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Properties

Label 22.10.349...224.1
Degree $22$
Signature $[10, 6]$
Discriminant $3.493\times 10^{29}$
Root discriminant \(22.02\)
Ramified primes $2,349,6615221297$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^{10}.S_{11}$ (as 22T51)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 9*x^20 + 34*x^18 - 69*x^16 + 75*x^14 - 25*x^12 - 38*x^10 + 52*x^8 - 24*x^6 - 4*x^4 + 9*x^2 - 1)
 
gp: K = bnfinit(y^22 - 9*y^20 + 34*y^18 - 69*y^16 + 75*y^14 - 25*y^12 - 38*y^10 + 52*y^8 - 24*y^6 - 4*y^4 + 9*y^2 - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 9*x^20 + 34*x^18 - 69*x^16 + 75*x^14 - 25*x^12 - 38*x^10 + 52*x^8 - 24*x^6 - 4*x^4 + 9*x^2 - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 9*x^20 + 34*x^18 - 69*x^16 + 75*x^14 - 25*x^12 - 38*x^10 + 52*x^8 - 24*x^6 - 4*x^4 + 9*x^2 - 1)
 

\( x^{22} - 9 x^{20} + 34 x^{18} - 69 x^{16} + 75 x^{14} - 25 x^{12} - 38 x^{10} + 52 x^{8} - 24 x^{6} + \cdots - 1 \) 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:  $[10, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(349316852822940057561884852224\) \(\medspace = 2^{16}\cdot 349^{2}\cdot 6615221297^{2}\) 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:  \(22.02\)
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:   \(2\), \(349\), \(6615221297\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{16}-\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{17}-\frac{1}{2}a^{15}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{18}-\frac{1}{2}a^{16}-\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$ 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

