):

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 252
The 21 conjugacy class representatives for $S_3\times F_7$
Character table for $S_3\times F_7$

Intermediate fields

3.1.5432.1, 7.1.96889010407000000.18

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 R ${\href{/padicField/3.6.0.1}{6} }^{3}{,}\,{\href{/padicField/3.3.0.1}{3} }$ R R ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.2.0.1}{2} }^{10}{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.3.0.1}{3} }$ ${\href{/padicField/19.6.0.1}{6} }^{3}{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.7.0.1}{7} }^{3}$ ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.3.0.1}{3} }^{7}$ ${\href{/padicField/41.6.0.1}{6} }^{3}{,}\,{\href{/padicField/41.3.0.1}{3} }$ ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.7.0.1}{7} }$ ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(2\) Copy content Toggle raw display 2.7.6.1$x^{7} + 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.14.27.213$x^{14} + 4 x^{11} + 4 x^{9} + 4 x^{8} + 4 x^{7} + 4 x^{4} + 2$$14$$1$$27$$(C_7:C_3) \times C_2$$[3]_{7}^{3}$
\(5\) Copy content Toggle raw display 5.7.6.1$x^{7} + 5$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
5.14.12.1$x^{14} + 28 x^{13} + 350 x^{12} + 2576 x^{11} + 12404 x^{10} + 41104 x^{9} + 96152 x^{8} + 160650 x^{7} + 192444 x^{6} + 165676 x^{5} + 106232 x^{4} + 65016 x^{3} + 59920 x^{2} + 57232 x + 27193$$7$$2$$12$$F_7$$[\ ]_{7}^{6}$
\(7\) Copy content Toggle raw display 7.7.13.2$x^{7} + 49 x^{2} + 98 x + 7$$7$$1$$13$$F_7$$[13/6]_{6}$
7.14.27.55$x^{14} + 35 x^{7} + 196 x^{2} + 245 x + 168$$14$$1$$27$$F_7 \times C_2$$[13/6]_{6}^{2}$
\(97\) Copy content Toggle raw display $\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$$[\ ]_{2}$
97.2.0.1$x^{2} + 96 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.1$x^{4} + 10474 x^{3} + 27919909 x^{2} + 2585716380 x + 183392493$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.2.1$x^{4} + 10474 x^{3} + 27919909 x^{2} + 2585716380 x + 183392493$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.2.1$x^{4} + 10474 x^{3} + 27919909 x^{2} + 2585716380 x + 183392493$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$