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Properties

Label 32.0.20328239244...0000.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{96}\cdot 3^{16}\cdot 5^{24}$
Root discriminant $46.33$
Ramified primes $2, 3, 5$
Class number $80$ (GRH)
Class group $[2, 2, 20]$ (GRH)
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![1, 0, -48, 0, 1988, 0, -13840, 0, 67325, 0, -146912, 0, 215964, 0, -220120, 0, 169224, 0, -98960, 0, 45376, 0, -16184, 0, 4525, 0, -960, 0, 152, 0, -16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 16*x^30 + 152*x^28 - 960*x^26 + 4525*x^24 - 16184*x^22 + 45376*x^20 - 98960*x^18 + 169224*x^16 - 220120*x^14 + 215964*x^12 - 146912*x^10 + 67325*x^8 - 13840*x^6 + 1988*x^4 - 48*x^2 + 1)
 
gp: K = bnfinit(x^32 - 16*x^30 + 152*x^28 - 960*x^26 + 4525*x^24 - 16184*x^22 + 45376*x^20 - 98960*x^18 + 169224*x^16 - 220120*x^14 + 215964*x^12 - 146912*x^10 + 67325*x^8 - 13840*x^6 + 1988*x^4 - 48*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{32} - 16 x^{30} + 152 x^{28} - 960 x^{26} + 4525 x^{24} - 16184 x^{22} + 45376 x^{20} - 98960 x^{18} + 169224 x^{16} - 220120 x^{14} + 215964 x^{12} - 146912 x^{10} + 67325 x^{8} - 13840 x^{6} + 1988 x^{4} - 48 x^{2} + 1 \)