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:  $15$
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:   $a$, $a^{2}-1$, $20a^{21}-151a^{19}+460a^{17}-705a^{15}+453a^{13}+196a^{11}-504a^{9}+305a^{7}-17a^{5}-118a^{3}+11a$, $4a^{21}-30a^{19}+91a^{17}-140a^{15}+94a^{13}+28a^{11}-88a^{9}+57a^{7}-7a^{5}-18a^{3}-a$, $13a^{21}-98a^{19}+298a^{17}-456a^{15}+293a^{13}+125a^{11}-322a^{9}+194a^{7}-9a^{5}-76a^{3}+5a$, $5a^{21}-37a^{19}+109a^{17}-157a^{15}+82a^{13}+72a^{11}-122a^{9}+57a^{7}+9a^{5}-32a^{3}-a$, $15a^{21}-114a^{19}+351a^{17}-548a^{15}+371a^{13}+124a^{11}-382a^{9}+247a^{7}-23a^{5}-88a^{3}+13a$, $11a^{21}+7a^{20}-\frac{165}{2}a^{19}-53a^{18}+249a^{17}+\frac{325}{2}a^{16}-376a^{15}-\frac{505}{2}a^{14}+232a^{13}+\frac{339}{2}a^{12}+117a^{11}+\frac{117}{2}a^{10}-269a^{9}-\frac{351}{2}a^{8}+\frac{305}{2}a^{7}+\frac{227}{2}a^{6}-a^{5}-11a^{4}-\frac{127}{2}a^{3}-41a^{2}+\frac{5}{2}a+\frac{11}{2}$, $15a^{21}-114a^{19}+351a^{17}-548a^{15}+371a^{13}+124a^{11}-382a^{9}+248a^{7}-26a^{5}-86a^{3}+13a$, $27a^{21}+\frac{37}{2}a^{20}-204a^{19}-140a^{18}+\frac{1245}{2}a^{17}+428a^{16}-\frac{1915}{2}a^{15}-660a^{14}+\frac{1245}{2}a^{13}+432a^{12}+\frac{509}{2}a^{11}+171a^{10}-\frac{1359}{2}a^{9}-\frac{933}{2}a^{8}+\frac{835}{2}a^{7}+289a^{6}-26a^{5}-\frac{41}{2}a^{4}-159a^{3}-\frac{217}{2}a^{2}+\frac{33}{2}a+11$, $a+1$, $\frac{11}{2}a^{21}-\frac{11}{2}a^{20}-42a^{19}+\frac{83}{2}a^{18}+\frac{261}{2}a^{17}-\frac{253}{2}a^{16}-\frac{415}{2}a^{15}+\frac{389}{2}a^{14}+\frac{297}{2}a^{13}-\frac{253}{2}a^{12}+\frac{67}{2}a^{11}-\frac{103}{2}a^{10}-138a^{9}+138a^{8}+\frac{195}{2}a^{7}-86a^{6}-\frac{31}{2}a^{5}+\frac{13}{2}a^{4}-\frac{59}{2}a^{3}+32a^{2}+\frac{11}{2}a-4$, $12a^{21}-8a^{20}-\frac{183}{2}a^{19}+61a^{18}+283a^{17}-\frac{377}{2}a^{16}-445a^{15}+\frac{591}{2}a^{14}+307a^{13}-\frac{403}{2}a^{12}+92a^{11}-\frac{131}{2}a^{10}-307a^{9}+\frac{413}{2}a^{8}+\frac{409}{2}a^{7}-\frac{269}{2}a^{6}-25a^{5}+15a^{4}-\frac{137}{2}a^{3}+46a^{2}+\frac{25}{2}a-\frac{15}{2}$, $12a^{21}+15a^{20}-\frac{183}{2}a^{19}-114a^{18}+283a^{17}+\frac{701}{2}a^{16}-445a^{15}-\frac{1091}{2}a^{14}+307a^{13}+\frac{733}{2}a^{12}+92a^{11}+\frac{253}{2}a^{10}-307a^{9}-\frac{757}{2}a^{8}+\frac{409}{2}a^{7}+\frac{485}{2}a^{6}-25a^{5}-24a^{4}-\frac{137}{2}a^{3}-85a^{2}+\frac{23}{2}a+\frac{21}{2}$, $\frac{57}{2}a^{21}+24a^{20}-\frac{429}{2}a^{19}-\frac{365}{2}a^{18}+651a^{17}+562a^{16}-993a^{15}-877a^{14}+632a^{13}+592a^{12}+281a^{11}+203a^{10}-\frac{1409}{2}a^{9}-617a^{8}+\frac{841}{2}a^{7}+\frac{799}{2}a^{6}-\frac{37}{2}a^{5}-40a^{4}-166a^{3}-\frac{281}{2}a^{2}+\frac{23}{2}a+\frac{41}{2}$ 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:  \( 3755049.20739 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{6}\cdot 3755049.20739 \cdot 1}{2\cdot\sqrt{349316852822940057561884852224}}\cr\approx \mathstrut & 0.200149741842 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^22 - 9*x^20 + 34*x^18 - 69*x^16 + 75*x^14 - 25*x^12 - 38*x^10 + 52*x^8 - 24*x^6 - 4*x^4 + 9*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 - 9*x^20 + 34*x^18 - 69*x^16 + 75*x^14 - 25*x^12 - 38*x^10 + 52*x^8 - 24*x^6 - 4*x^4 + 9*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 - 9*x^20 + 34*x^18 - 69*x^16 + 75*x^14 - 25*x^12 - 38*x^10 + 52*x^8 - 24*x^6 - 4*x^4 + 9*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 - 9*x^20 + 34*x^18 - 69*x^16 + 75*x^14 - 25*x^12 - 38*x^10 + 52*x^8 - 24*x^6 - 4*x^4 + 9*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

$C_2^{10}.S_{11}$ (as 22T51):

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 40874803200
The 376 conjugacy class representatives for $C_2^{10}.S_{11}$
Character table for $C_2^{10}.S_{11}$

Intermediate fields

11.7.2308712232653.1

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 sibling: data not computed
Minimal sibling: 22.10.650950215557845119749631763403230158533966338467386159913089772599843207665207111767636066068836035692839798145289168624501608976008019771392.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.10.0.1}{10} }$ ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{4}$ ${\href{/padicField/7.11.0.1}{11} }^{2}$ ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.8.0.1}{8} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.9.0.1}{9} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ ${\href{/padicField/29.11.0.1}{11} }^{2}$ ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.4.0.1}{4} }^{3}$ ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.10.0.1}{10} }$ ${\href{/padicField/41.11.0.1}{11} }^{2}$ ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ ${\href{/padicField/47.10.0.1}{10} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ $18{,}\,{\href{/padicField/53.4.0.1}{4} }$ ${\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$

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 2.3.0.1$x^{3} + x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} + x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $16$$2$$8$$16$
\(349\) Copy content Toggle raw display $\Q_{349}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{349}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $4$$2$$2$$2$
Deg $14$$1$$14$$0$$C_{14}$$[\ ]^{14}$
\(6615221297\) 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 $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$