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Properties

Label 32.0.48679729995...5376.7
Degree $32$
Signature $[0, 16]$
Discriminant $2^{128}\cdot 3^{16}\cdot 7^{16}$
Root discriminant $73.32$
Ramified primes $2, 3, 7$
Class number $2176$ (GRH)
Class group $[8, 272]$ (GRH)
Galois group $C_2^2\times C_8$ (as 32T37)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![923521, 0, -7872512, 0, 46443520, 0, -134830080, 0, 276113088, 0, -292113152, 0, 213070656, 0, -109410336, 0, 42535999, 0, -12541952, 0, 2891776, 0, -517120, 0, 72384, 0, -7680, 0, 608, 0, -32, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 32*x^30 + 608*x^28 - 7680*x^26 + 72384*x^24 - 517120*x^22 + 2891776*x^20 - 12541952*x^18 + 42535999*x^16 - 109410336*x^14 + 213070656*x^12 - 292113152*x^10 + 276113088*x^8 - 134830080*x^6 + 46443520*x^4 - 7872512*x^2 + 923521)
 
gp: K = bnfinit(x^32 - 32*x^30 + 608*x^28 - 7680*x^26 + 72384*x^24 - 517120*x^22 + 2891776*x^20 - 12541952*x^18 + 42535999*x^16 - 109410336*x^14 + 213070656*x^12 - 292113152*x^10 + 276113088*x^8 - 134830080*x^6 + 46443520*x^4 - 7872512*x^2 + 923521, 1)
 

Normalized defining polynomial

\( x^{32} - 32 x^{30} + 608 x^{28} - 7680 x^{26} + 72384 x^{24} - 517120 x^{22} + 2891776 x^{20} - 12541952 x^{18} + 42535999 x^{16} - 109410336 x^{14} + 213070656 x^{12} - 292113152 x^{10} + 276113088 x^{8} - 134830080 x^{6} + 46443520 x^{4} - 7872512 x^{2} + 923521 \)

