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Properties

Label 32.0.15729269092...0000.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{64}\cdot 3^{16}\cdot 5^{24}\cdot 7^{16}$
Root discriminant $61.29$
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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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![152587890625, 0, 0, 0, -17333984375, 0, 0, 0, 1725000000, 0, 0, 0, -168225625, 0, 0, 0, 16350431, 0, 0, 0, -269161, 0, 0, 0, 4416, 0, 0, 0, -71, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 71*x^28 + 4416*x^24 - 269161*x^20 + 16350431*x^16 - 168225625*x^12 + 1725000000*x^8 - 17333984375*x^4 + 152587890625)
 
gp: K = bnfinit(x^32 - 71*x^28 + 4416*x^24 - 269161*x^20 + 16350431*x^16 - 168225625*x^12 + 1725000000*x^8 - 17333984375*x^4 + 152587890625, 1)
 

Normalized defining polynomial

\( x^{32} - 71 x^{28} + 4416 x^{24} - 269161 x^{20} + 16350431 x^{16} - 168225625 x^{12} + 1725000000 x^{8} - 17333984375 x^{4} + 152587890625 \)

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:  \(1572926909253345615680132503044096000000000000000000000000=2^{64}\cdot 3^{16}\cdot 5^{24}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.29$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(840=2^{3}\cdot 3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{840}(1,·)$, $\chi_{840}(797,·)$, $\chi_{840}(671,·)$, $\chi_{840}(673,·)$, $\chi_{840}(547,·)$, $\chi_{840}(293,·)$, $\chi_{840}(167,·)$, $\chi_{840}(41,·)$, $\chi_{840}(43,·)$, $\chi_{840}(799,·)$, $\chi_{840}(629,·)$, $\chi_{840}(587,·)$, $\chi_{840}(211,·)$, $\chi_{840}(839,·)$, $\chi_{840}(713,·)$, $\chi_{840}(631,·)$, $\chi_{840}(589,·)$, $\chi_{840}(461,·)$, $\chi_{840}(337,·)$, $\chi_{840}(419,·)$, $\chi_{840}(169,·)$, $\chi_{840}(463,·)$, $\chi_{840}(421,·)$, $\chi_{840}(379,·)$, $\chi_{840}(209,·)$, $\chi_{840}(83,·)$, $\chi_{840}(757,·)$, $\chi_{840}(503,·)$, $\chi_{840}(377,·)$, $\chi_{840}(251,·)$, $\chi_{840}(253,·)$, $\chi_{840}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{11} a^{16} + \frac{3}{11} a^{12} - \frac{2}{11} a^{8} + \frac{5}{11} a^{4} + \frac{4}{11}$, $\frac{1}{55} a^{17} + \frac{14}{55} a^{13} - \frac{24}{55} a^{9} - \frac{6}{55} a^{5} + \frac{26}{55} a$, $\frac{1}{275} a^{18} - \frac{96}{275} a^{14} - \frac{134}{275} a^{10} - \frac{61}{275} a^{6} + \frac{81}{275} a^{2}$, $\frac{1}{1375} a^{19} + \frac{179}{1375} a^{15} + \frac{416}{1375} a^{11} + \frac{214}{1375} a^{7} - \frac{194}{1375} a^{3}$, $\frac{1}{112409213125} a^{20} + \frac{184}{6875} a^{16} + \frac{1311}{6875} a^{12} - \frac{1831}{6875} a^{8} - \frac{1874}{6875} a^{4} + \frac{32431701}{179854741}$, $\frac{1}{562046065625} a^{21} + \frac{184}{34375} a^{17} + \frac{8186}{34375} a^{13} - \frac