Properties

Label 22.12.237...576.1
Degree $22$
Signature $[12, 5]$
Discriminant $-2.379\times 10^{32}$
Root discriminant \(29.62\)
Ramified primes $2,29,131,5399,367163$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^{10}.(C_2\times S_{11})$ (as 22T53)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 12*x^20 + 59*x^18 - 151*x^16 + 204*x^14 - 106*x^12 - 64*x^10 + 101*x^8 - 19*x^6 - 16*x^4 + 3*x^2 + 1)
 
gp: K = bnfinit(y^22 - 12*y^20 + 59*y^18 - 151*y^16 + 204*y^14 - 106*y^12 - 64*y^10 + 101*y^8 - 19*y^6 - 16*y^4 + 3*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 12*x^20 + 59*x^18 - 151*x^16 + 204*x^14 - 106*x^12 - 64*x^10 + 101*x^8 - 19*x^6 - 16*x^4 + 3*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 12*x^20 + 59*x^18 - 151*x^16 + 204*x^14 - 106*x^12 - 64*x^10 + 101*x^8 - 19*x^6 - 16*x^4 + 3*x^2 + 1)
 

\( x^{22} - 12 x^{20} + 59 x^{18} - 151 x^{16} + 204 x^{14} - 106 x^{12} - 64 x^{10} + 101 x^{8} - 19 x^{6} - 16 x^{4} + 3 x^{2} + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $22$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[12, 5]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-237871803919090816246272099352576\) \(\medspace = -\,2^{22}\cdot 29^{2}\cdot 131^{2}\cdot 5399^{2}\cdot 367163^{2}\) 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:  \(29.62\)
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:  not computed
Ramified primes:   \(2\), \(29\), \(131\), \(5399\), \(367163\) 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:  \(\Q(\sqrt{-1}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$ 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:  $16$
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:   $a$, $a^{21}-11a^{19}+48a^{17}-103a^{15}+101a^{13}-5a^{11}-69a^{9}+32a^{7}+13a^{5}-3a^{3}$, $a^{3}-2a$, $3a^{21}-38a^{19}+201a^{17}-571a^{15}+915a^{13}-735a^{11}+51a^{9}+381a^{7}-228a^{5}+3a^{3}+18a$, $2a^{21}-25a^{19}+130a^{17}-361a^{15}+559a^{13}-417a^{11}-15a^{9}+249a^{7}-124a^{5}-10a^{3}+11a$, $a^{20}-11a^{18}+49a^{16}-112a^{14}+132a^{12}-55a^{10}-37a^{8}+41a^{6}-6a^{4}$, $2a^{20}-24a^{18}+117a^{16}-293a^{14}+378a^{12}-170a^{10}-137a^{8}+166a^{6}-15a^{4}-18a^{2}-1$, $a^{16}-10a^{14}+39a^{12}-73a^{10}+58a^{8}+9a^{6}-40a^{4}+10a^{2}+5$, $a^{20}-12a^{18}+59a^{16}-151a^{14}+205a^{12}-113a^{10}-47a^{8}+85a^{6}-19a^{4}-6a^{2}+1$, $a+1$, $a^{21}-a^{20}-11a^{19}+10a^{18}+48a^{17}-37a^{16}-103a^{15}+54a^{14}+101a^{13}+11a^{12}-5a^{11}-126a^{10}-69a^{9}+117a^{8}+32a^{7}+21a^{6}+13a^{5}-66a^{4}-3a^{3}+6a^{2}+7$, $4a^{21}+2a^{20}-47a^{19}-22a^{18}+225a^{17}+97a^{16}-557a^{15}-215a^{14}+722a^{13}+232a^{12}-354a^{11}-53a^{10}-211a^{9}-123a^{8}+304a^{7}+89a^{6}-58a^{5}+6a^{4}-25a^{3}-9a^{2}+5a$, $a^{2}-a-1$, $a^{21}-2a^{20}-11a^{19}+23a^{18}+49a^{17}-107a^{16}-112a^{15}+254a^{14}+132a^{13}-305a^{12}-55a^{11}+112a^{10}-37a^{9}+127a^{8}+41a^{7}-122a^{6}-6a^{5}+a^{4}+14a^{2}+a+1$, $a^{20}-11a^{18}+49a^{16}-112a^{14}+131a^{12}-48a^{10}-54a^{8}+57a^{6}-7a^{4}-6a^{2}+a$, $a^{21}-11a^{19}+3a^{18}+49a^{17}-32a^{16}-112a^{15}+137a^{14}+131a^{13}-297a^{12}-48a^{11}+321a^{10}-54a^{9}-93a^{8}+57a^{7}-135a^{6}-6a^{5}+111a^{4}-9a^{3}-6a^{2}+a-9$ 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:  \( 117361710.917 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{5}\cdot 117361710.917 \cdot 1}{2\cdot\sqrt{237871803919090816246272099352576}}\cr\approx \mathstrut & 0.152610439526 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^22 - 12*x^20 + 59*x^18 - 151*x^16 + 204*x^14 - 106*x^12 - 64*x^10 + 101*x^8 - 19*x^6 - 16*x^4 + 3*x^2 + 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^22 - 12*x^20 + 59*x^18 - 151*x^16 + 204*x^14 - 106*x^12 - 64*x^10 + 101*x^8 - 19*x^6 - 16*x^4 + 3*x^2 + 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^22 - 12*x^20 + 59*x^18 - 151*x^16 + 204*x^14 - 106*x^12 - 64*x^10 + 101*x^8 - 19*x^6 - 16*x^4 + 3*x^2 + 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^22 - 12*x^20 + 59*x^18 - 151*x^16 + 204*x^14 - 106*x^12 - 64*x^10 + 101*x^8 - 19*x^6 - 16*x^4 + 3*x^2 + 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

