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Properties

Label 32.0.15729269092...0000.7
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 $51200$ (GRH)
Class group $[4, 4, 20, 160]$ (GRH)
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![3481, 0, 16422920, 0, 140204289, 0, 498130816, 0, 987632472, 0, 1232817768, 0, 1035373423, 0, 609791200, 0, 258638727, 0, 80265176, 0, 18349567, 0, 3083136, 0, 375816, 0, 32312, 0, 1857, 0, 64, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 64*x^30 + 1857*x^28 + 32312*x^26 + 375816*x^24 + 3083136*x^22 + 18349567*x^20 + 80265176*x^18 + 258638727*x^16 + 609791200*x^14 + 1035373423*x^12 + 1232817768*x^10 + 987632472*x^8 + 498130816*x^6 + 140204289*x^4 + 16422920*x^2 + 3481)
 
gp: K = bnfinit(x^32 + 64*x^30 + 1857*x^28 + 32312*x^26 + 375816*x^24 + 3083136*x^22 + 18349567*x^20 + 80265176*x^18 + 258638727*x^16 + 609791200*x^14 + 1035373423*x^12 + 1232817768*x^10 + 987632472*x^8 + 498130816*x^6 + 140204289*x^4 + 16422920*x^2 + 3481, 1)
 

Normalized defining polynomial

\( x^{32} + 64 x^{30} + 1857 x^{28} + 32312 x^{26} + 375816 x^{24} + 3083136 x^{22} + 18349567 x^{20} + 80265176 x^{18} + 258638727 x^{16} + 609791200 x^{14} + 1035373423 x^{12} + 1232817768 x^{10} + 987632472 x^{8} + 498130816 x^{6} + 140204289 x^{4} + 16422920 x^{2} + 3481 \)