{8706}{34375} a^{9} + \frac{5001}{34375} a^{5} + \frac{32431701}{899273705} a$, $\frac{1}{2810230328125} a^{22} + \frac{184}{171875} a^{18} - \frac{26189}{171875} a^{14} + \frac{25669}{171875} a^{10} - \frac{63749}{171875} a^{6} - \frac{1766115709}{4496368525} a^{2}$, $\frac{1}{14051151640625} a^{23} + \frac{184}{859375} a^{19} + \frac{317561}{859375} a^{15} - \frac{318081}{859375} a^{11} + \frac{280001}{859375} a^{7} + \frac{7226621341}{22481842625} a^{3}$, $\frac{1}{70255758203125} a^{24} - \frac{71}{70255758203125} a^{20} - \frac{43689}{4296875} a^{16} - \frac{528706}{4296875} a^{12} + \frac{390626}{4296875} a^{8} + \frac{30656788964}{112409213125} a^{4} - \frac{32696446}{179854741}$, $\frac{1}{351278791015625} a^{25} - \frac{71}{351278791015625} a^{21} - \frac{43689}{21484375} a^{17} + \frac{3768169}{21484375} a^{13} - \frac{3906249}{21484375} a^{9} + \frac{255475215214}{562046065625} a^{5} + \frac{327013036}{899273705} a$, $\frac{1}{1756393955078125} a^{26} - \frac{71}{1756393955078125} a^{22} - \frac{43689}{107421875} a^{18} + \frac{25252544}{107421875} a^{14} - \frac{46874999}{107421875} a^{10} - \frac{306570850411}{2810230328125} a^{6} + \frac{2125560446}{4496368525} a^{2}$, $\frac{1}{8781969775390625} a^{27} - \frac{71}{8781969775390625} a^{23} - \frac{43689}{537109375} a^{19} + \frac{25252544}{537109375} a^{15} - \frac{154296874}{537109375} a^{11} - \frac{5927031506661}{14051151640625} a^{7} + \frac{11118297496}{22481842625} a^{3}$, $\frac{1}{43909848876953125} a^{28} - \frac{71}{43909848876953125} a^{24} + \frac{4416}{43909848876953125} a^{20} + \frac{76033794}{2685546875} a^{16} - \frac{732421874}{2685546875} a^{12} + \frac{12773773949589}{70255758203125} a^{8} - \frac{51095092459}{112409213125} a^{4} - \frac{65401795}{179854741}$, $\frac{1}{219549244384765625} a^{29} - \frac{71}{219549244384765625} a^{25} + \frac{4416}{219549244384765625} a^{21} + \frac{76033794}{13427734375} a^{17} + \frac{1953125001}{13427734375} a^{13} + \frac{12773773949589}{351278791015625} a^{9} - \frac{275913518709}{562046065625} a^{5} + \frac{114452946}{899273705} a$, $\frac{1}{1097746221923828125} a^{30} - \frac{71}{1097746221923828125} a^{26} + \frac{4416}{1097746221923828125} a^{22} + \frac{76033794}{67138671875} a^{18} + \frac{1953125001}{67138671875} a^{14} + \frac{364052564965214}{1756393955078125} a^{10} + \frac{286132546916}{2810230328125} a^{6} + \frac{1013726651}{4496368525} a^{2}$, $\frac{1}{5488731109619140625} a^{31} - \frac{71}{5488731109619140625} a^{27} + \frac{4416}{5488731109619140625} a^{23} + \frac{76033794}{335693359375} a^{19} + \frac{1953125001}{335693359375} a^{15} + \frac{364052564965214}{8781969775390625} a^{11} + \frac{5906593203166}{14051151640625} a^{7} + \frac{5510095176}{22481842625} a^{3}$

