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Properties

Label 32.0.15693807744...0000.5
Degree $32$
Signature $[0, 16]$
Discriminant $2^{96}\cdot 5^{24}\cdot 7^{16}$
Root discriminant $70.77$
Ramified primes $2, 5, 7$
Class number $800$ (GRH)
Class group $[2, 10, 40]$ (GRH)
Galois group $C_2\times C_4^2$ (as 32T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![22481, -429016, 3962076, -23591480, 101809462, -339412280, 909882172, -2015958408, 3765841276, -6020922808, 8336518556, -10090253496, 10757915718, -10166839032, 8560966476, -6450379432, 4363838181, -2657926856, 1460398932, -724763816, 325052602, -131698920, 48150868, -15845336, 4678388, -1231568, 287336, -58576, 10332, -1520, 184, -16, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 16*x^31 + 184*x^30 - 1520*x^29 + 10332*x^28 - 58576*x^27 + 287336*x^26 - 1231568*x^25 + 4678388*x^24 - 15845336*x^23 + 48150868*x^22 - 131698920*x^21 + 325052602*x^20 - 724763816*x^19 + 1460398932*x^18 - 2657926856*x^17 + 4363838181*x^16 - 6450379432*x^15 + 8560966476*x^14 - 10166839032*x^13 + 10757915718*x^12 - 10090253496*x^11 + 8336518556*x^10 - 6020922808*x^9 + 3765841276*x^8 - 2015958408*x^7 + 909882172*x^6 - 339412280*x^5 + 101809462*x^4 - 23591480*x^3 + 3962076*x^2 - 429016*x + 22481)
 
gp: K = bnfinit(x^32 - 16*x^31 + 184*x^30 - 1520*x^29 + 10332*x^28 - 58576*x^27 + 287336*x^26 - 1231568*x^25 + 4678388*x^24 - 15845336*x^23 + 48150868*x^22 - 131698920*x^21 + 325052602*x^20 - 724763816*x^19 + 1460398932*x^18 - 2657926856*x^17 + 4363838181*x^16 - 6450379432*x^15 + 8560966476*x^14 - 10166839032*x^13 + 10757915718*x^12 - 10090253496*x^11 + 8336518556*x^10 - 6020922808*x^9 + 3765841276*x^8 - 2015958408*x^7 + 909882172*x^6 - 339412280*x^5 + 101809462*x^4 - 23591480*x^3 + 3962076*x^2 - 429016*x + 22481, 1)
 

Normalized defining polynomial

\( x^{32} - 16 x^{31} + 184 x^{30} - 1520 x^{29} + 10332 x^{28} - 58576 x^{27} + 287336 x^{26} - 1231568 x^{25} + 4678388 x^{24} - 15845336 x^{23} + 48150868 x^{22} - 131698920 x^{21} + 325052602 x^{20} - 724763816 x^{19} + 1460398932 x^{18} - 2657926856 x^{17} + 4363838181 x^{16} - 6450379432 x^{15} + 8560966476 x^{14} - 10166839032 x^{13} + 10757915718 x^{12} - 10090253496 x^{11} + 8336518556 x^{10} - 6020922808 x^{9} + 3765841276 x^{8} - 2015958408 x^{7} + 909882172 x^{6} - 339412280 x^{5} + 101809462 x^{4} - 23591480 x^{3} + 3962076 x^{2} - 429016 x + 22481 \)