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$$\chi_{840}(1,·)$, $\chi_{840}(643,·)$, $\chi_{840}(391,·)$, $\chi_{840}(559,·)$, $\chi_{840}(797,·)$, $\chi_{840}(419,·)$, $\chi_{840}(293,·)$, $\chi_{840}(167,·)$, $\chi_{840}(41,·)$, $\chi_{840}(29,·)$, $\chi_{840}(433,·)$, $\chi_{840}(307,·)$, $\chi_{840}(181,·)$, $\chi_{840}(827,·)$, $\chi_{840}(701,·)$, $\chi_{840}(323,·)$, $\chi_{840}(71,·)$, $\chi_{840}(463,·)$, $\chi_{840}(209,·)$, $\chi_{840}(211,·)$, $\chi_{840}(169,·)$, $\chi_{840}(349,·)$, $\chi_{840}(97,·)$, $\chi_{840}(379,·)$, $\chi_{840}(617,·)$, $\chi_{840}(239,·)$, $\chi_{840}(113,·)$, $\chi_{840}(757,·)$, $\chi_{840}(503,·)$, $\chi_{840}(251,·)$, $\chi_{840}(253,·)$, $\chi_{840}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{14} + \frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{15} + \frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{18} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{19} - \frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{627} a^{20} + \frac{40}{627} a^{18} + \frac{53}{627} a^{16} + \frac{130}{627} a^{14} + \frac{34}{627} a^{12} + \frac{20}{57} a^{10} - \frac{2}{19} a^{8} - \frac{1}{19} a^{6} - \frac{3}{19} a^{4} - \frac{214}{627} a^{2} - \frac{40}{209}$, $\frac{1}{627} a^{21} + \frac{40}{627} a^{19} + \frac{53}{627} a^{17} + \frac{130}{627} a^{15} + \frac{34}{627} a^{13} + \frac{20}{57} a^{11} - \frac{2}{19} a^{9} - \frac{1}{19} a^{7} - \frac{3}{19} a^{5} - \frac{214}{627} a^{3} - \frac{40}{209} a$, $\frac{1}{627} a^{22} - \frac{28}{209} a^{18} + \frac{100}{627} a^{16} + \frac{268}{627} a^{14} - \frac{16}{33} a^{12} + \frac{11}{57} a^{10} + \frac{28}{57} a^{8} - \frac{1}{19} a^{6} - \frac{16}{627} a^{4} + \frac{289}{627} a^{2} + \frac{202}{627}$, $\frac{1}{627} a^{23} - \frac{28}{209} a^{19} + \frac{100}{627} a^{17} + \frac{268}{627} a^{15} - \frac{16}{33} a^{13} + \frac{11}{57} a^{11} + \frac{28}{57} a^{9} - \frac{1}{19} a^{7} - \frac{16}{627} a^{5} + \frac{289}{627} a^{3} + \frac{202}{627} a$, $\frac{1}{9405} a^{24} + \frac{1}{3135} a^{22} + \frac{1}{3135} a^{20} + \frac{164}{1881} a^{18} + \frac{111}{1045} a^{16} + \frac{244}{1045} a^{14} - \frac{14}{95} a^{12} + \frac{119}{285} a^{10} - \frac{128}{285} a^{8} - \frac{4658}{9405} a^{6} - \frac{489}{1045} a^{4} - \frac{137}{3135} a^{2} + \frac{151}{855}$, $\frac{1}{9405} a^{25} + \frac{1}{3135} a^{23} + \frac{1}{3135} a^{21} + \frac{164}{1881} a^{19} + \frac{111}{1045} a^{17} + \frac{244}{1045} a^{15} - \frac{14}{95} a^{13} + \frac{119}{285} a^{11} - \frac{128}{285} a^{9} - \frac{4658}{9405} a^{7} - \frac{489}{1045} a^{5} - \frac{137}{3135} a^{3} + \frac{151}{855} a$, $\frac{1}{4185225} a^{26} + \frac{52}{4185225} a^{24} - \frac{2}{3135} a^{22} - \frac{563}{4185225} a^{20} + \frac{357934}{4185225} a^{18} + \frac{21073}{465025} a^{16} - \frac{575414}{1395075} a^{14} - \frac{195258}{465025} a^{12} - \frac{44002}{126825} a^{10} - \frac{565394}{4185225} a^{8} + \frac{403102}{4185225} a^{6} + \frac{993}{8455} a^{4} - \frac{1509238}{4185225} a^{2} + \frac{1494839}{4185225}$, $\frac{1}{246928275} a^{27} - \frac{1082}{27436475} a^{25} + \frac{17}{61655} a^{23} - \frac{136733}{246928275} a^{21} + \frac{63751}{27436475} a^{19} - \frac{158741}{7482675} a^{17} + \frac{4945256}{82309425} a^{15} + \frac{14902006}{82309425} a^{13} + \frac{3645938}{7482675} a^{11} + \frac{32637391}{246928275} a^{9} + \frac{4206083}{27436475} a^{7} + \frac{345155}{1097459} a^{5} + \frac{121269377}{246928275} a^{3} + \frac{21437458}{82309425} a$, $\frac{1}{246928275} a^{28} - \frac{1}{82309425} a^{26} - \frac{661}{49385655} a^{24} + \frac{99562}{246928275} a^{22} + \frac{18383}{82309425} a^{20} + \frac{19233862}{246928275} a^{18} - \frac{168436}{4332075} a^{16} - \frac{11477779}{82309425} a^{14} - \frac{4478327}{82309425} a^{12} + \frac{8075986}{246928275} a^{10} + \frac{13559299}{82309425} a^{8} + \frac{1358638}{9877131} a^{6} + \frac{7519567}{22448025} a^{4} + \frac{25611118}{82309425} a^{2} + \frac{1259}{837045}$, $\frac{1}{246928275} a^{29} - \frac{696}{27436475} a^{25} - \frac{11243}{246928275} a^{23} + \frac{2612}{5487295} a^{21} + \frac{33742}{924825} a^{19} - \frac{1955249}{27436475} a^{17} + \frac{4460699}{82309425} a^{15} - \frac{172642}{1444025} a^{13} + \frac{99568783}{246928275} a^{11} + \frac{600230}{3292377} a^{9} + \frac{11949314}{27436475} a^{7} + \frac{2840282}{246928275} a^{5} + \frac{4475857}{16461885} a^{3} + \frac{9265669}{82309425} a$, $\frac{1}{246928275} a^{30} - \frac{2}{49385655} a^{26} - \frac{73}{16461885} a^{24} + \frac{13087}{16461885} a^{22} - \frac{182968}{246928275} a^{20} + \frac{7590004}{49385655} a^{18} + \frac{6226}{78765} a^{16} - \frac{114363}{498845} a^{14} - \frac{23307484}{49385655} a^{12} - \frac{38242753}{82309425} a^{10} - \frac{6243407}{49385655} a^{8} - \frac{11820091}{49385655} a^{6} + \frac{6850607}{16461885} a^{4} - \frac{20600467}{49385655} a^{2} - \frac{550906}{4185225}$, $\frac{1}{246928275} a^{31} + \frac{119}{4489605} a^{25} + \frac{716}{16461885} a^{23} - \frac{17921}{82309425} a^{21} + \frac{534157}{4489605} a^{19} + \frac{33373}{498845} a^{17} - \frac{5839994}{16461885} a^{15} + \frac{18089408}{49385655} a^{13} - \frac{36895693}{82309425} a^{11} - \frac{6430847}{16461885} a^{9} + \frac{5146142}{16461885} a^{7} + \frac{437419}{1496535} a^{5} - \frac{480228}{5487295} a^{3} - \frac{89232994}{246928275} a$

