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Properties

Label 32.0.59816430901...000.10
Degree $32$
Signature $[0, 16]$
Discriminant $2^{32}\cdot 3^{16}\cdot 5^{24}\cdot 13^{24}$
Root discriminant $79.30$
Ramified primes $2, 3, 5, 13$
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![815730721, 0, -815730721, 0, 62748517, 0, 627485170, 0, -627485170, 0, 574390271, 0, -43812574, 0, -441838670, 0, 441810109, 0, -33987590, 0, -259246, 0, 261443, 0, -21970, 0, 1690, 0, 13, 0, -13, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 13*x^30 + 13*x^28 + 1690*x^26 - 21970*x^24 + 261443*x^22 - 259246*x^20 - 33987590*x^18 + 441810109*x^16 - 441838670*x^14 - 43812574*x^12 + 574390271*x^10 - 627485170*x^8 + 627485170*x^6 + 62748517*x^4 - 815730721*x^2 + 815730721)
 
gp: K = bnfinit(x^32 - 13*x^30 + 13*x^28 + 1690*x^26 - 21970*x^24 + 261443*x^22 - 259246*x^20 - 33987590*x^18 + 441810109*x^16 - 441838670*x^14 - 43812574*x^12 + 574390271*x^10 - 627485170*x^8 + 627485170*x^6 + 62748517*x^4 - 815730721*x^2 + 815730721, 1)
 

Normalized defining polynomial

\( x^{32} - 13 x^{30} + 13 x^{28} + 1690 x^{26} - 21970 x^{24} + 261443 x^{22} - 259246 x^{20} - 33987590 x^{18} + 441810109 x^{16} - 441838670 x^{14} - 43812574 x^{12} + 574390271 x^{10} - 627485170 x^{8} + 627485170 x^{6} + 62748517 x^{4} - 815730721 x^{2} + 815730721 \)

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:  \(5981643090147991811559885370844487936000000000000000000000000=2^{32}\cdot 3^{16}\cdot 5^{24}\cdot 13^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.30$
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:  $2, 3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(780=2^{2}\cdot 3\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{780}(1,·)$, $\chi_{780}(619,·)$, $\chi_{780}(389,·)$, $\chi_{780}(649,·)$, $\chi_{780}(151,·)$, $\chi_{780}(157,·)$, $\chi_{780}(671,·)$, $\chi_{780}(677,·)$, $\chi_{780}(551,·)$, $\chi_{780}(47,·)$, $\chi_{780}(307,·)$, $\chi_{780}(181,·)$, $\chi_{780}(521,·)$, $\chi_{780}(313,·)$, $\chi_{780}(31,·)$, $\chi_{780}(701,·)$, $\chi_{780}(53,·)$, $\chi_{780}(707,·)$, $\chi_{780}(203,·)$, $\chi_{780}(77,·)$, $\chi_{780}(463,·)$, $\chi_{780}(337,·)$, $\chi_{780}(83,·)$, $\chi_{780}(469,·)$, $\chi_{780}(343,·)$, $\chi_{780}(187,·)$, $\chi_{780}(209,·)$, $\chi_{780}(233,·)$, $\chi_{780}(359,·)$, $\chi_{780}(493,·)$, $\chi_{780}(239,·)$, $\chi_{780}(499,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{13} a^{4}$, $\frac{1}{13} a^{5}$, $\frac{1}{13} a^{6}$, $\frac{1}{13} a^{7}$, $\frac{1}{169} a^{8}$, $\frac{1}{169} a^{9}$, $\frac{1}{169} a^{10}$, $\frac{1}{169} a^{11}$, $\frac{1}{2197} a^{12}$, $\frac{1}{2197} a^{13}$, $\frac{1}{2197} a^{14}$, $\frac{1}{2197} a^{15}$, $\frac{1}{85683} a^{16} + \frac{1}{6591} a^{14} - \frac{1}{507} a^{10} - \frac{1}{507} a^{8} - \frac{1}{39} a^{6} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{85683} a^{17} + \frac{1}{6591} a^{15} - \frac{1}{507} a^{11} - \frac{1}{507} a^{9} - \frac{1}{39} a^{7} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{85683} a^{18} - \frac{1}{6591} a^{14} - \frac{1}{6591} a^{12} + \frac{1}{39} a^{6} + \frac{1}{39} a^{4} - \frac{1}{3}$, $\frac{1}{85683} a^{19} - \frac{1}{6591} a^{15} - \frac{1}{6591} a^{13} + \frac{1}{39} a^{7} + \frac{1}{39} a^{5} - \frac{1}{3} a$, $\frac{1}{145918149} a^{20} - \frac{142}{66417} a^{10} + \frac{1}{393}$, $\frac{1}{145918149} a^{21} - \frac{142}{66417} a^{11} + \frac{1}{393} a$, $\frac{1}{145918149} a^{22} + \frac{119}{863421} a^{12} + \frac{1}{393} a^{2}$, $\frac{1}{145918149} a^{23} + \frac{119}{863421} a^{13} + \frac{1}{393} a^{3}$, $\frac{1}{22763231244} a^{24} + \frac{1}{1028196} a^{18} + \frac{47}{287807} a^{14} + \frac{11}{79092} a^{12} + \frac{1}{468} a^{6} - \frac{40}{1703} a^{4} + \frac{11}{36}$, $\frac{1}{22763231244} a^{25} + \frac{1}{1028196} a^{19} + \frac{47}{287807} a^{15} + \frac{11}{79092} a^{13} + \frac{1}{468} a^{7} - \frac{40}{1703} a^{5} + \frac{11}{36} a$, $\frac{1}{292985549341524} a^{26} - \frac{25}{5634337487337} a^{24} + \frac{1180}{1878112495779} a^{22} - \frac{14041}{22537349949348} a^{20} - \frac{295}{3308477679} a^{18} + \frac{11800}{11113091691} a^{16} - \frac{140411}{133357100292} a^{14} - \frac{295}{2564559621} a^{12} - \frac{608837}{284951069} a^{10} - \frac{140411}{78307164} a^{8} + \frac{295}{197273817} a^{6} + \frac{140411}{65757939} a^{4} - \frac{119}{60699636} a^{2} - \frac{421171}{15174909}$, $\frac{1}{292985549341524} a^{27} - \frac{25}{5634337487337} a^{25} + \frac{1180}{1878112495779} a^{23} - \frac{14041}{22537349949348} a^{21} - \frac{295}{3308477679} a^{19} + \frac{11800}{11113091691} a^{17} - \frac{140411}{133357100292} a^{15} - \frac{295}{2564559621} a^{13} - \frac{608837}{284951069} a^{11} - \frac{140411}{78307164} a^{9} + \frac{295}{197273817} a^{7} + \frac{140411}{65757939} a^{5} - \frac{119}{60699636} a^{3} - \frac{421171}{15174909} a$, $\frac{1}{3808812141439812} a^{28} - \frac{1}{292985549341524} a^{24} + \frac{12971}{22537349949348} a^{22} - \frac{1180}{1878112495779} a^{20} - \frac{141491}{1733642303796} a^{18} + \frac{14051}{13233910716} a^{16} - \frac{11800}{11113091691} a^{14} - \frac{14041}{133357100292} a^{12} - \frac{21918133}{10258238484} a^{10} - \frac{510940}{284951069} a^{8} - \frac{167413}{6023628} a^{6} + \frac{1684921}{789095268} a^{4} - \frac{128620}{5058303} a^{2} + \frac{119}{60699636}$, $\frac{1}{3808812141439812} a^{29} - \frac{1}{292985549341524} a^{25} + \frac{12971}{22537349949348} a^{23} - \frac{1180}{1878112495779} a^{21} - \frac{141491}{1733642303796} a^{19} + \frac{14051}{13233910716} a^{17} - \frac{11800}{11113091691} a^{15} - \frac{14041}{133357100292} a^{13} - \frac{21918133}{10258238484} a^{11} - \frac{510940}{284951069} a^{9} - \frac{167413}{6023628} a^{7} + \frac{1684921}{789095268} a^{5} - \frac{128620}{5058303} a^{3} + \frac{119}{60699636} a$, $\frac{1}{3808812141439812} a^{30} - \frac{18378371}{60699636}$, $\frac{1}{3808812141439812} a^{31} - \frac{18378371}{60699636} a$

