Properties

Label 32.0.366...000.1
Degree $32$
Signature $[0, 16]$
Discriminant $3.662\times 10^{47}$
Root discriminant $30.65$
Ramified primes $2, 3, 5, 7$
Class number not computed
Class group not computed
Galois group $C_2^3\times C_4$ (as 32T34)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 3*x^30 + 4*x^28 - 9*x^26 + 27*x^24 - 93*x^22 + 188*x^20 - 279*x^18 + 581*x^16 - 1116*x^14 + 3008*x^12 - 5952*x^10 + 6912*x^8 - 9216*x^6 + 16384*x^4 - 49152*x^2 + 65536)
 
gp: K = bnfinit(x^32 - 3*x^30 + 4*x^28 - 9*x^26 + 27*x^24 - 93*x^22 + 188*x^20 - 279*x^18 + 581*x^16 - 1116*x^14 + 3008*x^12 - 5952*x^10 + 6912*x^8 - 9216*x^6 + 16384*x^4 - 49152*x^2 + 65536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![65536, 0, -49152, 0, 16384, 0, -9216, 0, 6912, 0, -5952, 0, 3008, 0, -1116, 0, 581, 0, -279, 0, 188, 0, -93, 0, 27, 0, -9, 0, 4, 0, -3, 0, 1]);
 

\( x^{32} - 3 x^{30} + 4 x^{28} - 9 x^{26} + 27 x^{24} - 93 x^{22} + 188 x^{20} - 279 x^{18} + 581 x^{16} - 1116 x^{14} + 3008 x^{12} - 5952 x^{10} + 6912 x^{8} - 9216 x^{6} + 16384 x^{4} - 49152 x^{2} + 65536 \)

