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Properties

Label 32.0.47565240232...0000.6
Degree $32$
Signature $[0, 16]$
Discriminant $2^{48}\cdot 5^{24}\cdot 17^{28}$
Root discriminant $112.83$
Ramified primes $2, 5, 17$
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![5473632256, 0, 13684080640, 0, 28736569344, 0, 58841546752, 0, 120057683968, 0, 51808331776, 0, 16712689664, 0, 4764273664, 0, 1232647424, 0, 215959296, 0, 33625728, 0, 4670240, 0, 512992, 0, 21488, 0, 884, 0, 34, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 34*x^30 + 884*x^28 + 21488*x^26 + 512992*x^24 + 4670240*x^22 + 33625728*x^20 + 215959296*x^18 + 1232647424*x^16 + 4764273664*x^14 + 16712689664*x^12 + 51808331776*x^10 + 120057683968*x^8 + 58841546752*x^6 + 28736569344*x^4 + 13684080640*x^2 + 5473632256)
 
gp: K = bnfinit(x^32 + 34*x^30 + 884*x^28 + 21488*x^26 + 512992*x^24 + 4670240*x^22 + 33625728*x^20 + 215959296*x^18 + 1232647424*x^16 + 4764273664*x^14 + 16712689664*x^12 + 51808331776*x^10 + 120057683968*x^8 + 58841546752*x^6 + 28736569344*x^4 + 13684080640*x^2 + 5473632256, 1)
 

Normalized defining polynomial

\( x^{32} + 34 x^{30} + 884 x^{28} + 21488 x^{26} + 512992 x^{24} + 4670240 x^{22} + 33625728 x^{20} + 215959296 x^{18} + 1232647424 x^{16} + 4764273664 x^{14} + 16712689664 x^{12} + 51808331776 x^{10} + 120057683968 x^{8} + 58841546752 x^{6} + 28736569344 x^{4} + 13684080640 x^{2} + 5473632256 \)

