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Properties

Label 32.0.72578796740...0000.2
Degree $32$
Signature $[0, 16]$
Discriminant $2^{32}\cdot 5^{24}\cdot 17^{28}$
Root discriminant $79.78$
Ramified primes $2, 5, 17$
Class number $29200$ (GRH)
Class group $[2, 2, 10, 730]$ (GRH)
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![2027776, 0, 6817920, 0, 28638896, 0, 32864424, 0, 12169433, 0, -4579907, 0, -3440334, 0, 629743, 0, 517344, 0, -161567, 0, 61832, 0, -21915, 0, 1632, 0, 739, 0, -94, 0, -7, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 7*x^30 - 94*x^28 + 739*x^26 + 1632*x^24 - 21915*x^22 + 61832*x^20 - 161567*x^18 + 517344*x^16 + 629743*x^14 - 3440334*x^12 - 4579907*x^10 + 12169433*x^8 + 32864424*x^6 + 28638896*x^4 + 6817920*x^2 + 2027776)
 
gp: K = bnfinit(x^32 - 7*x^30 - 94*x^28 + 739*x^26 + 1632*x^24 - 21915*x^22 + 61832*x^20 - 161567*x^18 + 517344*x^16 + 629743*x^14 - 3440334*x^12 - 4579907*x^10 + 12169433*x^8 + 32864424*x^6 + 28638896*x^4 + 6817920*x^2 + 2027776, 1)
 

Normalized defining polynomial

\( x^{32} - 7 x^{30} - 94 x^{28} + 739 x^{26} + 1632 x^{24} - 21915 x^{22} + 61832 x^{20} - 161567 x^{18} + 517344 x^{16} + 629743 x^{14} - 3440334 x^{12} - 4579907 x^{10} + 12169433 x^{8} + 32864424 x^{6} + 28638896 x^{4} + 6817920 x^{2} + 2027776 \)

