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Properties

Label 32.0.20328239244...0000.3
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 $400$ (GRH)
Class group $[2, 10, 20]$ (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, 0, 0, 0, 0, 0, 0, 1124, 0, 0, 0, 0, 0, 0, 0, 5886, 0, 0, 0, 0, 0, 0, 0, 269, 0, 0, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 269*x^24 + 5886*x^16 + 1124*x^8 + 1)
 
gp: K = bnfinit(x^32 + 269*x^24 + 5886*x^16 + 1124*x^8 + 1, 1)
 

Normalized defining polynomial

\( x^{32} + 269 x^{24} + 5886 x^{16} + 1124 x^{8} + 1 \)

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$$\chi_{240}(1,·)$, $\chi_{240}(131,·)$, $\chi_{240}(133,·)$, $\chi_{240}(137,·)$, $\chi_{240}(11,·)$, $\chi_{240}(13,·)$, $\chi_{240}(143,·)$, $\chi_{240}(17,·)$, $\chi_{240}(149,·)$, $\chi_{240}(23,·)$, $\chi_{240}(29,·)$, $\chi_{240}(31,·)$, $\chi_{240}(163,·)$, $\chi_{240}(37,·)$, $\chi_{240}(167,·)$, $\chi_{240}(169,·)$, $\chi_{240}(43,·)$, $\chi_{240}(47,·)$, $\chi_{240}(49,·)$, $\chi_{240}(179,·)$, $\chi_{240}(59,·)$, $\chi_{240}(151,·)$, $\chi_{240}(67,·)$, $\chi_{240}(199,·)$, $\chi_{240}(79,·)$, $\chi_{240}(221,·)$, $\chi_{240}(187,·)$, $\chi_{240}(101,·)$, $\chi_{240}(233,·)$, $\chi_{240}(157,·)$, $\chi_{240}(113,·)$, $\chi_{240}(121,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{8} + \frac{1}{3}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{9} + \frac{1}{3} a$, $\frac{1}{3} a^{18} + \frac{1}{3} a^{10} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{19} + \frac{1}{3} a^{11} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{20} + \frac{1}{3} a^{12} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{21} + \frac{1}{3} a^{13} + \frac{1}{3} a^{5}$, $\frac{1}{3} a^{22} + \frac{1}{3} a^{14} + \frac{1}{3} a^{6}$, $\frac{1}{3} a^{23} + \frac{1}{3} a^{15} + \frac{1}{3} a^{7}$, $\frac{1}{14153403} a^{24} + \frac{111247}{4717801} a^{16} + \frac{1692283}{4717801} a^{8} - \frac{615799}{14153403}$, $\frac{1}{14153403} a^{25} + \frac{111247}{4717801} a^{17} + \frac{1692283}{4717801} a^{9} - \frac{615799}{14153403} a$, $\frac{1}{14153403} a^{26} + \frac{111247}{4717801} a^{18} + \frac{1692283}{4717801} a^{10} - \frac{615799}{14153403} a^{2}$, $\frac{1}{14153403} a^{27} + \frac{111247}{4717801} a^{19} + \frac{1692283}{4717801} a^{11} - \frac{615799}{14153403} a^{3}$, $\frac{1}{14153403} a^{28} + \frac{111247}{4717801} a^{20} + \frac{1692283}{4717801} a^{12} - \frac{615799}{14153403} a^{4}$, $\frac{1}{14153403} a^{29} + \frac{111247}{4717801} a^{21} + \frac{1692283}{4717801} a^{13} - \frac{615799}{14153403} a^{5}$, $\frac{1}{14153403} a^{30} + \frac{111247}{4717801} a^{22} + \frac{1692283}{4717801} a^{14} - \frac{615799}{14153403} a^{6}$, $\frac{1}{14153403} a^{31} + \frac{111247}{4717801} a^{23} + \frac{1692283}{4717801} a^{15} - \frac{615799}{14153403} a^{7}$

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

Class group and class number

$C_{2}\times C_{10}\times C_{20}$, which has order $400$ (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:  \( \frac{804857}{4717801} a^{30} + \frac{649525640}{14153403} a^{22} + \frac{14213793260}{14153403} a^{14} + \frac{2750540720}{14153403} a^{6} \) (order $8$)
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:  \( 631693271497.9135 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-10}) \), \(\Q(\zeta_{8})\), 4.4.18432.1, 4.0.18432.2, \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{10})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-5})\), 4.4.460800.1, 4.0.460800.2, \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), 4.4.72000.1, 4.0.72000.2, \(\Q(\zeta_{15})^+\), 4.0.18000.1, 4.4.256000.1, 4.0.256000.4, 4.4.256000.2, 4.0.256000.2, 8.0.1358954496.4, 8.0.40960000.1, 8.0.849346560000.6, 8.8.212336640000.1, 8.0.849346560000.5, 8.0.212336640000.5, 8.0.849346560000.3, 8.0.82944000000.7, 8.0.324000000.1, 8.0.262144000000.3, 8.0.262144000000.4, 8.8.5184000000.1, 8.0.82944000000.6, 8.8.65536000000.1, 8.0.65536000000.1, 8.0.82944000000.5, 8.0.5184000000.2, 8.0.262144000000.2, 8.0.262144000000.1, 16.0.721389578983833600000000.1, 16.0.6879707136000000000000.3, 16.0.68719476736000000000000.1, 16.16.28179280429056000000000000.1, 16.0.450868486864896000000000000.5, 16.0.28179280429056000000000000.4, 16.0.450868486864896000000000000.6

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$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5Data not computed