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:  \(156938077449417789520626992646455296000000000000000000000000=2^{96}\cdot 5^{24}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.77$
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:  \(560=2^{4}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{560}(1,·)$, $\chi_{560}(517,·)$, $\chi_{560}(321,·)$, $\chi_{560}(139,·)$, $\chi_{560}(13,·)$, $\chi_{560}(531,·)$, $\chi_{560}(533,·)$, $\chi_{560}(407,·)$, $\chi_{560}(281,·)$, $\chi_{560}(419,·)$, $\chi_{560}(293,·)$, $\chi_{560}(167,·)$, $\chi_{560}(41,·)$, $\chi_{560}(183,·)$, $\chi_{560}(447,·)$, $\chi_{560}(449,·)$, $\chi_{560}(197,·)$, $\chi_{560}(463,·)$, $\chi_{560}(209,·)$, $\chi_{560}(211,·)$, $\chi_{560}(169,·)$, $\chi_{560}(477,·)$, $\chi_{560}(223,·)$, $\chi_{560}(99,·)$, $\chi_{560}(379,·)$, $\chi_{560}(489,·)$, $\chi_{560}(491,·)$, $\chi_{560}(237,·)$, $\chi_{560}(503,·)$, $\chi_{560}(251,·)$, $\chi_{560}(253,·)$, $\chi_{560}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{14} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{15} - \frac{1}{2} a^{7}$, $\frac{1}{82} a^{24} - \frac{6}{41} a^{23} + \frac{15}{82} a^{22} + \frac{13}{82} a^{21} + \frac{7}{82} a^{20} - \frac{4}{41} a^{19} + \frac{19}{82} a^{18} - \frac{8}{41} a^{17} - \frac{3}{41} a^{16} - \frac{11}{41} a^{15} - \frac{15}{82} a^{14} - \frac{11}{82} a^{13} + \frac{17}{82} a^{12} + \frac{13}{41} a^{11} + \frac{13}{82} a^{10} - \frac{8}{41} a^{9} + \frac{18}{41} a^{8} - \frac{9}{41} a^{7} - \frac{1}{82} a^{6} - \frac{3}{82} a^{5} - \frac{19}{82} a^{4} + \frac{17}{41} a^{3} + \frac{13}{82} a^{2} - \frac{3}{41} a + \frac{19}{82}$, $\frac{1}{82} a^{25} - \frac{3}{41} a^{23} - \frac{6}{41} a^{22} - \frac{1}{82} a^{21} - \frac{3}{41} a^{20} + \frac{5}{82} a^{19} + \frac{7}{82} a^{18} + \frac{7}{82} a^{17} - \frac{6}{41} a^{16} + \frac{4}{41} a^{15} + \frac{7}{41} a^{14} - \frac{33}{82} a^{13} - \frac{8}{41} a^{12} - \frac{3}{82} a^{11} + \frac{17}{82} a^{10} - \frac{33}{82} a^{9} + \frac{2}{41} a^{8} - \frac{6}{41} a^{7} + \frac{13}{41} a^{6} + \frac{27}{82} a^{5} - \frac{15}{41} a^{4} + \frac{11}{82} a^{3} + \frac{27}{82} a^{2} - \frac{6}{41} a - \frac{9}{41}$, $\frac{1}{82} a^{26} - \frac{1}{41} a^{23} + \frac{7}{82} a^{22} - \frac{5}{41} a^{21} + \frac{3}{41} a^{20} - \frac{1}{41} a^{18} + \frac{15}{82} a^{17} + \frac{13}{82} a^{16} - \frac{18}{41} a^{15} - \frac{1}{2} a^{14} - \frac{12}{41} a^{12} - \frac{16}{41} a^{11} + \frac{2}{41} a^{10} + \frac{31}{82} a^{9} - \frac{1}{82} a^{8} + \frac{21}{82} a^{6} + \frac{17}{41} a^{5} + \frac{10}{41} a^{4} + \frac{13}{41} a^{3} + \frac{25}{82} a^{2} - \frac{13}{82} a - \frac{9}{82}$, $\frac{1}{82} a^{27} - \frac{17}{82} a^{23} + \frac{10}{41} a^{22} - \frac{9}{82} a^{21} + \frac{7}{41} a^{20} - \frac{9}{41} a^{19} + \frac{6}{41} a^{18} - \frac{19}{82} a^{17} - \frac{7}{82} a^{16} - \frac{3}{82} a^{15} - \frac{15}{41} a^{14} - \frac{5}{82} a^{13} + \frac{1}{41} a^{12} - \frac{13}{41} a^{11} + \frac{8}{41} a^{10} - \frac{33}{82} a^{9} + \frac{31}{82} a^{8} - \frac{15}{82} a^{7} + \frac{16}{41} a^{6} - \frac{27}{82} a^{5} - \frac{6}{41} a^{4} + \frac{11}{82} a^{3} - \frac{14}{41} a^{2} - \frac{21}{82} a - \frac{3}{82}$, $\frac{1}{246} a^{28} + \frac{1}{246} a^{27} - \frac{1}{246} a^{26} + \frac{1}{246} a^{25} - \frac{1}{246} a^{24} - \frac{29}{246} a^{23} + \frac{9}{82} a^{22} + \frac{17}{246} a^{21} + \frac{55}{246} a^{20} - \frac{1}{41} a^{19} + \frac{10}{41} a^{18} - \frac{22}{123} a^{17} - \frac{49}{246} a^{16} - \frac{95}{246} a^{15} - \frac{5}{82} a^{14} - \frac{7}{246} a^{13} - \frac{31}{246} a^{12} + \frac{11}{41} a^{11} + \frac{20}{123} a^{10} + \frac{1}{41} a^{9} + \frac{35}{82} a^{8} + \frac{15}{82} a^{7} - \frac{47}{246} a^{6} - \frac{53}{246} a^{5} + \frac{7}{123} a^{4} - \frac{7}{82} a^{3} + \frac{79}{246} a^{2} - \frac{37}{246} a - \frac{59}{123}$, $\frac{1}{246} a^{29} + \frac{1}{246} a^{27} - \frac{1}{246} a^{26} + \frac{1}{246} a^{25} - \frac{1}{246} a^{24} + \frac{19}{123} a^{23} + \frac{29}{246} a^{22} + \frac{10}{123} a^{21} + \frac{11}{246} a^{20} + \frac{19}{82} a^{19} - \frac{10}{123} a^{18} - \frac{13}{123} a^{17} - \frac{29}{123} a^{16} - \frac{11}{123} a^{15} + \frac{47}{246} a^{14} - \frac{11}{41} a^{13} - \frac{29}{246} a^{12} - \frac{53}{246} a^{11} - \frac{44}{123} a^{10} + \frac{19}{41} a^{9} + \frac{6}{41} a^{8} - \frac{22}{123} a^{7} - \frac{15}{82} a^{6} + \frac{7}{246} a^{5} - \frac{119}{246} a^{4} + \frac{11}{123} a^{3} - \frac{89}{246} a^{2} - \frac{19}{82} a + \frac{103}{246}$, $\frac{1}{714753319883723154640236426} a^{30} - \frac{5}{238251106627907718213412142} a^{29} + \frac{177460860321758470506364}{119125553313953859106706071} a^{28} + \frac{2526295535014316639423425}{714753319883723154640236426} a^{27} + \frac{1734702419678173289371270}{357376659941861577320118213} a^{26} - \frac{563732943753859574204206}{357376659941861577320118213} a^{25} - \frac{625504164017440135950685}{119125553313953859106706071} a^{24} + \frac{18235645843648694469274967}{357376659941861577320118213} a^{23} - \frac{6705013355835256272118453}{714753319883723154640236426} a^{22} + \frac{55286755215044967621026423}{238251106627907718213412142} a^{21} + \frac{36850315138588639891492378}{357376659941861577320118213} a^{20} + \frac{82559251479646209936004111}{714753319883723154640236426} a^{19} + \frac{1856904151280456661939829}{714753319883723154640236426} a^{18} - \frac{56733093506144183988682075}{357376659941861577320118213} a^{17} - \frac{243972828180838758339285}{238251106627907718213412142} a^{16} + \frac{178265286206547906568414340}{357376659941861577320118213} a^{15} - \frac{12512193698071553321674203}{238251106627907718213412142} a^{14} + \frac{100720348390355127345665459}{714753319883723154640236426} a^{13} - \frac{134262384821679195083612054}{357376659941861577320118213} a^{12} + \frac{301269228979109239553047913}{714753319883723154640236426} a^{11} - \frac{317707292200990663691269441}{714753319883723154640236426} a^{10} - \frac{24826765203455510126669163}{119125553313953859106706071} a^{9} - \frac{106648491054351015379650755}{714753319883723154640236426} a^{8} + \frac{22022105488139882805183115}{119125553313953859106706071} a^{7} + \frac{43914791233930700341983826}{119125553313953859106706071} a^{6} + \frac{54409701506985333607048416}{119125553313953859106706071} a^{5} + \frac{54383733227771459815735726}{357376659941861577320118213} a^{4} - \frac{57958043167978349885883418}{357376659941861577320118213} a^{3} + \frac{224713756622392988595897941}{714753319883723154640236426} a^{2} - \frac{68281468114779279102341828}{357376659941861577320118213} a + \frac{58182503276038640502928237}{714753319883723154640236426}$, $\frac{1}{15096304869264116749156433553546} a^{31} + \frac{3515}{5032101623088038916385477851182} a^{30} + \frac{2740420108801904034865679525}{7548152434632058374578216776773} a^{29} - \frac{6860637698199355576296676327}{15096304869264116749156433553546} a^{28} + \frac{38124183247519946720416745}{368202557786929676808693501306} a^{27} - \frac{8926991874433593602925369293}{7548152434632058374578216776773} a^{26} - \frac{61468928476970764574425253197}{15096304869264116749156433553546} a^{25} + \frac{47514361945045964762375214545}{15096304869264116749156433553546} a^{24} + \frac{193523946452260660406745951461}{15096304869264116749156433553546} a^{23} - \frac{557269871560777026119629077277}{15096304869264116749156433553546} a^{22} + \frac{1044948978808611434599665984893}{5032101623088038916385477851182} a^{21} - \frac{1816046938697625086550985833553}{7548152434632058374578216776773} a^{20} - \frac{3062854607415076624857312259169}{15096304869264116749156433553546} a^{19} - \frac{1100874576264626030240409382003}{15096304869264116749156433553546} a^{18} - \frac{1460979569862531535322679133859}{7548152434632058374578216776773} a^{17} + \frac{1477843376237862123628253630995}{7548152434632058374578216776773} a^{16} + \frac{1844048058336600683632949807661}{5032101623088038916385477851182} a^{15} + \frac{346773137545122273595907500021}{15096304869264116749156433553546} a^{14} - \frac{7018688322331887339305981478767}{15096304869264116749156433553546} a^{13} + \frac{471062423934444159152536521682}{7548152434632058374578216776773} a^{12} + \frac{47035276067300782260775833945}{5032101623088038916385477851182} a^{11} + \frac{522830858936250311727729726507}{5032101623088038916385477851182} a^{10} + \frac{2668565377182771524731455441806}{7548152434632058374578216776773} a^{9} + \frac{21640069376220540234607673266}{61367092964488279468115583551} a^{8} - \frac{1891538413601929844202298081564}{7548152434632058374578216776773} a^{7} + \frac{2306393001402525815309532069941}{7548152434632058374578216776773} a^{6} + \frac{3018484038442092307534484676583}{15096304869264116749156433553546} a^{5} + \frac{184152986610341465815835107201}{15096304869264116749156433553546} a^{4} + \frac{848278101521226363361148534954}{2516050811544019458192738925591} a^{3} + \frac{4865310469986697675780142682499}{15096304869264116749156433553546} a^{2} + \frac{465498231097277420273932579217}{5032101623088038916385477851182} a + \frac{141442144675134263215674728623}{5032101623088038916385477851182}$

