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Properties

Label 32.0.15693807744...0000.9
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 Not computed
Class group Not computed
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![214358881, 0, 0, 0, 0, 0, 0, 0, 2232620239, 0, 0, 0, 0, 0, 0, 0, 6799201, 0, 0, 0, 0, 0, 0, 0, 5599, 0, 0, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 5599*x^24 + 6799201*x^16 + 2232620239*x^8 + 214358881)
 
gp: K = bnfinit(x^32 + 5599*x^24 + 6799201*x^16 + 2232620239*x^8 + 214358881, 1)
 

Normalized defining polynomial

\( x^{32} + 5599 x^{24} + 6799201 x^{16} + 2232620239 x^{8} + 214358881 \)

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$$\chi_{560}(1,·)$, $\chi_{560}(363,·)$, $\chi_{560}(517,·)$, $\chi_{560}(519,·)$, $\chi_{560}(13,·)$, $\chi_{560}(153,·)$, $\chi_{560}(281,·)$, $\chi_{560}(27,·)$, $\chi_{560}(29,·)$, $\chi_{560}(421,·)$, $\chi_{560}(167,·)$, $\chi_{560}(169,·)$, $\chi_{560}(433,·)$, $\chi_{560}(307,·)$, $\chi_{560}(309,·)$, $\chi_{560}(351,·)$, $\chi_{560}(447,·)$, $\chi_{560}(449,·)$, $\chi_{560}(71,·)$, $\chi_{560}(141,·)$, $\chi_{560}(83,·)$, $\chi_{560}(223,·)$, $\chi_{560}(97,·)$, $\chi_{560}(99,·)$, $\chi_{560}(293,·)$, $\chi_{560}(491,·)$, $\chi_{560}(237,·)$, $\chi_{560}(239,·)$, $\chi_{560}(211,·)$, $\chi_{560}(503,·)$, $\chi_{560}(377,·)$, $\chi_{560}(379,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7} a^{16} + \frac{3}{7} a^{8} - \frac{3}{7}$, $\frac{1}{77} a^{17} + \frac{17}{77} a^{9} - \frac{17}{77} a$, $\frac{1}{77} a^{18} + \frac{17}{77} a^{10} - \frac{17}{77} a^{2}$, $\frac{1}{77} a^{19} + \frac{17}{77} a^{11} - \frac{17}{77} a^{3}$, $\frac{1}{77} a^{20} + \frac{17}{77} a^{12} - \frac{17}{77} a^{4}$, $\frac{1}{77} a^{21} + \frac{17}{77} a^{13} - \frac{17}{77} a^{5}$, $\frac{1}{77} a^{22} + \frac{17}{77} a^{14} - \frac{17}{77} a^{6}$, $\frac{1}{77} a^{23} + \frac{17}{77} a^{15} - \frac{17}{77} a^{7}$, $\frac{1}{18114614174277313} a^{24} + \frac{1158432924040211}{18114614174277313} a^{16} - \frac{6645503133536339}{18114614174277313} a^{8} - \frac{139147950106384}{1646783106752483}$, $\frac{1}{18114614174277313} a^{25} - \frac{17840723640134}{18114614174277313} a^{17} - \frac{1218220138546413}{2587802024896759} a^{9} + \frac{351410385118328}{18114614174277313} a$, $\frac{1}{199260755917050443} a^{26} - \frac{23008677561473}{18114614174277313} a^{18} + \frac{78046199499448501}{199260755917050443} a^{10} - \frac{31878487561323125}{199260755917050443} a^{2}$, $\frac{1}{2191868315087554873} a^{27} + \frac{447500781510665}{199260755917050443} a^{19} + \frac{109116248013869948}{313124045012507839} a^{11} - \frac{119863756407812931}{2191868315087554873} a^{3}$, $\frac{1}{24110551465963103603} a^{28} + \frac{10798708881097701}{2191868315087554873} a^{20} + \frac{9275094595982529987}{24110551465963103603} a^{12} - \frac{10823012931380808155}{24110551465963103603} a^{4}$, $\frac{1}{265216066125594139633} a^{29} + \frac{124661997976555097}{24110551465963103603} a^{21} - \frac{17653573275093144167}{265216066125594139633} a^{13} + \frac{64326757871621073205}{265216066125594139633} a^{5}$, $\frac{1}{2917376727381535535963} a^{30} + \frac{124661997976555097}{265216066125594139633} a^{22} - \frac{17653573275093144167}{2917376727381535535963} a^{14} + \frac{1125191022373997631737}{2917376727381535535963} a^{6}$, $\frac{1}{32091144001196890895593} a^{31} + \frac{124661997976555097}{2917376727381535535963} a^{23} + \frac{8734476608869513463722}{32091144001196890895593} a^{15} + \frac{9877321204518604239626}{32091144001196890895593} a^{7}$

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:  \( \frac{517354891400}{265216066125594139633} a^{29} + \frac{263279087968860}{24110551465963103603} a^{21} + \frac{3512907953406993230}{265216066125594139633} a^{13} + \frac{1147344951726799610803}{265216066125594139633} a^{5} \) (order $16$)
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

$C_2\times C_4^2$ (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\zeta_{8})\), \(\Q(\zeta_{16})^+\), 4.0.2048.2, \(\Q(i, \sqrt{10})\), \(\Q(i, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-5})\), 4.4.51200.1, 4.0.51200.2, \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{5})\), 4.4.12544000.1, 4.0.12544000.1, 4.4.12544000.2, 4.0.12544000.2, 4.4.6125.1, 4.0.98000.1, 4.4.392000.1, 4.0.392000.2, \(\Q(\zeta_{16})\), 8.0.40960000.1, 8.0.10485760000.3, 8.8.2621440000.1, 8.0.10485760000.2, 8.0.2621440000.1, 8.0.10485760000.1, 8.0.629407744000000.56, 8.0.629407744000000.57, 8.0.9604000000.1, 8.0.2458624000000.7, 8.8.157351936000000.4, 8.0.157351936000000.83, 8.8.153664000000.1, 8.0.2458624000000.6, 8.0.629407744000000.73, 8.0.629407744000000.72, 8.0.153664000000.3, 8.0.2458624000000.5, 16.0.109951162777600000000.1, 16.0.396154108207169536000000000000.16, 16.0.6044831973376000000000000.1, 16.16.24759631762948096000000000000.2, 16.0.396154108207169536000000000000.14, 16.0.396154108207169536000000000000.13, 16.0.24759631762948096000000000000.2

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{8}$ 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$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$