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:  \(475652402320384421216839760983630708473856000000000000000000000000=2^{48}\cdot 5^{24}\cdot 17^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $112.83$
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:  $2, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(680=2^{3}\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{680}(1,·)$, $\chi_{680}(389,·)$, $\chi_{680}(137,·)$, $\chi_{680}(273,·)$, $\chi_{680}(661,·)$, $\chi_{680}(89,·)$, $\chi_{680}(409,·)$, $\chi_{680}(33,·)$, $\chi_{680}(169,·)$, $\chi_{680}(349,·)$, $\chi_{680}(53,·)$, $\chi_{680}(441,·)$, $\chi_{680}(189,·)$, $\chi_{680}(117,·)$, $\chi_{680}(577,·)$, $\chi_{680}(77,·)$, $\chi_{680}(461,·)$, $\chi_{680}(81,·)$, $\chi_{680}(597,·)$, $\chi_{680}(217,·)$, $\chi_{680}(93,·)$, $\chi_{680}(353,·)$, $\chi_{680}(229,·)$, $\chi_{680}(361,·)$, $\chi_{680}(621,·)$, $\chi_{680}(637,·)$, $\chi_{680}(497,·)$, $\chi_{680}(501,·)$, $\chi_{680}(489,·)$, $\chi_{680}(633,·)$, $\chi_{680}(253,·)$, $\chi_{680}(213,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{272} a^{8}$, $\frac{1}{272} a^{9}$, $\frac{1}{544} a^{10}$, $\frac{1}{544} a^{11}$, $\frac{1}{1088} a^{12}$, $\frac{1}{1088} a^{13}$, $\frac{1}{2176} a^{14}$, $\frac{1}{2176} a^{15}$, $\frac{1}{73984} a^{16}$, $\frac{1}{73984} a^{17}$, $\frac{1}{147968} a^{18}$, $\frac{1}{147968} a^{19}$, $\frac{1}{295936} a^{20}$, $\frac{1}{295936} a^{21}$, $\frac{1}{591872} a^{22}$, $\frac{1}{591872} a^{23}$, $\frac{1}{1046429696} a^{24} - \frac{1}{7694336} a^{22} - \frac{5}{3847168} a^{20} - \frac{11}{7694336} a^{18} - \frac{3}{28288} a^{14} + \frac{1}{56576} a^{12} + \frac{3}{7072} a^{10} + \frac{21}{416} a^{6} - \frac{3}{26} a^{4} - \frac{2}{13} a^{2} + \frac{25}{52}$, $\frac{1}{1046429696} a^{25} - \frac{1}{7694336} a^{23} - \frac{5}{3847168} a^{21} - \frac{11}{7694336} a^{19} - \frac{3}{28288} a^{15} + \frac{1}{56576} a^{13} + \frac{3}{7072} a^{11} + \frac{21}{416} a^{7} - \frac{3}{26} a^{5} - \frac{2}{13} a^{3} + \frac{25}{52} a$, $\frac{1}{296941996194331983872} a^{26} + \frac{61991704677}{148470998097165991936} a^{24} - \frac{49831212731}{1091698515420338176} a^{22} - \frac{3636489284411}{2183397030840676352} a^{20} - \frac{3688650149179}{1091698515420338176} a^{18} + \frac{310171099973}{68231157213771136} a^{16} + \frac{2504304436085}{16054389932652032} a^{14} + \frac{674890205777}{8027194966326016} a^{12} - \frac{635819889691}{1003399370790752} a^{10} + \frac{3209236976705}{2006798741581504} a^{8} - \frac{1069851699583}{59023492399456} a^{6} + \frac{331588449241}{3688968274966} a^{4} - \frac{983561091411}{14755873099864} a^{2} - \frac{3511760935003}{7377936549932}$, $\frac{1}{296941996194331983872} a^{27} + \frac{61991704677}{148470998097165991936} a^{25} - \frac{49831212731}{1091698515420338176} a^{23} - \frac{3636489284411}{2183397030840676352} a^{21} - \frac{3688650149179}{1091698515420338176} a^{19} + \frac{310171099973}{68231157213771136} a^{17} + \frac{2504304436085}{16054389932652032} a^{15} + \frac{674890205777}{8027194966326016} a^{13} - \frac{635819889691}{1003399370790752} a^{11} + \frac{3209236976705}{2006798741581504} a^{9} - \frac{1069851699583}{59023492399456} a^{7} + \frac{331588449241}{3688968274966} a^{5} - \frac{983561091411}{14755873099864} a^{3} - \frac{3511760935003}{7377936549932} a$, $\frac{1}{593883992388663967744} a^{28} + \frac{65120407865}{148470998097165991936} a^{24} - \frac{1874032384883}{4366794061681352704} a^{22} - \frac{325910746843}{272924628855084544} a^{20} - \frac{495001280931}{1091698515420338176} a^{18} - \frac{2349029184811}{545849257710169088} a^{16} + \frac{718231177373}{4013597483163008} a^{14} - \frac{529975297435}{8027194966326016} a^{12} - \frac{1658444996947}{4013597483163008} a^{10} + \frac{24813687473}{38592283491952} a^{8} - \frac{2724415687387}{59023492399456} a^{6} + \frac{327096262257}{29511746199728} a^{4} + \frac{122595807743}{3688968274966} a^{2} - \frac{3299769935687}{7377936549932}$, $\frac{1}{593883992388663967744} a^{29} + \frac{65120407865}{148470998097165991936} a^{25} - \frac{1874032384883}{4366794061681352704} a^{23} - \frac{325910746843}{272924628855084544} a^{21} - \frac{495001280931}{1091698515420338176} a^{19} - \frac{2349029184811}{545849257710169088} a^{17} + \frac{718231177373}{4013597483163008} a^{15} - \frac{529975297435}{8027194966326016} a^{13} - \frac{1658444996947}{4013597483163008} a^{11} + \frac{24813687473}{38592283491952} a^{9} - \frac{2724415687387}{59023492399456} a^{7} + \frac{327096262257}{29511746199728} a^{5} + \frac{122595807743}{3688968274966} a^{3} - \frac{3299769935687}{7377936549932} a$, $\frac{1}{1187767984777327935488} a^{30} - \frac{173352617025}{136462314427542272} a^{20} - \frac{563057580707}{1003399370790752} a^{10} + \frac{568026535487}{7377936549932}$, $\frac{1}{1187767984777327935488} a^{31} - \frac{173352617025}{136462314427542272} a^{21} - \frac{563057580707}{1003399370790752} a^{11} + \frac{568026535487}{7377936549932} a$

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:  \( \frac{30271229}{1427605750934288384} a^{30} + \frac{1907087427}{2855211501868576768} a^{28} + \frac{756780725}{44612679716696512} a^{26} + \frac{1169193691501}{2855211501868576768} a^{24} + \frac{2028172343}{207863388313088} a^{22} + \frac{189709792143}{2624275277452736} a^{20} + \frac{2499949446965}{5248550554905472} a^{18} + \frac{447015238643}{154369133967808} a^{16} + \frac{148658843813}{9648070872988} a^{14} + \frac{1553670828425}{38592283491952} a^{12} + \frac{2469133335843}{19296141745976} a^{10} + \frac{44498706630}{141883395191} a^{8} + \frac{43620840989}{283766790382} a^{6} - \frac{579387875453}{141883395191} a^{4} + \frac{5085566472}{141883395191} a^{2} + \frac{2058443572}{141883395191} \) (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

\(\Q(\sqrt{85}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{5}) \), 4.0.614125.2, \(\Q(\sqrt{5}, \sqrt{17})\), 4.0.614125.1, 4.4.122825.1, 4.4.4913.1, 4.0.36125.1, \(\Q(\zeta_{5})\), 8.0.377149515625.1, 8.8.15085980625.1, 8.0.1305015625.1, 8.0.26261675072000000.10, 8.0.26261675072000000.7, 8.8.1680747204608.1, 8.8.1050467002880000.1, 16.0.142241757136172119140625.1, 16.0.689675577587306205184000000000000.9, 16.16.1103480924139689928294400000000.2

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
$2$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
5Data not computed
17Data not computed