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

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -15, 48, -113, 261, -656, 1630, -3576, 6680, -10594, 14374, -16811, 17039, -15003, 11469, -7582, 4301, -2067, 825, -265, 65, -11, 1]);
sage: x = polygen(QQ); K.<a> = 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)
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)

Normalized defining 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:  \( -1 \) (order $2$)
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

22T53:

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

11.1.5960386319.1

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

Frobenius cycle types

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

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$$0$Trivial$[\ ]$
$\Q_{149}$$x + 2$$1$$1$$0$Trivial$[\ ]$
149.2.1.2$x^{2} + 298$$2$$1$$1$$C_2$$[\ ]_{2}$
149.3.0.1$x^{3} - x + 18$$1$$3$$0$$C_3$$[\ ]^{3}$
149.3.0.1$x^{3} - x + 18$$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