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Properties

Label 32.0.42127744383...0000.5
Degree $32$
Signature $[0, 16]$
Discriminant $2^{124}\cdot 5^{24}\cdot 7^{16}$
Root discriminant $129.80$
Ramified primes $2, 5, 7$
Class number Not computed
Class group Not computed
Galois group $C_4\times C_8$ (as 32T43)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![531726889113616, 0, 607687873272704, 0, 585984734941536, 0, 551879803278272, 0, 518107062507832, 0, 99340895568320, 0, 14110203638048, 0, 1779432654272, 0, 204604317092, 0, 14547063552, 0, 875308560, 0, 47059600, 0, 2184910, 0, 71344, 0, 2156, 0, 56, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 56*x^30 + 2156*x^28 + 71344*x^26 + 2184910*x^24 + 47059600*x^22 + 875308560*x^20 + 14547063552*x^18 + 204604317092*x^16 + 1779432654272*x^14 + 14110203638048*x^12 + 99340895568320*x^10 + 518107062507832*x^8 + 551879803278272*x^6 + 585984734941536*x^4 + 607687873272704*x^2 + 531726889113616)
 
gp: K = bnfinit(x^32 + 56*x^30 + 2156*x^28 + 71344*x^26 + 2184910*x^24 + 47059600*x^22 + 875308560*x^20 + 14547063552*x^18 + 204604317092*x^16 + 1779432654272*x^14 + 14110203638048*x^12 + 99340895568320*x^10 + 518107062507832*x^8 + 551879803278272*x^6 + 585984734941536*x^4 + 607687873272704*x^2 + 531726889113616, 1)
 

Normalized defining polynomial

\( x^{32} + 56 x^{30} + 2156 x^{28} + 71344 x^{26} + 2184910 x^{24} + 47059600 x^{22} + 875308560 x^{20} + 14547063552 x^{18} + 204604317092 x^{16} + 1779432654272 x^{14} + 14110203638048 x^{12} + 99340895568320 x^{10} + 518107062507832 x^{8} + 551879803278272 x^{6} + 585984734941536 x^{4} + 607687873272704 x^{2} + 531726889113616 \)