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:  \(486797299958529611528622959063550881784243829894751113445376=2^{128}\cdot 3^{16}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.32$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(672=2^{5}\cdot 3\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{672}(1,·)$, $\chi_{672}(517,·)$, $\chi_{672}(391,·)$, $\chi_{672}(13,·)$, $\chi_{672}(659,·)$, $\chi_{672}(281,·)$, $\chi_{672}(155,·)$, $\chi_{672}(671,·)$, $\chi_{672}(547,·)$, $\chi_{672}(293,·)$, $\chi_{672}(167,·)$, $\chi_{672}(169,·)$, $\chi_{672}(43,·)$, $\chi_{672}(559,·)$, $\chi_{672}(181,·)$, $\chi_{672}(55,·)$, $\chi_{672}(449,·)$, $\chi_{672}(323,·)$, $\chi_{672}(461,·)$, $\chi_{672}(335,·)$, $\chi_{672}(337,·)$, $\chi_{672}(211,·)$, $\chi_{672}(349,·)$, $\chi_{672}(223,·)$, $\chi_{672}(617,·)$, $\chi_{672}(491,·)$, $\chi_{672}(113,·)$, $\chi_{672}(629,·)$, $\chi_{672}(503,·)$, $\chi_{672}(505,·)$, $\chi_{672}(379,·)$, $\chi_{672}(125,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{31} a^{15} + \frac{4}{31} a^{13} - \frac{12}{31} a^{11} - \frac{10}{31} a^{9} - \frac{14}{31} a^{7} + \frac{7}{31} a^{5} + \frac{7}{31} a^{3} - \frac{13}{31} a$, $\frac{1}{93} a^{16} + \frac{35}{93} a^{14} - \frac{43}{93} a^{12} - \frac{10}{93} a^{10} - \frac{15}{31} a^{8} - \frac{8}{31} a^{6} - \frac{8}{31} a^{4} - \frac{13}{93} a^{2} + \frac{1}{3}$, $\frac{1}{93} a^{17} - \frac{1}{93} a^{15} - \frac{1}{93} a^{13} - \frac{43}{93} a^{11} + \frac{12}{31} a^{9} + \frac{5}{31} a^{7} + \frac{1}{31} a^{5} + \frac{14}{93} a^{3} + \frac{34}{93} a$, $\frac{1}{93} a^{18} + \frac{34}{93} a^{14} + \frac{7}{93} a^{12} + \frac{26}{93} a^{10} - \frac{10}{31} a^{8} - \frac{7}{31} a^{6} - \frac{10}{93} a^{4} + \frac{7}{31} a^{2} + \frac{1}{3}$, $\frac{1}{93} a^{19} + \frac{1}{93} a^{15} - \frac{32}{93} a^{13} - \frac{43}{93} a^{11} + \frac{7}{31} a^{9} - \frac{8}{31} a^{7} + \frac{38}{93} a^{5} - \frac{8}{31} a^{3} - \frac{5}{93} a$, $\frac{1}{93} a^{20} + \frac{26}{93} a^{14} + \frac{1}{3} a^{10} + \frac{7}{31} a^{8} - \frac{1}{3} a^{6} + \frac{8}{93} a^{2} - \frac{1}{3}$, $\frac{1}{93} a^{21} - \frac{1}{93} a^{15} - \frac{5}{31} a^{13} - \frac{17}{93} a^{11} + \frac{4}{31} a^{9} - \frac{25}{93} a^{7} - \frac{1}{31} a^{5} + \frac{5}{93} a^{3} + \frac{41}{93} a$, $\frac{1}{93} a^{22} + \frac{20}{93} a^{14} + \frac{11}{31} a^{12} + \frac{2}{93} a^{10} + \frac{23}{93} a^{8} - \frac{9}{31} a^{6} - \frac{19}{93} a^{4} + \frac{28}{93} a^{2} + \frac{1}{3}$, $\frac{1}{93} a^{23} - \frac{1}{93} a^{15} + \frac{14}{31} a^{13} - \frac{25}{93} a^{11} - \frac{46}{93} a^{9} - \frac{4}{31} a^{7} + \frac{20}{93} a^{5} - \frac{26}{93} a^{3} + \frac{25}{93} a$, $\frac{1}{279} a^{24} + \frac{1}{279} a^{18} + \frac{2}{31} a^{14} - \frac{61}{279} a^{12} + \frac{7}{31} a^{10} + \frac{11}{31} a^{8} + \frac{68}{279} a^{6} + \frac{14}{31} a^{4} + \frac{14}{31} a^{2} + \frac{2}{9}$, $\frac{1}{279} a^{25} + \frac{1}{279} a^{19} - \frac{133}{279} a^{13} + \frac{41}{279} a^{7} + \frac{17}{279} a$, $\frac{1}{8649} a^{26} - \frac{5}{2883} a^{24} - \frac{2}{2883} a^{22} - \frac{41}{8649} a^{20} + \frac{10}{2883} a^{18} + \frac{10}{2883} a^{16} + \frac{686}{8649} a^{14} + \frac{937}{2883} a^{12} - \frac{835}{2883} a^{10} + \frac{3206}{8649} a^{8} - \frac{128}{2883} a^{6} + \frac{1157}{2883} a^{4} + \frac{2363}{8649} a^{2}$, $\frac{1}{8649} a^{27} - \frac{5}{2883} a^{25} - \frac{2}{2883} a^{23} - \frac{41}{8649} a^{21} + \frac{10}{2883} a^{19} + \frac{10}{2883} a^{17} + \frac{128}{8649} a^{15} + \frac{193}{2883} a^{13} + \frac{1397}{2883} a^{11} + \frac{137}{8649} a^{9} - \frac{407}{2883} a^{7} - \frac{145}{2883} a^{5} - \frac{1543}{8649} a^{3} - \frac{5}{31} a$, $\frac{1}{147033} a^{28} - \frac{7}{147033} a^{26} + \frac{215}{147033} a^{24} + \frac{190}{147033} a^{22} + \frac{446}{147033} a^{20} + \frac{146}{147033} a^{18} - \frac{376}{147033} a^{16} - \frac{65264}{147033} a^{14} + \frac{43822}{147033} a^{12} - \frac{48640}{147033} a^{10} + \frac{67114}{147033} a^{8} + \frac{42652}{147033} a^{6} + \frac{34502}{147033} a^{4} - \frac{12344}{147033} a^{2} + \frac{31}{153}$, $\frac{1}{4558023} a^{29} - \frac{194}{4558023} a^{27} + \frac{5128}{4558023} a^{25} - \frac{12917}{4558023} a^{23} + \frac{6532}{4558023} a^{21} - \frac{22328}{4558023} a^{19} - \frac{2824}{4558023} a^{17} + \frac{13565}{4558023} a^{15} + \frac{1591910}{4558023} a^{13} - \frac{1667125}{4558023} a^{11} + \frac{183785}{4558023} a^{9} - \frac{2221204}{4558023} a^{7} - \frac{55780}{147033} a^{5} - \frac{598096}{4558023} a^{3} + \frac{1544}{4743} a$, $\frac{1}{67478933098214527960073784590552506427127} a^{30} - \frac{3191763078727318409138764104619528}{22492977699404842653357928196850835475709} a^{28} - \frac{3245587234287141647190809568313835387}{67478933098214527960073784590552506427127} a^{26} + \frac{39106171427689363309812454268109474577}{67478933098214527960073784590552506427127} a^{24} + \frac{75272230258174081201106086764577127726}{22492977699404842653357928196850835475709} a^{22} + \frac{284944743146123956470776791638429295414}{67478933098214527960073784590552506427127} a^{20} - \frac{213506227400128850735148565063049448784}{67478933098214527960073784590552506427127} a^{18} - \frac{87469702763927056579043546490067458227}{22492977699404842653357928196850835475709} a^{16} - \frac{9108059882413009646889262096314175574974}{67478933098214527960073784590552506427127} a^{14} - \frac{1088843541087851292995349497328800089828}{67478933098214527960073784590552506427127} a^{12} + \frac{9723212046107850473464620677122228450313}{22492977699404842653357928196850835475709} a^{10} + \frac{21888970199743075336647789447009063212267}{67478933098214527960073784590552506427127} a^{8} - \frac{164796153111647147407139649151081169413}{2176739777361758966453993051308145368617} a^{6} + \frac{3154643499797247601703481075380492908206}{7497659233134947551119309398950278491903} a^{4} - \frac{29965246878514576250754128990692170778}{2176739777361758966453993051308145368617} a^{2} - \frac{68004113203534768614538333672021138}{755025937343655555481787392059710499}$, $\frac{1}{67478933098214527960073784590552506427127} a^{31} + \frac{5229140907811290958501107530449465}{67478933098214527960073784590552506427127} a^{29} + \frac{1684288003662609332719684579840744930}{67478933098214527960073784590552506427127} a^{27} - \frac{64701002650963914467102099984967016}{2176739777361758966453993051308145368617} a^{25} - \frac{394314242282101994195084251706378603}{2176739777361758966453993051308145368617} a^{23} + \frac{61767958725425770218071988985485456739}{67478933098214527960073784590552506427127} a^{21} - \frac{103333834359558943124986059082749770722}{22492977699404842653357928196850835475709} a^{19} - \frac{70158778441884874766807285092018050367}{67478933098214527960073784590552506427127} a^{17} + \frac{798368961730503018155923761704088337523}{67478933098214527960073784590552506427127} a^{15} - \frac{332664683991932709737290783182665083528}{3969349005777325174121987328856029789831} a^{13} + \frac{182132587910599584228201531616040475518}{67478933098214527960073784590552506427127} a^{11} + \frac{6088009449467211770217731666981106509930}{67478933098214527960073784590552506427127} a^{9} + \frac{33020869310900743644518230045324079350147}{67478933098214527960073784590552506427127} a^{7} - \frac{7131793938660021645685933265588990794822}{67478933098214527960073784590552506427127} a^{5} - \frac{1667764568061217964454975829689090453386}{3969349005777325174121987328856029789831} a^{3} + \frac{27859046674551671962131211209009319307}{70217412172959966659806227461553076407} a$

