# Properties

 Label 22.2.178901193768541138691065033.1 Degree 22 Signature $[2, 10]$ Discriminant $149\cdot 33797\cdot 64661^{2}\cdot 92179^{2}$ Ramified primes $149, 33797, 64661, 92179$ Class number 1 (GRH) Class group Trivial (GRH) Galois Group 22T53

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: K<a> := NumberField(PolynomialRing(Rationals())![1, -15, 48, -113, 261, -656, 1630, -3576, 6680, -10594, 14374, -16811, 17039, -15003, 11469, -7582, 4301, -2067, 825, -265, 65, -11, 1]);
sage: K = NumberField(x^22 - 11*x^21 + 65*x^20 - 265*x^19 + 825*x^18 - 2067*x^17 + 4301*x^16 - 7582*x^15 + 11469*x^14 - 15003*x^13 + 17039*x^12 - 16811*x^11 + 14374*x^10 - 10594*x^9 + 6680*x^8 - 3576*x^7 + 1630*x^6 - 656*x^5 + 261*x^4 - 113*x^3 + 48*x^2 - 15*x + 1,"a")
gp: K = bnfinit(x^22 - 11*x^21 + 65*x^20 - 265*x^19 + 825*x^18 - 2067*x^17 + 4301*x^16 - 7582*x^15 + 11469*x^14 - 15003*x^13 + 17039*x^12 - 16811*x^11 + 14374*x^10 - 10594*x^9 + 6680*x^8 - 3576*x^7 + 1630*x^6 - 656*x^5 + 261*x^4 - 113*x^3 + 48*x^2 - 15*x + 1, 1)

## Normalizeddefining polynomial

$x^{22}$ $\mathstrut -\mathstrut 11 x^{21}$ $\mathstrut +\mathstrut 65 x^{20}$ $\mathstrut -\mathstrut 265 x^{19}$ $\mathstrut +\mathstrut 825 x^{18}$ $\mathstrut -\mathstrut 2067 x^{17}$ $\mathstrut +\mathstrut 4301 x^{16}$ $\mathstrut -\mathstrut 7582 x^{15}$ $\mathstrut +\mathstrut 11469 x^{14}$ $\mathstrut -\mathstrut 15003 x^{13}$ $\mathstrut +\mathstrut 17039 x^{12}$ $\mathstrut -\mathstrut 16811 x^{11}$ $\mathstrut +\mathstrut 14374 x^{10}$ $\mathstrut -\mathstrut 10594 x^{9}$ $\mathstrut +\mathstrut 6680 x^{8}$ $\mathstrut -\mathstrut 3576 x^{7}$ $\mathstrut +\mathstrut 1630 x^{6}$ $\mathstrut -\mathstrut 656 x^{5}$ $\mathstrut +\mathstrut 261 x^{4}$ $\mathstrut -\mathstrut 113 x^{3}$ $\mathstrut +\mathstrut 48 x^{2}$ $\mathstrut -\mathstrut 15 x$ $\mathstrut +\mathstrut 1$

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

## Invariants

 Degree: $22$ magma: Degree(K); sage: K.degree() gp: poldegree(K.pol) Signature: $[2, 10]$ magma: Signature(K); sage: K.signature() gp: K.sign Discriminant: $178901193768541138691065033=149\cdot 33797\cdot 64661^{2}\cdot 92179^{2}$ magma: Discriminant(K); sage: K.disc() gp: K.disc Ramified primes: $149, 33797, 64661, 92179$ magma: PrimeDivisors(Discriminant(K)); sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ This field is not Galois over $\Q$.

