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Properties

Label 32.0.53503546288...0000.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{64}\cdot 5^{24}\cdot 17^{16}$
Root discriminant $55.15$
Ramified primes $2, 5, 17$
Class number Not computed
Class group Not computed
Galois group $C_2^3\times C_4$ (as 32T34)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4294967296, 0, 0, 0, -822083584, 0, 0, 0, 140574720, 0, 0, 0, -23695616, 0, 0, 0, 3986369, 0, 0, 0, -92561, 0, 0, 0, 2145, 0, 0, 0, -49, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 49*x^28 + 2145*x^24 - 92561*x^20 + 3986369*x^16 - 23695616*x^12 + 140574720*x^8 - 822083584*x^4 + 4294967296)
 
gp: K = bnfinit(x^32 - 49*x^28 + 2145*x^24 - 92561*x^20 + 3986369*x^16 - 23695616*x^12 + 140574720*x^8 - 822083584*x^4 + 4294967296, 1)
 

Normalized defining polynomial

\( x^{32} - 49 x^{28} + 2145 x^{24} - 92561 x^{20} + 3986369 x^{16} - 23695616 x^{12} + 140574720 x^{8} - 822083584 x^{4} + 4294967296 \)

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:  \(53503546288734745169056818528256000000000000000000000000=2^{64}\cdot 5^{24}\cdot 17^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.15$
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$$\chi_{680}(1,·)$, $\chi_{680}(647,·)$, $\chi_{680}(137,·)$, $\chi_{680}(271,·)$, $\chi_{680}(273,·)$, $\chi_{680}(67,·)$, $\chi_{680}(407,·)$, $\chi_{680}(409,·)$, $\chi_{680}(543,·)$, $\chi_{680}(33,·)$, $\chi_{680}(679,·)$, $\chi_{680}(169,·)$, $\chi_{680}(171,·)$, $\chi_{680}(307,·)$, $\chi_{680}(441,·)$, $\chi_{680}(443,·)$, $\chi_{680}(577,·)$, $\chi_{680}(579,·)$, $\chi_{680}(69,·)$, $\chi_{680}(203,·)$, $\chi_{680}(339,·)$, $\chi_{680}(341,·)$, $\chi_{680}(477,·)$, $\chi_{680}(101,·)$, $\chi_{680}(611,·)$, $\chi_{680}(613,·)$, $\chi_{680}(103,·)$, $\chi_{680}(237,·)$, $\chi_{680}(239,·)$, $\chi_{680}(373,·)$, $\chi_{680}(509,·)$, $\chi_{680}(511,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{9} a^{16} - \frac{2}{9} a^{12} + \frac{4}{9} a^{8} + \frac{1}{9} a^{4} - \frac{2}{9}$, $\frac{1}{36} a^{17} + \frac{7}{36} a^{13} + \frac{13}{36} a^{9} - \frac{17}{36} a^{5} - \frac{11}{36} a$, $\frac{1}{144} a^{18} - \frac{65}{144} a^{14} + \frac{49}{144} a^{10} - \frac{17}{144} a^{6} - \frac{47}{144} a^{2}$, $\frac{1}{576} a^{19} + \frac{79}{576} a^{15} - \frac{95}{576} a^{11} - \frac{17}{576} a^{7} - \frac{191}{576} a^{3}$, $\frac{1}{9184594176} a^{20} - \frac{113}{2304} a^{16} - \frac{863}{2304} a^{12} - \frac{209}{2304} a^{8} + \frac{769}{2304} a^{4} + \frac{11866546}{35877321}$, $\frac{1}{36738376704} a^{21} - \frac{113}{9216} a^{17} - \frac{863}{9216} a^{13} - \frac{2513}{9216} a^{9} + \frac{3073}{9216} a^{5} + \frac{47743867}{143509284} a$, $\frac{1}{146953506816} a^{22} - \frac{113}{36864} a^{18} + \frac{17569}{36864} a^{14} + \frac{15919}{36864} a^{10} - \frac{6143}{36864} a^{6} - \frac{95765417}{574037136} a^{2}$, $\frac{1}{587814027264} a^{23} - \frac{113}{147456} a^{19} + \frac{54433}{147456} a^{15} + \frac{15919}{147456} a^{11} + \frac{30721}{147456} a^{7} + \frac{1052308855}{2296148544} a^{3}$, $\frac{1}{2351256109056} a^{24} - \frac{49}{2351256109056} a^{20} - \frac{9055}{589824} a^{16} + \frac{13871}{589824} a^{12} - \frac{131071}{589824} a^{8} + \frac{1360649765}{3061531392} a^{4} + \frac{3988514}{35877321}$, $\frac{1}{9405024436224} a^{25} - \frac{49}{9405024436224} a^{21} - \frac{9055}{2359296} a^{17} - \frac{575953}{2359296} a^{13} - \frac{131071}{2359296} a^{9} - \frac{4762413019}{12246125568} a^{5} + \frac{39865835}{143509284} a$, $\frac{1}{37620097744896} a^{26} - \frac{49}{37620097744896} a^{22} - \frac{9055}{9437184} a^{18} + \frac{4142639}{9437184} a^{14} - \frac{2490367}{9437184} a^{10} + \frac{7483712549}{48984502272} a^{6} + \frac{39865835}{574037136} a^{2}$, $\frac{1}{150480390979584} a^{27} - \frac{49}{150480390979584} a^{23} - \frac{9055}{37748736} a^{19} + \frac{13579823}{37748736} a^{15} + \frac{16384001}{37748736} a^{11} + \frac{56468214821}{195938009088} a^{7} - \frac{534171301}{2296148544} a^{3}$, $\frac{1}{601921563918336} a^{28} - \frac{49}{601921563918336} a^{24} + \frac{715}{200640521306112} a^{20} + \frac{6895151}{150994944} a^{16} + \frac{16777217}{150994944} a^{12} - \frac{58055716681}{261250678784} a^{8} + \frac{4082044001}{9184594176} a^{4} + \frac{3986320}{35877321}$, $\frac{1}{2407686255673344} a^{29} - \frac{49}{2407686255673344} a^{25} + \frac{715}{802562085224448} a^{21} + \frac{6895151}{603979776} a^{17} + \frac{167772161}{603979776} a^{13} - \frac{58055716681}{1045002715136} a^{9} - \frac{14287144351}{36738376704} a^{5} + \frac{39863641}{143509284} a$, $\frac{1}{9630745022693376} a^{30} - \frac{49}{9630745022693376} a^{26} + \frac{715}{3210248340897792} a^{22} + \frac{6895151}{2415919104} a^{18} - \frac{436207615}{2415919104} a^{14} - \frac{1103058431817}{4180010860544} a^{10} + \frac{22451232353}{146953506816} a^{6} + \frac{39863641}{574037136} a^{2}$, $\frac{1}{38522980090773504} a^{31} - \frac{49}{38522980090773504} a^{27} + \frac{715}{12840993363591168} a^{23} + \frac{6895151}{9663676416} a^{19} - \frac{436207615}{9663676416} a^{15} + \frac{7256963289271}{16720043442176} a^{11} + \frac{169404739169}{587814027264} a^{7} - \frac{534173495}{2296148544} a^{3}$

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:  \( -\frac{181}{587814027264} a^{27} - \frac{25963647845}{587814027264} a^{7} \) (order $40$)
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