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:  \( -\frac{142681}{97661849780508} a^{30} + \frac{142681}{8138487481709} a^{28} + \frac{1309}{97661849780508} a^{26} - \frac{142681}{57346946436} a^{24} + \frac{1426810}{48156730661} a^{22} - \frac{15552229}{44452366764} a^{20} - \frac{142681}{44452366764} a^{18} + \frac{2409608499}{48156730661} a^{16} - \frac{15552229}{26102388} a^{14} + \frac{142681}{3419412828} a^{12} + \frac{15552229}{21919313} a^{10} - \frac{203891149}{263031756} a^{8} + \frac{222581051}{263031756} a^{6} - \frac{20403383}{20233212} a^{2} + \frac{1854853}{1686101} \) (order $30$)
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

\(\Q(\sqrt{-195}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{-3}, \sqrt{65})\), 4.0.4394000.2, 4.4.39546000.2, \(\Q(\sqrt{13}, \sqrt{-15})\), \(\Q(\sqrt{5}, \sqrt{-39})\), \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\sqrt{-15}, \sqrt{-39})\), 4.0.4394000.1, 4.4.39546000.1, \(\Q(\sqrt{-3}, \sqrt{13})\), \(\Q(\sqrt{-3}, \sqrt{5})\), 4.0.316368.2, 4.4.878800.1, 4.0.7909200.1, 4.4.35152.1, \(\Q(\zeta_{15})^+\), 4.0.21125.1, 4.4.190125.1, \(\Q(\zeta_{5})\), 8.0.1563886116000000.48, 8.0.1445900625.1, 8.0.1563886116000000.39, 8.0.19307236000000.5, 8.0.1563886116000000.66, 8.8.1563886116000000.7, 8.0.1563886116000000.47, 8.0.62555444640000.22, 8.0.62555444640000.40, 8.0.36147515625.1, 8.0.36147515625.3, 8.0.62555444640000.63, 8.8.772289440000.1, 8.8.36147515625.1, 8.0.446265625.1, 8.0.100088711424.2, 8.0.62555444640000.70, \(\Q(\zeta_{15})\), 8.0.36147515625.2, 16.0.2445739783817565456000000000000.12, 16.0.3913183654108104729600000000.6, 16.0.1306642885859619140625.1, 16.0.2445739783817565456000000000000.15, 16.0.372769361959696000000000000.3, 16.0.2445739783817565456000000000000.14, 16.16.2445739783817565456000000000000.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
$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}$
5Data not computed
13Data not computed