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Properties

Label 22.22.6913172200...7888.1
Degree $22$
Signature $[22, 0]$
Discriminant $2^{32}\cdot 7^{11}\cdot 11^{22}$
Root discriminant $79.76$
Ramified primes $2, 7, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times F_{11}$ (as 22T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3592, 43648, 123904, -109120, -619520, 76384, 1208064, -21824, -1208064, 2728, 704704, -124, -256256, 0, 59840, 0, -8976, 0, 836, 0, -44, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 44*x^20 + 836*x^18 - 8976*x^16 + 59840*x^14 - 256256*x^12 - 124*x^11 + 704704*x^10 + 2728*x^9 - 1208064*x^8 - 21824*x^7 + 1208064*x^6 + 76384*x^5 - 619520*x^4 - 109120*x^3 + 123904*x^2 + 43648*x + 3592)
 
gp: K = bnfinit(x^22 - 44*x^20 + 836*x^18 - 8976*x^16 + 59840*x^14 - 256256*x^12 - 124*x^11 + 704704*x^10 + 2728*x^9 - 1208064*x^8 - 21824*x^7 + 1208064*x^6 + 76384*x^5 - 619520*x^4 - 109120*x^3 + 123904*x^2 + 43648*x + 3592, 1)
 

Normalized defining polynomial

\( x^{22} - 44 x^{20} + 836 x^{18} - 8976 x^{16} + 59840 x^{14} - 256256 x^{12} - 124 x^{11} + 704704 x^{10} + 2728 x^{9} - 1208064 x^{8} - 21824 x^{7} + 1208064 x^{6} + 76384 x^{5} - 619520 x^{4} - 109120 x^{3} + 123904 x^{2} + 43648 x + 3592 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[22, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(691317220063227578535949337667555335077888=2^{32}\cdot 7^{11}\cdot 11^{22}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.76$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{10} - \frac{1}{6} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{6} a^{13} + \frac{1}{6} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{14} + \frac{1}{6} a^{10} + \frac{1}{6} a^{8} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{12} a^{15} + \frac{1}{6} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{12} a^{16} + \frac{1}{6} a^{10} - \frac{1}{6} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{5388} a^{17} - \frac{5}{898} a^{16} - \frac{17}{2694} a^{15} + \frac{31}{2694} a^{14} - \frac{211}{2694} a^{13} + \frac{23}{1347} a^{12} + \frac{14}{1347} a^{11} + \frac{161}{898} a^{10} - \frac{51}{898} a^{9} + \frac{91}{449} a^{8} + \frac{4}{1347} a^{7} - \frac{151}{1347} a^{6} - \frac{78}{449} a^{5} - \frac{352}{1347} a^{4} + \frac{47}{1347} a^{3} - \frac{182}{1347} a^{2} - \frac{388}{1347} a + \frac{1}{3}$, $\frac{1}{5388} a^{18} - \frac{3}{449} a^{16} - \frac{5}{449} a^{15} - \frac{179}{2694} a^{14} + \frac{1}{1347} a^{13} + \frac{61}{2694} a^{12} - \frac{4}{449} a^{11} - \frac{80}{449} a^{10} + \frac{223}{1347} a^{9} - \frac{75}{898} a^{8} - \frac{31}{1347} a^{7} + \frac{175}{1347} a^{6} - \frac{188}{1347} a^{5} - \frac{635}{1347} a^{4} - \frac{119}{1347} a^{3} - \frac{460}{1347} a^{2} + \frac{34}{1347} a - \frac{1}{3}$, $\frac{1}{5388} a^{19} + \frac{69}{1796} a^{16} + \frac{107}{2694} a^{15} + \frac{110}{1347} a^{14} + \frac{49}{1347} a^{13} - \frac{82}{1347} a^{12} + \frac{79}{2694} a^{11} - \frac{21}{449} a^{10} + \frac{52}{1347} a^{9} + \frac{287}{2694} a^{8} - \frac{193}{449} a^{7} + \frac{662}{1347} a^{6} + \frac{370}{1347} a^{5} + \frac{230}{1347} a^{4} + \frac{334}{1347} a^{3} - \frac{232}{1347} a^{2} - \frac{49}{1347} a + \frac{1}{3}$, $\frac{1}{5388} a^{20} + \frac{23}{898} a^{16} - \frac{155}{5388} a^{15} - \frac{11}{898} a^{14} - \frac{20}{1347} a^{13} - \frac{7}{1347} a^{12} - \frac{85}{2694} a^{11} - \frac{199}{2694} a^{10} + \frac{88}{449} a^{9} - \frac{583}{2694} a^{8} - \frac{166}{1347} a^{7} - \frac{84}{449} a^{6} + \frac{176}{1347} a^{5} + \frac{11}{1347} a^{4} + \frac{122}{449} a^{3} - \frac{180}{449} a^{2} + \frac{394}{1347} a - \frac{1}{3}$, $\frac{1}{5388} a^{21} - \frac{14}{1347} a^{16} + \frac{34}{1347} a^{15} + \frac{86}{1347} a^{14} - \frac{27}{898} a^{13} - \frac{49}{898} a^{12} - \frac{11}{1347} a^{11} - \frac{286}{1347} a^{10} + \frac{163}{1347} a^{9} + \frac{325}{1347} a^{8} - \frac{355}{1347} a^{7} + \frac{120}{449} a^{6} - \frac{158}{449} a^{5} + \frac{150}{449} a^{4} + \frac{158}{1347} a^{3} + \frac{122}{449} a^{2} + \frac{562}{1347} a$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $21$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 398119325026000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_{11}$ (as 22T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 220
The 22 conjugacy class representatives for $C_2\times F_{11}$
Character table for $C_2\times F_{11}$ is not computed

Intermediate fields

\(\Q(\sqrt{7}) \), 11.11.4910318845910094848.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 22 sibling: data not computed
Degree 44 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11Data not computed