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:  \(203282392447840896882957090816000000000000000000000000=2^{96}\cdot 3^{16}\cdot 5^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.33$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(240=2^{4}\cdot 3\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{240}(1,·)$, $\chi_{240}(131,·)$, $\chi_{240}(133,·)$, $\chi_{240}(7,·)$, $\chi_{240}(11,·)$, $\chi_{240}(13,·)$, $\chi_{240}(143,·)$, $\chi_{240}(19,·)$, $\chi_{240}(23,·)$, $\chi_{240}(157,·)$, $\chi_{240}(161,·)$, $\chi_{240}(37,·)$, $\chi_{240}(167,·)$, $\chi_{240}(41,·)$, $\chi_{240}(173,·)$, $\chi_{240}(47,·)$, $\chi_{240}(49,·)$, $\chi_{240}(179,·)$, $\chi_{240}(53,·)$, $\chi_{240}(59,·)$, $\chi_{240}(139,·)$, $\chi_{240}(197,·)$, $\chi_{240}(77,·)$, $\chi_{240}(209,·)$, $\chi_{240}(211,·)$, $\chi_{240}(89,·)$, $\chi_{240}(91,·)$, $\chi_{240}(223,·)$, $\chi_{240}(103,·)$, $\chi_{240}(169,·)$, $\chi_{240}(121,·)$, $\chi_{240}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{4} a^{24} + \frac{1}{4}$, $\frac{1}{4} a^{25} + \frac{1}{4} a$, $\frac{1}{4} a^{26} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{27} + \frac{1}{4} a^{3}$, $\frac{1}{796} a^{28} - \frac{21}{398} a^{26} - \frac{37}{398} a^{24} - \frac{49}{199} a^{22} - \frac{27}{199} a^{20} - \frac{54}{199} a^{18} - \frac{23}{199} a^{16} + \frac{66}{199} a^{14} + \frac{60}{199} a^{12} - \frac{99}{199} a^{10} - \frac{28}{199} a^{8} - \frac{43}{199} a^{6} - \frac{283}{796} a^{4} - \frac{139}{398} a^{2} + \frac{65}{398}$, $\frac{1}{796} a^{29} - \frac{21}{398} a^{27} - \frac{37}{398} a^{25} - \frac{49}{199} a^{23} - \frac{27}{199} a^{21} - \frac{54}{199} a^{19} - \frac{23}{199} a^{17} + \frac{66}{199} a^{15} + \frac{60}{199} a^{13} - \frac{99}{199} a^{11} - \frac{28}{199} a^{9} - \frac{43}{199} a^{7} - \frac{283}{796} a^{5} - \frac{139}{398} a^{3} + \frac{65}{398} a$, $\frac{1}{333597497567043560726236} a^{30} - \frac{87796179254378890773}{333597497567043560726236} a^{28} + \frac{19497399130015925822625}{166798748783521780363118} a^{26} - \frac{29958214754697442399175}{333597497567043560726236} a^{24} - \frac{34875794024636335040530}{83399374391760890181559} a^{22} + \frac{13120024048666742360491}{83399374391760890181559} a^{20} - \frac{21381019663800047697501}{83399374391760890181559} a^{18} - \frac{24075340688324064899797}{83399374391760890181559} a^{16} + \frac{24568004350965208923350}{83399374391760890181559} a^{14} - \frac{28151629611750261381421}{83399374391760890181559} a^{12} - \frac{38332562459166449982257}{83399374391760890181559} a^{10} - \frac{6405834782648254221753}{83399374391760890181559} a^{8} + \frac{15800649469478718075009}{333597497567043560726236} a^{6} - \frac{166302798908739396243241}{333597497567043560726236} a^{4} - \frac{72674952880381685031985}{166798748783521780363118} a^{2} - \frac{161424751778120864892991}{333597497567043560726236}$, $\frac{1}{333597497567043560726236} a^{31} - \frac{87796179254378890773}{333597497567043560726236} a^{29} + \frac{19497399130015925822625}{166798748783521780363118} a^{27} - \frac{29958214754697442399175}{333597497567043560726236} a^{25} - \frac{34875794024636335040530}{83399374391760890181559} a^{23} + \frac{13120024048666742360491}{83399374391760890181559} a^{21} - \frac{21381019663800047697501}{83399374391760890181559} a^{19} - \frac{24075340688324064899797}{83399374391760890181559} a^{17} + \frac{24568004350965208923350}{83399374391760890181559} a^{15} - \frac{28151629611750261381421}{83399374391760890181559} a^{13} - \frac{38332562459166449982257}{83399374391760890181559} a^{11} - \frac{6405834782648254221753}{83399374391760890181559} a^{9} + \frac{15800649469478718075009}{333597497567043560726236} a^{7} - \frac{166302798908739396243241}{333597497567043560726236} a^{5} - \frac{72674952880381685031985}{166798748783521780363118} a^{3} - \frac{161424751778120864892991}{333597497567043560726236} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{20}$, which has order $80$ (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:  \( \frac{4366251400675137246941}{166798748783521780363118} a^{30} - \frac{139419596465868338323055}{333597497567043560726236} a^{28} + \frac{330649311330808604060600}{83399374391760890181559} a^{26} - \frac{4169267722004324704631585}{166798748783521780363118} a^{24} + \frac{9808880326525928344379500}{83399374391760890181559} a^{22} - \frac{35006374090433504431991038}{83399374391760890181559} a^{20} + \frac{97913368924490331152866680}{83399374391760890181559} a^{18} - \frac{212870332447547083767334180}{83399374391760890181559} a^{16} + \frac{362653308169359271622909280}{83399374391760890181559} a^{14} - \frac{469216208282469529517110860}{83399374391760890181559} a^{12} + \frac{457227468859736880583674912}{83399374391760890181559} a^{10} - \frac{307351339084823514267820220}{83399374391760890181559} a^{8} + \frac{277212667649755983508367585}{166798748783521780363118} a^{6} - \frac{107263433800862459772757275}{333597497567043560726236} a^{4} + \frac{4042882751195651204983510}{83399374391760890181559} a^{2} - \frac{28386804048725376514543}{166798748783521780363118} \) (order $6$)
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:  \( 1296765452391.0793 \) (assuming GRH)
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

\(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{2}, \sqrt{-3})\), 4.0.2048.2, 4.4.18432.1, \(\Q(\sqrt{-6}, \sqrt{10})\), \(\Q(\sqrt{5}, \sqrt{-6})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-15})\), 4.0.51200.2, 4.4.460800.1, \(\Q(\sqrt{-3}, \sqrt{10})\), \(\Q(\sqrt{-3}, \sqrt{5})\), 4.0.256000.2, 4.4.2304000.1, 4.0.256000.4, 4.4.2304000.2, \(\Q(\zeta_{20})^+\), 4.0.72000.2, 4.4.8000.1, 4.0.18000.1, 8.0.339738624.1, 8.0.207360000.1, 8.0.212336640000.6, 8.0.2621440000.1, 8.0.212336640000.1, 8.8.212336640000.1, 8.0.212336640000.3, 8.0.5308416000000.4, 8.0.5308416000000.8, 8.0.82944000000.2, 8.0.82944000000.3, 8.0.65536000000.1, 8.8.5308416000000.1, \(\Q(\zeta_{40})^+\), 8.0.82944000000.6, 8.0.5308416000000.5, 8.0.5308416000000.1, 8.0.324000000.2, 8.0.5184000000.4, 16.0.45086848686489600000000.2, 16.0.28179280429056000000000000.2, 16.0.6879707136000000000000.5, 16.0.68719476736000000000000.2, 16.0.450868486864896000000000000.12, 16.0.450868486864896000000000000.5, 16.16.450868486864896000000000000.2

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
$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