# Properties

 Label 32.32.6208007170...4288.1 Degree $32$ Signature $[32, 0]$ Discriminant $2^{191}\cdot 3^{16}\cdot 11^{16}$ Root discriminant $359.77$ Ramified primes $2, 3, 11$ Class number Not computed Class group Not computed Galois group $C_{32}$ (as 32T33)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3955970402925117755868162, 0, -15344370047709547659124992, 0, 9880844348903875386557760, 0, -2515124016084622825669248, 0, 336166467517804024859472, 0, -27164967072145779786624, 0, 1440566435644094382624, 0, -52768001305644482880, 0, 1379163670488435348, 0, -26222957489976192, 0, 365950363695840, 0, -3744474402240, 0, 27750551400, 0, -144897984, 0, 505296, 0, -1056, 0, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 1056*x^30 + 505296*x^28 - 144897984*x^26 + 27750551400*x^24 - 3744474402240*x^22 + 365950363695840*x^20 - 26222957489976192*x^18 + 1379163670488435348*x^16 - 52768001305644482880*x^14 + 1440566435644094382624*x^12 - 27164967072145779786624*x^10 + 336166467517804024859472*x^8 - 2515124016084622825669248*x^6 + 9880844348903875386557760*x^4 - 15344370047709547659124992*x^2 + 3955970402925117755868162)

gp: K = bnfinit(x^32 - 1056*x^30 + 505296*x^28 - 144897984*x^26 + 27750551400*x^24 - 3744474402240*x^22 + 365950363695840*x^20 - 26222957489976192*x^18 + 1379163670488435348*x^16 - 52768001305644482880*x^14 + 1440566435644094382624*x^12 - 27164967072145779786624*x^10 + 336166467517804024859472*x^8 - 2515124016084622825669248*x^6 + 9880844348903875386557760*x^4 - 15344370047709547659124992*x^2 + 3955970402925117755868162, 1)

## Normalizeddefining polynomial

$$x^{32} - 1056 x^{30} + 505296 x^{28} - 144897984 x^{26} + 27750551400 x^{24} - 3744474402240 x^{22} + 365950363695840 x^{20} - 26222957489976192 x^{18} + 1379163670488435348 x^{16} - 52768001305644482880 x^{14} + 1440566435644094382624 x^{12} - 27164967072145779786624 x^{10} + 336166467517804024859472 x^{8} - 2515124016084622825669248 x^{6} + 9880844348903875386557760 x^{4} - 15344370047709547659124992 x^{2} + 3955970402925117755868162$$

magma: DefiningPolynomial(K);

sage: K.defining_polynomial()

gp: K.pol

## Invariants

 Degree: $32$ magma: Degree(K);  sage: K.degree()  gp: poldegree(K.pol) Signature: $[32, 0]$ magma: Signature(K);  sage: K.signature()  gp: K.sign Discriminant: $$6208007170334900849388551577113100621683540552916899074163477799595412799816204288=2^{191}\cdot 3^{16}\cdot 11^{16}$$ magma: Discriminant(Integers(K));  sage: K.disc()  gp: K.disc Root discriminant: $359.77$ 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: $2, 3, 11$ magma: PrimeDivisors(Discriminant(Integers(K)));  sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~ This field is Galois and abelian over $\Q$. Conductor: $$4224=2^{7}\cdot 3\cdot 11$$ Dirichlet character group: $\lbrace$$\chi_{4224}(1,·), \chi_{4224}(2309,·), \chi_{4224}(265,·), \chi_{4224}(2573,·), \chi_{4224}(529,·), \chi_{4224}(2837,·), \chi_{4224}(793,·), \chi_{4224}(3101,·), \chi_{4224}(1057,·), \chi_{4224}(3365,·), \chi_{4224}(1321,·), \chi_{4224}(3629,·), \chi_{4224}(1585,·), \chi_{4224}(3893,·), \chi_{4224}(1849,·), \chi_{4224}(4157,·), \chi_{4224}(2113,·), \chi_{4224}(197,·), \chi_{4224}(2377,·), \chi_{4224}(461,·), \chi_{4224}(2641,·), \chi_{4224}(725,·), \chi_{4224}(2905,·), \chi_{4224}(989,·), \chi_{4224}(3169,·), \chi_{4224}(1253,·), \chi_{4224}(3433,·), \chi_{4224}(1517,·), \chi_{4224}(3697,·), \chi_{4224}(1781,·), \chi_{4224}(3961,·), \chi_{4224}(2045,·)$$\rbrace$ This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $\frac{1}{33} a^{2}$, $\frac{1}{33} a^{3}$, $\frac{1}{1089} a^{4}$, $\frac{1}{1089} a^{5}$, $\frac{1}{35937} a^{6}$, $\frac{1}{35937} a^{7}$, $\frac{1}{1185921} a^{8}$, $\frac{1}{1185921} a^{9}$, $\frac{1}{39135393} a^{10}$, $\frac{1}{39135393} a^{11}$, $\frac{1}{1291467969} a^{12}$, $\frac{1}{1291467969} a^{13}$, $\frac{1}{42618442977} a^{14}$, $\frac{1}{42618442977} a^{15}$, $\frac{1}{1406408618241} a^{16}$, $\frac{1}{1406408618241} a^{17}$, $\frac{1}{46411484401953} a^{18}$, $\frac{1}{46411484401953} a^{19}$, $\frac{1}{1531578985264449} a^{20}$, $\frac{1}{1531578985264449} a^{21}$, $\frac{1}{50542106513726817} a^{22}$, $\frac{1}{50542106513726817} a^{23}$, $\frac{1}{1667889514952984961} a^{24}$, $\frac{1}{1667889514952984961} a^{25}$, $\frac{1}{55040353993448503713} a^{26}$, $\frac{1}{55040353993448503713} a^{27}$, $\frac{1}{1816331681783800622529} a^{28}$, $\frac{1}{1816331681783800622529} a^{29}$, $\frac{1}{59938945498865420543457} a^{30}$, $\frac{1}{59938945498865420543457} a^{31}$

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: $31$ 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: 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

$C_{32}$ (as 32T33):

magma: GaloisGroup(K);

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

 A cyclic group of order 32 The 32 conjugacy class representatives for $C_{32}$ Character table for $C_{32}$ is not computed

## Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R R $32$ $16^{2}$ R $32$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{4}$ $32$ $16^{2}$ $32$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ $32$ $16^{2}$ $32$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{4}$ $32$ $32$

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
3Data not computed
11Data not computed