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

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 16]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(42127744383897781264481528176959874165374976000000000000000000000000=2^{124}\cdot 5^{24}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $129.80$
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:  $2, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1120=2^{5}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{1120}(897,·)$, $\chi_{1120}(643,·)$, $\chi_{1120}(1,·)$, $\chi_{1120}(393,·)$, $\chi_{1120}(139,·)$, $\chi_{1120}(531,·)$, $\chi_{1120}(281,·)$, $\chi_{1120}(923,·)$, $\chi_{1120}(673,·)$, $\chi_{1120}(27,·)$, $\chi_{1120}(113,·)$, $\chi_{1120}(169,·)$, $\chi_{1120}(811,·)$, $\chi_{1120}(561,·)$, $\chi_{1120}(307,·)$, $\chi_{1120}(57,·)$, $\chi_{1120}(699,·)$, $\chi_{1120}(449,·)$, $\chi_{1120}(1091,·)$, $\chi_{1120}(841,·)$, $\chi_{1120}(587,·)$, $\chi_{1120}(337,·)$, $\chi_{1120}(419,·)$, $\chi_{1120}(953,·)$, $\chi_{1120}(729,·)$, $\chi_{1120}(979,·)$, $\chi_{1120}(867,·)$, $\chi_{1120}(617,·)$, $\chi_{1120}(363,·)$, $\chi_{1120}(1009,·)$, $\chi_{1120}(83,·)$, $\chi_{1120}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{7} a^{2}$, $\frac{1}{7} a^{3}$, $\frac{1}{49} a^{4}$, $\frac{1}{49} a^{5}$, $\frac{1}{343} a^{6}$, $\frac{1}{343} a^{7}$, $\frac{1}{4802} a^{8}$, $\frac{1}{4802} a^{9}$, $\frac{1}{33614} a^{10}$, $\frac{1}{33614} a^{11}$, $\frac{1}{235298} a^{12}$, $\frac{1}{235298} a^{13}$, $\frac{1}{1647086} a^{14}$, $\frac{1}{1647086} a^{15}$, $\frac{1}{23059204} a^{16}$, $\frac{1}{23059204} a^{17}$, $\frac{1}{5003847268} a^{18} - \frac{15}{714835324} a^{16} + \frac{2}{25529833} a^{14} + \frac{1}{3647119} a^{12} + \frac{1}{1042034} a^{10} + \frac{13}{148862} a^{8} + \frac{11}{10633} a^{6} - \frac{10}{1519} a^{4} - \frac{5}{217} a^{2} + \frac{13}{31}$, $\frac{1}{5003847268} a^{19} - \frac{15}{714835324} a^{17} + \frac{2}{25529833} a^{15} + \frac{1}{3647119} a^{13} + \frac{1}{1042034} a^{11} + \frac{13}{148862} a^{9} + \frac{11}{10633} a^{7} - \frac{10}{1519} a^{5} - \frac{5}{217} a^{3} + \frac{13}{31} a$, $\frac{1}{35026930876} a^{20} - \frac{3}{1042034} a^{10} + \frac{9}{31}$, $\frac{1}{35026930876} a^{21} - \frac{3}{1042034} a^{11} + \frac{9}{31} a$, $\frac{1}{245188516132} a^{22} - \frac{3}{7294238} a^{12} + \frac{9}{217} a^{2}$, $\frac{1}{245188516132} a^{23} - \frac{3}{7294238} a^{13} + \frac{9}{217} a^{3}$, $\frac{1}{3432639225848} a^{24} + \frac{1}{3647119} a^{14} - \frac{11}{1519} a^{4}$, $\frac{1}{3432639225848} a^{25} + \frac{1}{3647119} a^{15} - \frac{11}{1519} a^{5}$, $\frac{1}{9917776911755914936} a^{26} - \frac{3173}{22852020533999804} a^{24} + \frac{85069}{101201805221999132} a^{22} + \frac{85657}{7228700372999938} a^{20} + \frac{79539}{1032671481857134} a^{18} - \frac{411907}{295048994816324} a^{16} - \frac{1671}{61442939362} a^{14} + \frac{31366}{1505352014369} a^{12} + \frac{3144819}{430100575534} a^{10} - \frac{3841175}{61442939362} a^{8} - \frac{5143468}{4388781383} a^{6} - \frac{3127171}{626968769} a^{4} - \frac{1006672}{89566967} a^{2} + \frac{4928301}{12795281}$, $\frac{1}{9917776911755914936} a^{27} - \frac{3173}{22852020533999804} a^{25} + \frac{85069}{101201805221999132} a^{23} + \frac{85657}{7228700372999938} a^{21} + \frac{79539}{1032671481857134} a^{19} - \frac{411907}{295048994816324} a^{17} - \frac{1671}{61442939362} a^{15} + \frac{31366}{1505352014369} a^{13} + \frac{3144819}{430100575534} a^{11} - \frac{3841175}{61442939362} a^{9} - \frac{5143468}{4388781383} a^{7} - \frac{3127171}{626968769} a^{5} - \frac{1006672}{89566967} a^{3} + \frac{4928301}{12795281} a$, $\frac{1}{2152157589851033541112} a^{28} - \frac{15}{307451084264433363016} a^{26} - \frac{1353545}{43921583466347623288} a^{24} + \frac{295935}{224089711562998078} a^{22} - \frac{4501075}{448179423125996156} a^{20} - \frac{1974767}{64025631875142308} a^{18} + \frac{44767237}{9146518839306044} a^{16} + \frac{122974903}{653322774236146} a^{14} + \frac{11094585}{46665912445439} a^{12} - \frac{181713821}{13333117841554} a^{10} - \frac{35684806}{952365560111} a^{8} - \frac{174213533}{136052222873} a^{6} - \frac{90689607}{19436031839} a^{4} - \frac{125886043}{2776575977} a^{2} + \frac{9949030}{396653711}$, $\frac{1}{2152157589851033541112} a^{29} - \frac{15}{307451084264433363016} a^{27} - \frac{1353545}{43921583466347623288} a^{25} + \frac{295935}{224089711562998078} a^{23} - \frac{4501075}{448179423125996156} a^{21} - \frac{1974767}{64025631875142308} a^{19} + \frac{44767237}{9146518839306044} a^{17} + \frac{122974903}{653322774236146} a^{15} + \frac{11094585}{46665912445439} a^{13} - \frac{181713821}{13333117841554} a^{11} - \frac{35684806}{952365560111} a^{9} - \frac{174213533}{136052222873} a^{7} - \frac{90689607}{19436031839} a^{5} - \frac{125886043}{2776575977} a^{3} + \frac{9949030}{396653711} a$, $\frac{1}{15065103128957234787784} a^{30} - \frac{57208}{112044855781499039} a^{20} - \frac{54751068}{6666558920777} a^{10} + \frac{130294233}{396653711}$, $\frac{1}{15065103128957234787784} a^{31} - \frac{57208}{112044855781499039} a^{21} - \frac{54751068}{6666558920777} a^{11} + \frac{130294233}{396653711} a$

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

Class group and class number

Not computed

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{93249}{3766275782239308696946} a^{30} - \frac{1025739}{1076078794925516770556} a^{28} - \frac{1212237}{38431385533054170377} a^{26} - \frac{6061185}{6274511923763946184} a^{24} - \frac{22194523}{784313990470493273} a^{22} - \frac{1398735}{3614350186499969} a^{20} - \frac{102946896}{16006407968785577} a^{18} - \frac{827398377}{9146518839306044} a^{16} - \frac{256994244}{326661387118073} a^{14} + \frac{521935826}{46665912445439} a^{12} - \frac{292801860}{6666558920777} a^{10} - \frac{436312071}{1904731120222} a^{8} - \frac{33196644}{136052222873} a^{6} - \frac{5035446}{19436031839} a^{4} - \frac{2606246504}{2776575977} a^{2} - \frac{93249}{396653711} \) (order $10$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times C_8$ (as 32T43):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_4\times C_8$
Character table for $C_4\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\zeta_{5})\), \(\Q(\sqrt{2}, \sqrt{5})\), 4.0.8000.2, \(\Q(\zeta_{16})^+\), 4.4.51200.1, 4.0.256000.4, 4.0.256000.2, 8.0.64000000.2, 8.8.2621440000.1, 8.0.65536000000.1, 8.8.3222567649280000.4, 8.8.5156108238848.1, 8.0.80564191232000000.92, 8.0.80564191232000000.78, 16.0.4294967296000000000000.1, 16.16.10384942254186025084518400000000.1, 16.0.6490588908866265677824000000000000.9

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ R R ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{32}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
5Data not computed
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$