$C_2^{10}.(C_2\times S_{11})$ (as 22T53):

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 81749606400
The 752 conjugacy class representatives for $C_2^{10}.(C_2\times S_{11})$ are not computed
Character table for $C_2^{10}.(C_2\times S_{11})$ is not computed

Intermediate fields

11.9.7530807227563.1

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 22 sibling: data not computed
Degree 44 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 ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.6.0.1}{6} }^{2}$ ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.10.0.1}{10} }$ ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.6.0.1}{6} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }^{2}{,}\,{\href{/padicField/11.6.0.1}{6} }$ ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ $16{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ $18{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ R ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.5.0.1}{5} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}$ ${\href{/padicField/41.11.0.1}{11} }^{2}$ $16{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ $22$ $16{,}\,{\href{/padicField/53.6.0.1}{6} }$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$

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
\(2\) Copy content Toggle raw display 2.6.6.1$x^{6} + 6 x^{5} + 34 x^{4} + 80 x^{3} + 204 x^{2} + 216 x + 216$$2$$3$$6$$A_4$$[2, 2]^{3}$
Deg $16$$2$$8$$16$
\(29\) Copy content Toggle raw display 29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} + 2 x^{2} + 15 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} + 2 x^{2} + 15 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
29.5.0.1$x^{5} + 3 x + 27$$1$$5$$0$$C_5$$[\ ]^{5}$
29.5.0.1$x^{5} + 3 x + 27$$1$$5$$0$$C_5$$[\ ]^{5}$
\(131\) Copy content Toggle raw display 131.2.1.2$x^{2} + 131$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 131$$2$$1$$1$$C_2$$[\ ]_{2}$
131.4.0.1$x^{4} + 9 x^{2} + 109 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
131.4.0.1$x^{4} + 9 x^{2} + 109 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
131.10.0.1$x^{10} + 124 x^{5} + 97 x^{4} + 9 x^{3} + 126 x^{2} + 44 x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
\(5399\) Copy content Toggle raw display 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 $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
\(367163\) Copy content Toggle raw display Deg $4$$2$$2$$2$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$