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

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 $C_2^3\times C_4$
Character table for $C_2^3\times C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-170}) \), \(\Q(\sqrt{170}) \), \(\Q(\sqrt{-85}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-34}) \), \(\Q(\sqrt{34}) \), \(\Q(\sqrt{-17}) \), \(\Q(\sqrt{17}) \), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{170})\), \(\Q(i, \sqrt{85})\), \(\Q(\sqrt{2}, \sqrt{-85})\), \(\Q(\sqrt{2}, \sqrt{85})\), \(\Q(\sqrt{-2}, \sqrt{85})\), \(\Q(\sqrt{-2}, \sqrt{-85})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{10})\), \(\Q(i, \sqrt{34})\), \(\Q(i, \sqrt{17})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{2}, \sqrt{-17})\), \(\Q(\sqrt{2}, \sqrt{17})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{17})\), \(\Q(\sqrt{-2}, \sqrt{-17})\), \(\Q(\sqrt{5}, \sqrt{-34})\), \(\Q(\sqrt{-5}, \sqrt{34})\), \(\Q(\sqrt{10}, \sqrt{-17})\), \(\Q(\sqrt{-10}, \sqrt{17})\), \(\Q(\sqrt{5}, \sqrt{34})\), \(\Q(\sqrt{-5}, \sqrt{-34})\), \(\Q(\sqrt{10}, \sqrt{17})\), \(\Q(\sqrt{-10}, \sqrt{-17})\), \(\Q(\sqrt{5}, \sqrt{-17})\), \(\Q(\sqrt{-5}, \sqrt{17})\), \(\Q(\sqrt{10}, \sqrt{-34})\), \(\Q(\sqrt{-10}, \sqrt{34})\), \(\Q(\sqrt{5}, \sqrt{17})\), \(\Q(\sqrt{-5}, \sqrt{-17})\), \(\Q(\sqrt{10}, \sqrt{34})\), \(\Q(\sqrt{-10}, \sqrt{-34})\), 4.4.2312000.1, 4.0.2312000.1, 4.4.578000.1, 4.0.36125.1, \(\Q(\zeta_{5})\), \(\Q(\zeta_{20})^+\), 4.0.8000.2, 4.4.8000.1, 8.0.3421020160000.10, 8.0.40960000.1, 8.0.5473632256.1, 8.0.3421020160000.9, 8.0.3421020160000.6, 8.0.13363360000.1, 8.0.3421020160000.1, 8.0.3421020160000.8, 8.0.3421020160000.5, 8.8.213813760000.1, 8.0.3421020160000.4, 8.0.213813760000.1, 8.0.3421020160000.2, 8.0.3421020160000.7, 8.0.3421020160000.3, 8.0.85525504000000.31, 8.0.334084000000.2, \(\Q(\zeta_{20})\), 8.0.1024000000.2, 8.8.85525504000000.3, 8.0.5345344000000.5, 8.0.64000000.2, \(\Q(\zeta_{40})^+\), 8.0.5345344000000.1, 8.0.85525504000000.26, 8.0.64000000.1, 8.0.1024000000.1, 8.0.5345344000000.6, 8.0.85525504000000.22, 8.0.85525504000000.5, 8.0.5345344000000.4, 8.8.85525504000000.2, 8.0.5345344000000.7, 8.8.85525504000000.1, 8.0.5345344000000.3, 8.0.85525504000000.17, 8.0.85525504000000.30, 8.0.334084000000.3, 8.0.334084000000.1, 8.8.5345344000000.2, 8.0.5345344000000.2, 8.8.334084000000.1, 8.0.1305015625.1, 16.0.11703378935126425600000000.1, 16.0.7314611834454016000000000000.9, \(\Q(\zeta_{40})\), 16.0.7314611834454016000000000000.10, 16.0.7314611834454016000000000000.4, 16.0.7314611834454016000000000000.3, 16.0.111612119056000000000000.1, 16.0.7314611834454016000000000000.7, 16.0.7314611834454016000000000000.6, 16.16.7314611834454016000000000000.1, 16.0.28572702478336000000000000.1, 16.0.28572702478336000000000000.2, 16.0.7314611834454016000000000000.5, 16.0.7314611834454016000000000000.11, 16.0.7314611834454016000000000000.8

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

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

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
$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}$
$17$17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$