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

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

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

## Class group and class number

Trivial Abelian group, 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: $11$ magma: UnitRank(K); sage: UK.rank() gp: #K.fu Torsion generator: $-1$ magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); sage: UK.torsion_generator() gp: K.tu[2] Fundamental units: $a^{2} - a + 1$,  $2 a^{20} - 20 a^{19} + 107 a^{18} - 393 a^{17} + 1096 a^{16} - 2444 a^{15} + 4492 a^{14} - 6932 a^{13} + 9080 a^{12} - 10150 a^{11} + 9691 a^{10} - 7877 a^{9} + 5413 a^{8} - 3114 a^{7} + 1492 a^{6} - 606 a^{5} + 232 a^{4} - 98 a^{3} + 45 a^{2} - 16 a + 2$,  $a^{18} - 9 a^{17} + 43 a^{16} - 140 a^{15} + 344 a^{14} - 672 a^{13} + 1076 a^{12} - 1438 a^{11} + 1620 a^{10} - 1544 a^{9} + 1243 a^{8} - 840 a^{7} + 473 a^{6} - 221 a^{5} + 90 a^{4} - 36 a^{3} + 17 a^{2} - 7 a + 2$,  $a^{20} - 10 a^{19} + 54 a^{18} - 201 a^{17} + 569 a^{16} - 1288 a^{15} + 2400 a^{14} - 3746 a^{13} + 4946 a^{12} - 5548 a^{11} + 5284 a^{10} - 4250 a^{9} + 2858 a^{8} - 1584 a^{7} + 718 a^{6} - 275 a^{5} + 107 a^{4} - 52 a^{3} + 26 a^{2} - 9 a - 1$,  $2 a^{20} - 20 a^{19} + 106 a^{18} - 384 a^{17} + 1052 a^{16} - 2296 a^{15} + 4115 a^{14} - 6169 a^{13} + 7819 a^{12} - 8421 a^{11} + 7709 a^{10} - 5975 a^{9} + 3893 a^{8} - 2115 a^{7} + 962 a^{6} - 384 a^{5} + 156 a^{4} - 72 a^{3} + 31 a^{2} - 9 a$,  $a^{19} - 9 a^{18} + 43 a^{17} - 140 a^{16} + 342 a^{15} - 658 a^{14} + 1023 a^{13} - 1302 a^{12} + 1360 a^{11} - 1157 a^{10} + 785 a^{9} - 406 a^{8} + 146 a^{7} - 29 a^{6} + 3 a^{5} - 5 a^{4} + 4 a^{3} - a^{2} - 3 a$,  $a^{21} - 11 a^{20} + 63 a^{19} - 245 a^{18} + 719 a^{17} - 1682 a^{16} + 3240 a^{15} - 5244 a^{14} + 7222 a^{13} - 8524 a^{12} + 8647 a^{11} - 7532 a^{10} + 5609 a^{9} - 3546 a^{8} + 1892 a^{7} - 858 a^{6} + 349 a^{5} - 144 a^{4} + 65 a^{3} - 26 a^{2} + 7 a - 1$,  $2 a^{21} - 20 a^{20} + 107 a^{19} - 394 a^{18} + 1104 a^{17} - 2480 a^{16} + 4606 a^{15} - 7211 a^{14} + 9631 a^{13} - 11049 a^{12} + 10918 a^{11} - 9285 a^{10} + 6772 a^{9} - 4213 a^{8} + 2231 a^{7} - 1017 a^{6} + 421 a^{5} - 175 a^{4} + 74 a^{3} - 29 a^{2} + 6 a$,  $a^{20} - 10 a^{19} + 53 a^{18} - 191 a^{17} + 518 a^{16} - 1113 a^{15} + 1952 a^{14} - 2843 a^{13} + 3469 a^{12} - 3553 a^{11} + 3040 a^{10} - 2146 a^{9} + 1223 a^{8} - 544 a^{7} + 185 a^{6} - 55 a^{5} + 27 a^{4} - 17 a^{3} + 7 a^{2} - 1$,  $a^{17} - 9 a^{16} + 43 a^{15} - 139 a^{14} + 336 a^{13} - 639 a^{12} + 984 a^{11} - 1247 a^{10} + 1310 a^{9} - 1141 a^{8} + 819 a^{7} - 479 a^{6} + 227 a^{5} - 89 a^{4} + 34 a^{3} - 16 a^{2} + 8 a - 2$,  $a^{18} - 9 a^{17} + 43 a^{16} - 140 a^{15} + 343 a^{14} - 665 a^{13} + 1049 a^{12} - 1367 a^{11} + 1480 a^{10} - 1328 a^{9} + 977 a^{8} - 577 a^{7} + 266 a^{6} - 94 a^{5} + 31 a^{4} - 15 a^{3} + 9 a^{2} - 3 a - 1$ (assuming GRH) magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $11377.8928745$ (assuming GRH) magma: Regulator(K); sage: K.regulator() gp: K.reg

## Galois group

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 A non-solvable group of order 81749606400 Conjugacy class representatives for 22T53 Character table for 22T53

## Intermediate fields

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

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ $18{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $20{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ $22$

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
$149$$\Q_{149}$$x + 2$$1$$1$$0Trivial[\ ] \Q_{149}$$x + 2$$1$$1$$0Trivial[\ ] 149.2.1.2x^{2} + 298$$2$$1$$1$$C_2$$[\ ]_{2}$
149.3.0.1$x^{3} - x + 18$$1$$3$$0$$C_3$$[\ ]^{3} 149.3.0.1x^{3} - x + 18$$1$$3$$0$$C_3$$[\ ]^{3}$
149.12.0.1$x^{12} - x + 89$$1$$12$$0$$C_{12}$$[\ ]^{12}$
33797Data not computed
64661Data not computed
92179Data not computed