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

Class group and class number

$C_{2}\times C_{10}\times C_{40}$, which has order $800$ (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:  \( 35034437667347.797 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_4^2$ (as 32T36):

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 $C_2\times C_4^2$
Character table for $C_2\times C_4^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-70}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{2}, \sqrt{-35})\), 4.4.2508800.1, 4.0.2048.2, \(\Q(\sqrt{10}, \sqrt{-14})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-7})\), 4.4.100352.1, 4.0.51200.2, \(\Q(\sqrt{-7}, \sqrt{10})\), \(\Q(\sqrt{5}, \sqrt{-14})\), 4.0.256000.4, 4.4.12544000.2, 4.0.256000.2, 4.4.12544000.1, 4.0.392000.2, 4.4.8000.1, 4.0.98000.1, \(\Q(\zeta_{20})^+\), 8.0.6294077440000.4, 8.0.6146560000.2, 8.0.6294077440000.1, 8.8.6294077440000.1, 8.0.6294077440000.3, 8.0.2621440000.1, 8.0.10070523904.1, 8.0.157351936000000.61, 8.0.157351936000000.59, 8.0.153664000000.4, 8.0.9604000000.2, 8.0.65536000000.1, 8.8.157351936000000.4, 8.0.2458624000000.6, \(\Q(\zeta_{40})^+\), 8.0.157351936000000.14, 8.0.157351936000000.37, 8.0.2458624000000.4, 8.0.2458624000000.2, 16.0.39615410820716953600000000.2, 16.0.24759631762948096000000000000.6, 16.0.6044831973376000000000000.2, 16.0.396154108207169536000000000000.5, 16.16.396154108207169536000000000000.1, 16.0.68719476736000000000000.2, 16.0.396154108207169536000000000000.13

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{8}$ R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
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}$