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

Class group and class number

$C_{8}\times C_{272}$, which has order $2176$ (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:  \( -\frac{5588037799199692437815130897092704}{22492977699404842653357928196850835475709} a^{30} + \frac{177618892499354894089021836294903008}{22492977699404842653357928196850835475709} a^{28} - \frac{3359818823788169317875884207981913088}{22492977699404842653357928196850835475709} a^{26} + \frac{42207471233935779536540190945386453056}{22492977699404842653357928196850835475709} a^{24} - \frac{395654112907150740008079308234422739968}{22492977699404842653357928196850835475709} a^{22} + \frac{2807565089990419228479456289271439104000}{22492977699404842653357928196850835475709} a^{20} - \frac{5194140049637997435822538867828664786944}{7497659233134947551119309398950278491903} a^{18} + \frac{66918149107693139002028386432941961832509}{22492977699404842653357928196850835475709} a^{16} - \frac{224292926005368763396551248537197387974560}{22492977699404842653357928196850835475709} a^{14} + \frac{567263872466338905650167640529114951191552}{22492977699404842653357928196850835475709} a^{12} - \frac{1081942377975156221823331566175726979455232}{22492977699404842653357928196850835475709} a^{10} + \frac{1431747474514333755889302315491639489730496}{22492977699404842653357928196850835475709} a^{8} - \frac{1346157327645763751100320295352499408896}{23405804057653322219935409153851025469} a^{6} + \frac{547297140164574031721754548533415853675520}{22492977699404842653357928196850835475709} a^{4} - \frac{6782803115177087593084347673249231364096}{725579925787252988817997683769381789539} a^{2} + \frac{1189415419557514628270232866191986688}{755025937343655555481787392059710499} \) (order $6$)
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:  \( 56808797853492.48 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_8$ (as 32T37):

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 $C_2^2\times C_8$
Character table for $C_2^2\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{14}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-42}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-6}, \sqrt{14})\), \(\Q(\sqrt{2}, \sqrt{7})\), \(\Q(\sqrt{-3}, \sqrt{14})\), \(\Q(\sqrt{2}, \sqrt{-21})\), \(\Q(\sqrt{-3}, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-6}, \sqrt{7})\), \(\Q(\zeta_{16})^+\), 4.4.100352.1, 4.0.903168.5, 4.0.18432.2, 8.0.12745506816.4, 8.8.40282095616.1, 8.0.3262849744896.6, 8.0.3262849744896.5, 8.0.3262849744896.1, 8.0.339738624.2, 8.0.815712436224.6, 8.0.5156108238848.1, 8.0.2147483648.1, 8.8.173946175488.1, 8.8.417644767346688.4, 16.0.10646188457767892278050816.1, 16.0.106341808682864896865468416.1, 16.16.697708606768276588334338277376.3, 16.0.697708606768276588334338277376.3, 16.0.697708606768276588334338277376.5, 16.0.174427151692069147083584569344.3, 16.0.30257271966902092038144.2

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{4}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
3Data 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}$