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:  \( -\frac{9704351}{219549244384765625} a^{29} + \frac{603583296}{219549244384765625} a^{25} - \frac{36789194641}{219549244384765625} a^{21} + \frac{136681}{13427734375} a^{17} - \frac{8275601}{13427734375} a^{13} + \frac{603583296}{562046065625} a^{9} - \frac{9704351}{899273705} a^{5} + \frac{17085125}{179854741} a \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-210}) \), \(\Q(\sqrt{210}) \), \(\Q(\sqrt{-105}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-42}) \), \(\Q(\sqrt{42}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{21}) \), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{210})\), \(\Q(i, \sqrt{105})\), \(\Q(\sqrt{2}, \sqrt{-105})\), \(\Q(\sqrt{2}, \sqrt{105})\), \(\Q(\sqrt{-2}, \sqrt{105})\), \(\Q(\sqrt{-2}, \sqrt{-105})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{10})\), \(\Q(i, \sqrt{42})\), \(\Q(i, \sqrt{21})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{2}, \sqrt{-21})\), \(\Q(\sqrt{2}, \sqrt{21})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{21})\), \(\Q(\sqrt{-2}, \sqrt{-21})\), \(\Q(\sqrt{5}, \sqrt{-42})\), \(\Q(\sqrt{-5}, \sqrt{42})\), \(\Q(\sqrt{10}, \sqrt{-21})\), \(\Q(\sqrt{-10}, \sqrt{21})\), \(\Q(\sqrt{5}, \sqrt{42})\), \(\Q(\sqrt{-5}, \sqrt{-42})\), \(\Q(\sqrt{10}, \sqrt{21})\), \(\Q(\sqrt{-10}, \sqrt{-21})\), \(\Q(\sqrt{5}, \sqrt{-21})\), \(\Q(\sqrt{-5}, \sqrt{21})\), \(\Q(\sqrt{10}, \sqrt{-42})\), \(\Q(\sqrt{-10}, \sqrt{42})\), \(\Q(\sqrt{5}, \sqrt{21})\), \(\Q(\sqrt{-5}, \sqrt{-21})\), \(\Q(\sqrt{10}, \sqrt{42})\), \(\Q(\sqrt{-10}, \sqrt{-42})\), 4.4.3528000.1, 4.0.3528000.1, 4.4.882000.1, 4.0.55125.1, \(\Q(\zeta_{5})\), \(\Q(\zeta_{20})^+\), 4.0.8000.2, 4.4.8000.1, 8.0.7965941760000.69, 8.0.40960000.1, 8.0.12745506816.8, 8.0.7965941760000.68, 8.0.7965941760000.39, 8.0.31116960000.8, 8.0.7965941760000.13, 8.0.7965941760000.62, 8.0.7965941760000.40, 8.8.497871360000.1, 8.0.7965941760000.33, 8.0.497871360000.13, 8.0.7965941760000.4, 8.0.7965941760000.55, 8.0.7965941760000.12, 8.0.199148544000000.177, 8.0.777924000000.8, \(\Q(\zeta_{20})\), 8.0.1024000000.2, 8.8.199148544000000.8, 8.0.12446784000000.14, 8.0.64000000.2, \(\Q(\zeta_{40})^+\), 8.0.12446784000000.7, 8.0.199148544000000.169, 8.0.64000000.1, 8.0.1024000000.1, 8.0.12446784000000.19, 8.0.199148544000000.160, 8.0.199148544000000.59, 8.0.12446784000000.12, 8.8.199148544000000.7, 8.0.12446784000000.20, 8.8.199148544000000.4, 8.0.12446784000000.11, 8.0.199148544000000.138, 8.0.199148544000000.224, 8.0.777924000000.10, 8.0.777924000000.6, 8.8.12446784000000.2, 8.0.12446784000000.5, 8.8.777924000000.2, 8.0.3038765625.3, 16.0.63456228123711897600000000.6, 16.0.39660142577319936000000000000.60, \(\Q(\zeta_{40})\), 16.0.39660142577319936000000000000.66, 16.0.39660142577319936000000000000.35, 16.0.39660142577319936000000000000.25, 16.0.605165749776000000000000.6, 16.0.39660142577319936000000000000.58, 16.0.39660142577319936000000000000.50, 16.16.39660142577319936000000000000.5, 16.0.154922431942656000000000000.20, 16.0.154922431942656000000000000.19, 16.0.39660142577319936000000000000.42, 16.0.39660142577319936000000000000.12, 16.0.39660142577319936000000000000.59

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 ${\href{/LocalNumberField/11.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{32}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

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
$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}$