Properties

Label 33.33.271...161.1
Degree $33$
Signature $[33, 0]$
Discriminant $2.719\times 10^{58}$
Root discriminant $58.98$
Ramified prime $67$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{33}$ (as 33T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^33 - x^32 - 32*x^31 + 31*x^30 + 465*x^29 - 435*x^28 - 4060*x^27 + 3654*x^26 + 23751*x^25 - 20475*x^24 - 98280*x^23 + 80730*x^22 + 296010*x^21 - 230230*x^20 - 657800*x^19 + 480700*x^18 + 1081575*x^17 - 735471*x^16 - 1307504*x^15 + 817190*x^14 + 1144066*x^13 - 646646*x^12 - 705432*x^11 + 352716*x^10 + 293930*x^9 - 125970*x^8 - 77520*x^7 + 27132*x^6 + 11628*x^5 - 3060*x^4 - 816*x^3 + 136*x^2 + 17*x - 1)
 
gp: K = bnfinit(x^33 - x^32 - 32*x^31 + 31*x^30 + 465*x^29 - 435*x^28 - 4060*x^27 + 3654*x^26 + 23751*x^25 - 20475*x^24 - 98280*x^23 + 80730*x^22 + 296010*x^21 - 230230*x^20 - 657800*x^19 + 480700*x^18 + 1081575*x^17 - 735471*x^16 - 1307504*x^15 + 817190*x^14 + 1144066*x^13 - 646646*x^12 - 705432*x^11 + 352716*x^10 + 293930*x^9 - 125970*x^8 - 77520*x^7 + 27132*x^6 + 11628*x^5 - 3060*x^4 - 816*x^3 + 136*x^2 + 17*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 17, 136, -816, -3060, 11628, 27132, -77520, -125970, 293930, 352716, -705432, -646646, 1144066, 817190, -1307504, -735471, 1081575, 480700, -657800, -230230, 296010, 80730, -98280, -20475, 23751, 3654, -4060, -435, 465, 31, -32, -1, 1]);
 

\( x^{33} - x^{32} - 32 x^{31} + 31 x^{30} + 465 x^{29} - 435 x^{28} - 4060 x^{27} + 3654 x^{26} + 23751 x^{25} - 20475 x^{24} - 98280 x^{23} + 80730 x^{22} + 296010 x^{21} - 230230 x^{20} - 657800 x^{19} + 480700 x^{18} + 1081575 x^{17} - 735471 x^{16} - 1307504 x^{15} + 817190 x^{14} + 1144066 x^{13} - 646646 x^{12} - 705432 x^{11} + 352716 x^{10} + 293930 x^{9} - 125970 x^{8} - 77520 x^{7} + 27132 x^{6} + 11628 x^{5} - 3060 x^{4} - 816 x^{3} + 136 x^{2} + 17 x - 1 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $33$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[33, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(27189028279553414235049966267283185807800188603627566700161\)\(\medspace = 67^{32}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $58.98$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $67$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $33$
This field is Galois and abelian over $\Q$.
Conductor:  \(67\)
Dirichlet character group:    $\lbrace$$\chi_{67}(1,·)$, $\chi_{67}(4,·)$, $\chi_{67}(6,·)$, $\chi_{67}(9,·)$, $\chi_{67}(10,·)$, $\chi_{67}(14,·)$, $\chi_{67}(15,·)$, $\chi_{67}(16,·)$, $\chi_{67}(17,·)$, $\chi_{67}(19,·)$, $\chi_{67}(21,·)$, $\chi_{67}(22,·)$, $\chi_{67}(23,·)$, $\chi_{67}(24,·)$, $\chi_{67}(25,·)$, $\chi_{67}(26,·)$, $\chi_{67}(29,·)$, $\chi_{67}(33,·)$, $\chi_{67}(35,·)$, $\chi_{67}(36,·)$, $\chi_{67}(37,·)$, $\chi_{67}(39,·)$, $\chi_{67}(40,·)$, $\chi_{67}(47,·)$, $\chi_{67}(49,·)$, $\chi_{67}(54,·)$, $\chi_{67}(55,·)$, $\chi_{67}(56,·)$, $\chi_{67}(59,·)$, $\chi_{67}(60,·)$, $\chi_{67}(62,·)$, $\chi_{67}(64,·)$, $\chi_{67}(65,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $32$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 3985748844865106400 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{33}\cdot(2\pi)^{0}\cdot 3985748844865106400 \cdot 1}{2\sqrt{27189028279553414235049966267283185807800188603627566700161}}\approx 0.103818069031133$ (assuming GRH)

Galois group

$C_{33}$ (as 33T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 33
The 33 conjugacy class representatives for $C_{33}$
Character table for $C_{33}$ is not computed

Intermediate fields

3.3.4489.1, 11.11.1822837804551761449.1

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 $33$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{3}$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{3}$ $33$ $33$ $33$ $33$ $33$ $33$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{11}$ $33$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{11}$ $33$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{3}$ $33$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{3}$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{3}$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
67Data not computed