# Properties

 Label 22.14.126...600.1 Degree $22$ Signature $[14, 4]$ Discriminant $1.269\times 10^{76}$ Root discriminant $2879.05$ Ramified primes $2, 3, 5, 7, 23, 59, 137, 1033$ Class number $1$ (GRH) Class group trivial (GRH) Galois group 22T52

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 74*x^20 + 2449*x^18 - 47772*x^16 + 608919*x^14 - 5307720*x^12 + 32127975*x^10 - 134044248*x^8 + 373194336*x^6 - 645946216*x^4 + 597000440*x^2 - 195030400)

gp: K = bnfinit(x^22 - 74*x^20 + 2449*x^18 - 47772*x^16 + 608919*x^14 - 5307720*x^12 + 32127975*x^10 - 134044248*x^8 + 373194336*x^6 - 645946216*x^4 + 597000440*x^2 - 195030400, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-195030400, 0, 597000440, 0, -645946216, 0, 373194336, 0, -134044248, 0, 32127975, 0, -5307720, 0, 608919, 0, -47772, 0, 2449, 0, -74, 0, 1]);

$$x^{22} - 74 x^{20} + 2449 x^{18} - 47772 x^{16} + 608919 x^{14} - 5307720 x^{12} + 32127975 x^{10} - 134044248 x^{8} + 373194336 x^{6} - 645946216 x^{4} + 597000440 x^{2} - 195030400$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $22$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[14, 4]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$126\!\cdots\!600$$$$\medspace = 2^{45}\cdot 3^{28}\cdot 5^{2}\cdot 7^{4}\cdot 23^{4}\cdot 59\cdot 137^{16}\cdot 1033$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $2879.05$ sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol))  magma: Abs(Discriminant(Integers(K)))^(1/Degree(K)); Ramified primes: $2, 3, 5, 7, 23, 59, 137, 1033$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $2$ This field is not Galois over $\Q$. This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{12} a^{18} - \frac{1}{4} a^{14} - \frac{1}{2} a^{12} + \frac{1}{4} a^{10} - \frac{1}{2} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{3}$, $\frac{1}{12} a^{19} - \frac{1}{4} a^{15} - \frac{1}{2} a^{13} + \frac{1}{4} a^{11} - \frac{1}{2} a^{9} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{3} a$, $\frac{1}{27000} a^{20} - \frac{527}{13500} a^{18} - \frac{153}{1000} a^{16} + \frac{64}{375} a^{14} - \frac{1807}{9000} a^{12} + \frac{9}{50} a^{10} + \frac{1}{40} a^{8} - \frac{229}{2250} a^{6} + \frac{191}{750} a^{4} + \frac{1663}{3375} a^{2} + \frac{163}{675}$, $\frac{1}{216000} a^{21} - \frac{1}{54000} a^{20} + \frac{1723}{108000} a^{19} + \frac{527}{27000} a^{18} + \frac{1347}{8000} a^{17} - \frac{347}{2000} a^{16} - \frac{247}{6000} a^{15} - \frac{32}{375} a^{14} - \frac{6307}{72000} a^{13} + \frac{6307}{18000} a^{12} - \frac{83}{200} a^{11} - \frac{9}{100} a^{10} - \frac{19}{320} a^{9} + \frac{19}{80} a^{8} - \frac{677}{9000} a^{7} - \frac{2021}{4500} a^{6} - \frac{23}{750} a^{5} + \frac{46}{375} a^{4} + \frac{1663}{27000} a^{3} - \frac{1663}{6750} a^{2} - \frac{1637}{5400} a - \frac{163}{1350}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

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

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

## Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $17$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-1$$ (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); 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) sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$171825983862000000000000000000000$$ (assuming GRH) sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{14}\cdot(2\pi)^{4}\cdot 171825983862000000000000000000000 \cdot 1}{2\sqrt{12690240639024486099207064421849753745845788293805139479005039009036081561600}}\approx 19.4743659906574$ (assuming GRH)

## Galois group

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A non-solvable group of order 40874803200 The 400 conjugacy class representatives for t22n52 are not computed Character table for t22n52 is not computed

## Intermediate fields

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

## Sibling fields

 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 R R R ${\href{/padicField/11.11.0.1}{11} }^{2}$ $18{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ ${\href{/padicField/17.11.0.1}{11} }^{2}$ ${\href{/padicField/19.5.0.1}{5} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ R ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.11.0.1}{11} }^{2}$ ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{5}$ ${\href{/padicField/41.7.0.1}{7} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ $18{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{3}$ R

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

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]])

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];

## Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2} 2.8.21.1x^{8} + 2 x^{6} + 4 x^{2} + 2$$8$$1$$21$$C_2 \wr C_2\wr C_2$$[2, 2, 3, 7/2, 7/2, 15/4]^{2}$
2.12.24.424$x^{12} + 2 x^{10} + 4 x^{7} - 2 x^{6} + 4 x^{4} + 4 x^{3} - 2 x^{2} + 4 x + 2$$12$$1$$2412T149[4/3, 4/3, 2, 7/3, 7/3, 3]_{3}^{2} 33.2.1.1x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2} 3.9.13.2x^{9} + 6 x^{5} + 3 x^{3} + 3$$9$$1$$13$$C_3^2 : D_{6}$$[3/2, 3/2, 5/3]_{2}^{2}$
3.9.13.2$x^{9} + 6 x^{5} + 3 x^{3} + 3$$9$$1$$13$$C_3^2 : D_{6} $$[3/2, 3/2, 5/3]_{2}^{2} 5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2} 5.6.0.1x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
5.8.0.1$x^{8} + x^{2} - 2 x + 3$$1$$8$$0$$C_8$$[\ ]^{8} 77.4.2.1x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2} 7.14.0.1x^{14} + 2 x^{2} - 2 x + 3$$1$$14$$0$$C_{14}$$[\ ]^{14}$
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2} 23.2.1.2x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2} 23.2.1.2x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.14.0.1$x^{14} - x + 7$$1$$14$$0$$C_{14}$$[\ ]^{14} 5959.2.1.2x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.6.0.1$x^{6} - x + 23$$1$$6$$0$$C_6$$[\ ]^{6} 59.6.0.1x^{6} - x + 23$$1$$6$$0$$C_6$$[\ ]^{6}$
59.8.0.1$x^{8} - x + 14$$1$$8$$0$$C_8$$[\ ]^{8} 137137.2.0.1x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
137.5.4.1$x^{5} - 137$$5$$1$$4$$F_5$$[\ ]_{5}^{4} 137.5.4.1x^{5} - 137$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
137.10.8.1$x^{10} - 137 x^{5} + 112614$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
1033Data not computed