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:  \(7257879674078131427258907485712138496000000000000000000000000=2^{32}\cdot 5^{24}\cdot 17^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.78$
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:  \(340=2^{2}\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{340}(1,·)$, $\chi_{340}(179,·)$, $\chi_{340}(257,·)$, $\chi_{340}(19,·)$, $\chi_{340}(149,·)$, $\chi_{340}(151,·)$, $\chi_{340}(291,·)$, $\chi_{340}(169,·)$, $\chi_{340}(47,·)$, $\chi_{340}(307,·)$, $\chi_{340}(53,·)$, $\chi_{340}(183,·)$, $\chi_{340}(59,·)$, $\chi_{340}(67,·)$, $\chi_{340}(69,·)$, $\chi_{340}(327,·)$, $\chi_{340}(203,·)$, $\chi_{340}(77,·)$, $\chi_{340}(81,·)$, $\chi_{340}(213,·)$, $\chi_{340}(89,·)$, $\chi_{340}(331,·)$, $\chi_{340}(219,·)$, $\chi_{340}(93,·)$, $\chi_{340}(101,·)$, $\chi_{340}(103,·)$, $\chi_{340}(111,·)$, $\chi_{340}(117,·)$, $\chi_{340}(297,·)$, $\chi_{340}(123,·)$, $\chi_{340}(253,·)$, $\chi_{340}(21,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} + \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} + \frac{1}{4} a^{4}$, $\frac{1}{4} a^{11} + \frac{1}{4} a^{5}$, $\frac{1}{4} a^{12} + \frac{1}{4} a^{6}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{3}$, $\frac{1}{32} a^{16} - \frac{1}{8} a^{14} - \frac{1}{16} a^{10} - \frac{1}{2} a^{6} + \frac{13}{32} a^{4} + \frac{1}{8} a^{2} - \frac{1}{2}$, $\frac{1}{32} a^{17} - \frac{1}{16} a^{11} + \frac{13}{32} a^{5}$, $\frac{1}{32} a^{18} - \frac{1}{16} a^{12} + \frac{13}{32} a^{6}$, $\frac{1}{32} a^{19} - \frac{1}{16} a^{13} - \frac{3}{32} a^{7} - \frac{1}{2} a$, $\frac{1}{128} a^{20} - \frac{1}{128} a^{18} + \frac{1}{128} a^{16} - \frac{7}{64} a^{14} - \frac{3}{64} a^{12} - \frac{5}{64} a^{10} + \frac{13}{128} a^{8} + \frac{59}{128} a^{6} + \frac{5}{128} a^{4} - \frac{13}{32} a^{2} - \frac{3}{8}$, $\frac{1}{128} a^{21} - \frac{1}{128} a^{19} + \frac{1}{128} a^{17} + \frac{1}{64} a^{15} - \frac{3}{64} a^{13} - \frac{5}{64} a^{11} + \frac{13}{128} a^{9} - \frac{5}{128} a^{7} + \frac{5}{128} a^{5} + \frac{15}{32} a^{3} + \frac{1}{8} a$, $\frac{1}{128} a^{22} - \frac{1}{128} a^{16} - \frac{1}{32} a^{14} - \frac{1}{8} a^{12} + \frac{11}{128} a^{10} + \frac{1}{16} a^{8} + \frac{13}{128} a^{4} - \frac{13}{32} a^{2} + \frac{1}{8}$, $\frac{1}{11392} a^{23} - \frac{31}{11392} a^{21} + \frac{95}{11392} a^{19} - \frac{9}{2848} a^{17} - \frac{217}{5696} a^{15} + \frac{501}{5696} a^{13} - \frac{1047}{11392} a^{11} - \frac{203}{11392} a^{9} + \frac{1499}{11392} a^{7} - \frac{1873}{5696} a^{5} - \frac{9}{1424} a^{3} + \frac{43}{356} a$, $\frac{1}{11392} a^{24} - \frac{31}{11392} a^{22} + \frac{3}{5696} a^{20} + \frac{53}{11392} a^{18} - \frac{167}{11392} a^{16} + \frac{103}{1424} a^{14} - \frac{513}{11392} a^{12} - \frac{25}{11392} a^{10} + \frac{171}{5696} a^{8} - \frac{3301}{11392} a^{6} + \frac{4111}{11392} a^{4} - \frac{991}{2848} a^{2} - \frac{1}{8}$, $\frac{1}{22784} a^{25} - \frac{65}{22784} a^{21} - \frac{7}{5696} a^{19} - \frac{37}{22784} a^{17} + \frac{271}{11392} a^{15} + \frac{1001}{22784} a^{13} + \frac{313}{11392} a^{11} - \frac{77}{22784} a^{9} + \frac{2627}{11392} a^{7} + \frac{8135}{22784} a^{5} - \frac{1015}{5696} a^{3} - \frac{93}{1424} a$, $\frac{1}{12348928} a^{26} - \frac{251}{6174464} a^{24} - \frac{34521}{12348928} a^{22} - \frac{3697}{3087232} a^{20} + \frac{40641}{12348928} a^{18} + \frac{37115}{6174464} a^{16} - \frac{1232871}{12348928} a^{14} - \frac{18183}{192952} a^{12} + \frac{434155}{12348928} a^{10} - \frac{393}{69376} a^{8} - \frac{4474695}{12348928} a^{6} - \frac{696325}{3087232} a^{4} + \frac{228897}{771808} a^{2} - \frac{239}{542}$, $\frac{1}{24697856} a^{27} - \frac{251}{12348928} a^{25} + \frac{167}{24697856} a^{23} - \frac{1805}{1543616} a^{21} - \frac{40659}{24697856} a^{19} - \frac{56651}{12348928} a^{17} - \frac{272447}{24697856} a^{15} + \frac{123097}{3087232} a^{13} - \frac{188061}{24697856} a^{11} + \frac{254993}{12348928} a^{9} - \frac{233003}{24697856} a^{7} + \frac{333163}{1543616} a^{5} - \frac{36123}{192952} a^{3} - \frac{135219}{385904} a$, $\frac{1}{2494483456} a^{28} + \frac{39}{1247241728} a^{26} - \frac{56849}{2494483456} a^{24} - \frac{25823}{623620864} a^{22} - \frac{7019883}{2494483456} a^{20} - \frac{2229429}{1247241728} a^{18} - \frac{14844487}{2494483456} a^{16} + \frac{29381343}{623620864} a^{14} - \frac{293195789}{2494483456} a^{12} - \frac{11350877}{1247241728} a^{10} - \frac{36945115}{2494483456} a^{8} - \frac{151742105}{623620864} a^{6} - \frac{15453489}{38976304} a^{4} + \frac{13897585}{38976304} a^{2} + \frac{10159}{109484}$, $\frac{1}{2494483456} a^{29} - \frac{23}{2494483456} a^{27} - \frac{6147}{2494483456} a^{25} + \frac{98809}{2494483456} a^{23} + \frac{8597141}{2494483456} a^{21} + \frac{961509}{2494483456} a^{19} + \frac{8204319}{2494483456} a^{17} + \frac{88986711}{2494483456} a^{15} - \frac{290181141}{2494483456} a^{13} + \frac{195772255}{2494483456} a^{11} + \frac{120441771}{2494483456} a^{9} - \frac{352642845}{2494483456} a^{7} + \frac{108484531}{311810432} a^{5} + \frac{483861}{77952608} a^{3} - \frac{12122731}{38976304} a$, $\frac{1}{551107338489787903679561737200833536} a^{30} - \frac{33426120723022209934278705}{275553669244893951839780868600416768} a^{28} + \frac{8514009375840411336909516153}{551107338489787903679561737200833536} a^{26} - \frac{2064620027696687062443008191317}{68888417311223487959945217150104192} a^{24} - \frac{1925040015305416971791718807732909}{551107338489787903679561737200833536} a^{22} + \frac{959002368601018176698460245834823}{275553669244893951839780868600416768} a^{20} - \frac{5424142441525206131090966847780141}{551107338489787903679561737200833536} a^{18} + \frac{8193977220301395973212665674566}{538190760243933499687072008985189} a^{16} + \frac{8721893348915772610552412518070285}{551107338489787903679561737200833536} a^{14} + \frac{17201895811751541345068693525011467}{275553669244893951839780868600416768} a^{12} - \frac{4006046235884328251471820179030725}{551107338489787903679561737200833536} a^{10} - \frac{7845887064152525422685028821622675}{68888417311223487959945217150104192} a^{8} - \frac{92225121562848252117369292542405291}{275553669244893951839780868600416768} a^{6} - \frac{7224663265676308749019289574804789}{17222104327805871989986304287526048} a^{4} - \frac{695702097730407663753245720009945}{2152763040975733998748288035940756} a^{2} + \frac{4151748860656034332676485636381}{48376697550016494353894113167208}$, $\frac{1}{551107338489787903679561737200833536} a^{31} - \frac{33426120723022209934278705}{275553669244893951839780868600416768} a^{29} + \frac{8514009375840411336909516153}{551107338489787903679561737200833536} a^{27} + \frac{1917847138358687669350747763267}{137776834622446975919890434300208384} a^{25} + \frac{10027886695242802364045718955411}{551107338489787903679561737200833536} a^{23} + \frac{318011126063299626509363246369317}{275553669244893951839780868600416768} a^{21} - \frac{3102060959124414402104049415754157}{551107338489787903679561737200833536} a^{19} - \frac{472353888947468893408182349319029}{137776834622446975919890434300208384} a^{17} - \frac{8548587676440115873787785882622971}{551107338489787903679561737200833536} a^{15} - \frac{25938024228475667495016381891846267}{275553669244893951839780868600416768} a^{13} + \frac{105973055867073768609179440181955}{551107338489787903679561737200833536} a^{11} + \frac{12650923548810862784217655908592637}{137776834622446975919890434300208384} a^{9} - \frac{63755435054663545190102606943503383}{275553669244893951839780868600416768} a^{7} + \frac{11206005842494307141881399070437999}{137776834622446975919890434300208384} a^{5} + \frac{2009086908336707658824556968883753}{34444208655611743979972608575052096} a^{3} + \frac{878094302653071532483860018631541}{8611052163902935994993152143763024} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{10}\times C_{730}$, which has order $29200$ (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:  $15$
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:  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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 39925225682933.99 \) (assuming GRH)
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

\(\Q(\sqrt{85}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{5}) \), 4.4.9826000.2, \(\Q(\sqrt{5}, \sqrt{17})\), 4.4.9826000.1, 4.4.4913.1, 4.4.122825.1, \(\Q(\zeta_{20})^+\), 4.4.578000.1, 8.8.96550276000000.1, 8.8.15085980625.1, 8.8.334084000000.1, 8.0.105046700288.1, 8.0.65654187680000.2, 8.0.6411541765625.1, 8.0.6411541765625.2, 16.16.9321955795676176000000000000.1, 16.0.4310472359920663782400000000.4, 16.0.41107867812353742431640625.1

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$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
5Data not computed
17Data not computed