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

Invariants

Degree:  $32$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 16]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(366225584701948244050176000000000000000000000000\)\(\medspace = 2^{32}\cdot 3^{16}\cdot 5^{24}\cdot 7^{16}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $30.65$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 3, 5, 7$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $32$
This field is Galois and abelian over $\Q$.
Conductor:  \(420=2^{2}\cdot 3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{420}(1,·)$, $\chi_{420}(391,·)$, $\chi_{420}(139,·)$, $\chi_{420}(13,·)$, $\chi_{420}(407,·)$, $\chi_{420}(281,·)$, $\chi_{420}(29,·)$, $\chi_{420}(419,·)$, $\chi_{420}(293,·)$, $\chi_{420}(167,·)$, $\chi_{420}(41,·)$, $\chi_{420}(43,·)$, $\chi_{420}(307,·)$, $\chi_{420}(181,·)$, $\chi_{420}(323,·)$, $\chi_{420}(197,·)$, $\chi_{420}(71,·)$, $\chi_{420}(209,·)$, $\chi_{420}(211,·)$, $\chi_{420}(349,·)$, $\chi_{420}(223,·)$, $\chi_{420}(97,·)$, $\chi_{420}(251,·)$, $\chi_{420}(337,·)$, $\chi_{420}(239,·)$, $\chi_{420}(113,·)$, $\chi_{420}(83,·)$, $\chi_{420}(169,·)$, $\chi_{420}(377,·)$, $\chi_{420}(379,·)$, $\chi_{420}(253,·)$, $\chi_{420}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{18} + \frac{1}{4} a^{16} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{19} + \frac{1}{8} a^{17} - \frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{8} a^{9} + \frac{1}{8} a^{5} + \frac{1}{8} a^{3}$, $\frac{1}{176} a^{20} - \frac{1}{16} a^{18} + \frac{1}{4} a^{16} + \frac{5}{16} a^{14} - \frac{7}{16} a^{12} + \frac{43}{176} a^{10} - \frac{1}{4} a^{8} - \frac{5}{16} a^{6} + \frac{7}{16} a^{4} + \frac{1}{4} a^{2} - \frac{2}{11}$, $\frac{1}{352} a^{21} - \frac{1}{32} a^{19} + \frac{1}{8} a^{17} - \frac{11}{32} a^{15} + \frac{9}{32} a^{13} - \frac{133}{352} a^{11} - \frac{1}{8} a^{9} + \frac{11}{32} a^{7} - \frac{9}{32} a^{5} + \frac{1}{8} a^{3} + \frac{9}{22} a$, $\frac{1}{704} a^{22} + \frac{1}{704} a^{20} - \frac{1}{8} a^{18} + \frac{5}{64} a^{16} + \frac{5}{64} a^{14} + \frac{351}{704} a^{12} - \frac{29}{88} a^{10} + \frac{27}{64} a^{8} + \frac{27}{64} a^{6} - \frac{1}{8} a^{4} + \frac{5}{11} a^{2} + \frac{5}{11}$, $\frac{1}{1408} a^{23} + \frac{1}{1408} a^{21} - \frac{1}{16} a^{19} + \frac{5}{128} a^{17} - \frac{59}{128} a^{15} + \frac{351}{1408} a^{13} - \frac{29}{176} a^{11} + \frac{27}{128} a^{9} - \frac{37}{128} a^{7} + \frac{7}{16} a^{5} - \frac{3}{11} a^{3} + \frac{5}{22} a$, $\frac{1}{14080} a^{24} + \frac{1}{2816} a^{22} - \frac{1}{704} a^{20} - \frac{27}{1280} a^{18} - \frac{123}{256} a^{16} - \frac{1153}{2816} a^{14} - \frac{59}{3520} a^{12} + \frac{565}{2816} a^{10} + \frac{27}{256} a^{8} - \frac{7}{320} a^{6} - \frac{31}{88} a^{4} - \frac{3}{22} a^{2} - \frac{14}{55}$, $\frac{1}{28160} a^{25} + \frac{1}{5632} a^{23} - \frac{1}{1408} a^{21} - \frac{27}{2560} a^{19} - \frac{123}{512} a^{17} - \frac{1153}{5632} a^{15} - \frac{59}{7040} a^{13} + \frac{565}{5632} a^{11} + \frac{27}{512} a^{9} - \frac{7}{640} a^{7} + \frac{57}{176} a^{5} - \frac{3}{44} a^{3} - \frac{7}{55} a$, $\frac{1}{25062400} a^{26} + \frac{193}{25062400} a^{24} - \frac{237}{626560} a^{22} + \frac{27463}{25062400} a^{20} + \frac{48389}{2278400} a^{18} + \frac{1266179}{5012480} a^{16} + \frac{767593}{3132800} a^{14} - \frac{1794663}{25062400} a^{12} - \frac{1911467}{5012480} a^{10} + \frac{141669}{284800} a^{8} + \frac{411321}{1566400} a^{6} - \frac{20597}{78320} a^{4} - \frac{8869}{97900} a^{2} + \frac{11667}{24475}$, $\frac{1}{50124800} a^{27} + \frac{193}{50124800} a^{25} - \frac{237}{1253120} a^{23} + \frac{27463}{50124800} a^{21} + \frac{48389}{4556800} a^{19} + \frac{1266179}{10024960} a^{17} + \frac{767593}{6265600} a^{15} - \frac{1794663}{50124800} a^{13} - \frac{1911467}{10024960} a^{11} + \frac{141669}{569600} a^{9} + \frac{411321}{3132800} a^{7} - \frac{20597}{156640} a^{5} - \frac{8869}{195800} a^{3} + \frac{11667}{48950} a$, $\frac{1}{100249600} a^{28} + \frac{1}{100249600} a^{26} + \frac{413}{12531200} a^{24} - \frac{39177}{100249600} a^{22} + \frac{100983}{100249600} a^{20} - \frac{10419553}{100249600} a^{18} - \frac{13933}{140800} a^{16} + \frac{5837289}{100249600} a^{14} + \frac{12111721}{100249600} a^{12} + \frac{2466949}{12531200} a^{10} - \frac{628117}{1566400} a^{8} + \frac{741977}{1566400} a^{6} - \frac{143039}{391600} a^{4} + \frac{19079}{97900} a^{2} - \frac{651}{24475}$, $\frac{1}{200499200} a^{29} + \frac{1}{200499200} a^{27} + \frac{413}{25062400} a^{25} - \frac{39177}{200499200} a^{23} + \frac{100983}{200499200} a^{21} - \frac{10419553}{200499200} a^{19} - \frac{13933}{281600} a^{17} - \frac{94412311}{200499200} a^{15} - \frac{88137879}{200499200} a^{13} + \frac{2466949}{25062400} a^{11} - \frac{628117}{3132800} a^{9} - \frac{824423}{3132800} a^{7} + \frac{248561}{783200} a^{5} - \frac{78821}{195800} a^{3} - \frac{651}{48950} a$, $\frac{1}{400998400} a^{30} + \frac{1}{400998400} a^{28} - \frac{1}{50124800} a^{26} + \frac{5127}{400998400} a^{24} + \frac{170743}{400998400} a^{22} + \frac{581391}{400998400} a^{20} + \frac{3644777}{50124800} a^{18} - \frac{128966951}{400998400} a^{16} - \frac{162762007}{400998400} a^{14} + \frac{9092791}{50124800} a^{12} + \frac{9525477}{25062400} a^{10} - \frac{1900949}{6265600} a^{8} + \frac{655029}{1566400} a^{6} + \frac{55237}{195800} a^{4} - \frac{6217}{24475} a^{2} + \frac{11961}{24475}$, $\frac{1}{801996800} a^{31} + \frac{1}{801996800} a^{29} - \frac{1}{100249600} a^{27} + \frac{5127}{801996800} a^{25} + \frac{170743}{801996800} a^{23} + \frac{581391}{801996800} a^{21} + \frac{3644777}{100249600} a^{19} - \frac{128966951}{801996800} a^{17} - \frac{162762007}{801996800} a^{15} - \frac{41032009}{100249600} a^{13} + \frac{9525477}{50124800} a^{11} - \frac{1900949}{12531200} a^{9} - \frac{911371}{3132800} a^{7} - \frac{140563}{391600} a^{5} + \frac{9129}{24475} a^{3} + \frac{11961}{48950} a$