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

Class group and class number

$C_{4}\times C_{4}\times C_{20}\times C_{160}$, which has order $51200$ (assuming GRH)

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:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 69548659497.34634 \) (assuming GRH)
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

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{-210}) \), \(\Q(\sqrt{-70}) \), \(\Q(\sqrt{35}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-42}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-2}, \sqrt{105})\), \(\Q(\sqrt{-2}, \sqrt{35})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{-6}, \sqrt{-70})\), \(\Q(\sqrt{3}, \sqrt{35})\), \(\Q(\sqrt{3}, \sqrt{-70})\), \(\Q(\sqrt{-6}, \sqrt{35})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{21})\), \(\Q(\sqrt{-2}, \sqrt{7})\), \(\Q(\sqrt{-2}, \sqrt{15})\), \(\Q(\sqrt{5}, \sqrt{21})\), \(\Q(\sqrt{-10}, \sqrt{-42})\), \(\Q(\sqrt{-14}, \sqrt{-30})\), \(\Q(\sqrt{7}, \sqrt{15})\), \(\Q(\sqrt{5}, \sqrt{-42})\), \(\Q(\sqrt{-10}, \sqrt{21})\), \(\Q(\sqrt{-14}, \sqrt{15})\), \(\Q(\sqrt{7}, \sqrt{-30})\), \(\Q(\sqrt{5}, \sqrt{-14})\), \(\Q(\sqrt{7}, \sqrt{-10})\), \(\Q(\sqrt{21}, \sqrt{-30})\), \(\Q(\sqrt{15}, \sqrt{-42})\), \(\Q(\sqrt{5}, \sqrt{7})\), \(\Q(\sqrt{-10}, \sqrt{-14})\), \(\Q(\sqrt{15}, \sqrt{21})\), \(\Q(\sqrt{-30}, \sqrt{35})\), \(\Q(\sqrt{5}, \sqrt{-6})\), \(\Q(\sqrt{-6}, \sqrt{-10})\), \(\Q(\sqrt{-6}, \sqrt{-14})\), \(\Q(\sqrt{-6}, \sqrt{7})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{-10})\), \(\Q(\sqrt{3}, \sqrt{7})\), \(\Q(\sqrt{3}, \sqrt{-14})\), 4.0.8000.2, \(\Q(\zeta_{20})^+\), 4.0.3528000.1, 4.4.882000.1, 4.4.6125.1, 4.0.392000.2, \(\Q(\zeta_{15})^+\), 4.0.72000.2, 8.0.7965941760000.57, 8.0.497871360000.13, 8.0.7965941760000.24, 8.0.98344960000.6, 8.0.7965941760000.28, 8.0.3317760000.7, 8.0.12745506816.3, 8.0.497871360000.6, 8.0.7965941760000.20, 8.8.31116960000.1, 8.0.7965941760000.22, 8.0.7965941760000.53, 8.0.7965941760000.60, 8.0.7965941760000.59, 8.0.7965941760000.46, 8.0.1024000000.1, 8.0.199148544000000.169, 8.0.153664000000.3, 8.0.5184000000.2, 8.0.12446784000000.5, 8.8.777924000000.2, 8.8.3038765625.1, 8.0.12446784000000.1, 8.0.199148544000000.59, 8.0.199148544000000.160, 8.0.12446784000000.2, 8.0.12446784000000.13, 8.0.153664000000.1, 8.0.2458624000000.4, 8.0.12446784000000.16, 8.0.199148544000000.16, 8.0.2458624000000.3, 8.8.9604000000.1, 8.0.199148544000000.157, 8.8.777924000000.3, 8.0.5184000000.1, 8.0.82944000000.2, 8.0.12446784000000.6, 8.0.199148544000000.193, 8.0.82944000000.1, \(\Q(\zeta_{60})^+\), 8.0.199148544000000.137, 8.8.777924000000.1, 16.0.63456228123711897600000000.22, 16.0.39660142577319936000000000000.42, 16.0.154922431942656000000000000.9, 16.0.6044831973376000000000000.3, 16.0.39660142577319936000000000000.19, 16.0.6879707136000000000000.1, 16.0.39660142577319936000000000000.21, 16.0.154922431942656000000000000.4, 16.0.39660142577319936000000000000.56, 16.0.39660142577319936000000000000.11, 16.16.605165749776000000000000.1, 16.0.39660142577319936000000000000.57, 16.0.39660142577319936000000000000.48, 16.0.39660142577319936000000000000.64, 16.0.39660142577319936000000000000.51

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 ${\href{/LocalNumberField/11.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{32}$

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}$