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

Class group and class number

not computed

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{1}{48950} a^{31} - \frac{7193}{48950} a \) (order $60$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

Galois group

$C_2^3\times C_4$ (as 32T34):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2^3\times C_4$
Character table for $C_2^3\times C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{35}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{-105}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-15}) \), \(\Q(i, \sqrt{35})\), \(\Q(i, \sqrt{105})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{3}, \sqrt{35})\), \(\Q(\sqrt{-3}, \sqrt{35})\), \(\Q(\sqrt{-3}, \sqrt{-35})\), \(\Q(\sqrt{3}, \sqrt{-35})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{7})\), \(\Q(i, \sqrt{21})\), \(\Q(i, \sqrt{15})\), \(\Q(\sqrt{5}, \sqrt{7})\), \(\Q(\sqrt{-5}, \sqrt{-7})\), \(\Q(\sqrt{15}, \sqrt{21})\), \(\Q(\sqrt{-15}, \sqrt{-21})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{-5}, \sqrt{7})\), \(\Q(\sqrt{-15}, \sqrt{21})\), \(\Q(\sqrt{15}, \sqrt{-21})\), \(\Q(\sqrt{5}, \sqrt{21})\), \(\Q(\sqrt{-5}, \sqrt{-21})\), \(\Q(\sqrt{7}, \sqrt{15})\), \(\Q(\sqrt{-7}, \sqrt{-15})\), \(\Q(\sqrt{5}, \sqrt{-21})\), \(\Q(\sqrt{-5}, \sqrt{21})\), \(\Q(\sqrt{7}, \sqrt{-15})\), \(\Q(\sqrt{-7}, \sqrt{15})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{-5})\), \(\Q(\sqrt{3}, \sqrt{7})\), \(\Q(\sqrt{3}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-5})\), \(\Q(\sqrt{-3}, \sqrt{7})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), 4.4.882000.1, 4.0.55125.1, \(\Q(\zeta_{15})^+\), 4.0.18000.1, \(\Q(\zeta_{20})^+\), \(\Q(\zeta_{5})\), 4.4.6125.1, 4.0.98000.1, 8.0.31116960000.10, 8.0.384160000.1, 8.0.31116960000.4, 8.0.31116960000.8, 8.0.31116960000.7, 8.0.12960000.1, 8.0.49787136.1, 8.8.31116960000.1, 8.0.31116960000.1, 8.0.31116960000.9, 8.0.31116960000.3, 8.0.121550625.1, 8.0.31116960000.2, 8.0.31116960000.6, 8.0.31116960000.5, 8.0.777924000000.8, 8.0.324000000.1, \(\Q(\zeta_{20})\), 8.0.9604000000.1, 8.8.777924000000.3, 8.0.777924000000.3, 8.8.9604000000.1, 8.0.9604000000.3, 8.0.777924000000.4, 8.0.3038765625.2, 8.0.9604000000.2, 8.0.37515625.1, 8.8.777924000000.2, 8.0.3038765625.3, 8.8.3038765625.1, 8.0.777924000000.1, 8.0.777924000000.10, 8.0.777924000000.6, 8.0.777924000000.7, 8.0.777924000000.2, 8.8.777924000000.1, 8.0.777924000000.5, \(\Q(\zeta_{60})^+\), 8.0.324000000.3, 8.0.777924000000.9, 8.0.3038765625.1, \(\Q(\zeta_{15})\), 8.0.324000000.2, 16.0.968265199641600000000.1, 16.0.605165749776000000000000.4, 16.0.92236816000000000000.1, 16.0.605165749776000000000000.6, 16.0.605165749776000000000000.3, 16.0.605165749776000000000000.7, \(\Q(\zeta_{60})\), 16.16.605165749776000000000000.1, 16.0.605165749776000000000000.5, 16.0.605165749776000000000000.9, 16.0.605165749776000000000000.1, 16.0.605165749776000000000000.10, 16.0.9234096523681640625.1, 16.0.605165749776000000000000.8, 16.0.605165749776000000000000.2

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 R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$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}$