Properties

Label 44.0.488...561.1
Degree $44$
Signature $[0, 22]$
Discriminant $4.889\times 10^{67}$
Root discriminant $34.55$
Ramified primes $3, 23$
Class number $69$ (GRH)
Class group $[69]$ (GRH)
Galois group $C_2\times C_{22}$ (as 44T2)

Related objects

Downloads

Learn more about

Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 - x^43 + x^41 - x^40 + x^38 - x^37 + x^35 - x^34 + x^32 - x^31 + x^29 - x^28 + x^26 - x^25 + x^23 - x^22 + x^21 - x^19 + x^18 - x^16 + x^15 - x^13 + x^12 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1)
 
gp: K = bnfinit(x^44 - x^43 + x^41 - x^40 + x^38 - x^37 + x^35 - x^34 + x^32 - x^31 + x^29 - x^28 + x^26 - x^25 + x^23 - x^22 + x^21 - x^19 + x^18 - x^16 + x^15 - x^13 + x^12 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1]);
 

\(x^{44} - x^{43} + x^{41} - x^{40} + x^{38} - x^{37} + x^{35} - x^{34} + x^{32} - x^{31} + x^{29} - x^{28} + x^{26} - x^{25} + x^{23} - x^{22} + x^{21} - x^{19} + x^{18} - x^{16} + x^{15} - x^{13} + x^{12} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1\)  Toggle raw display

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

Invariants

Degree:  $44$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 22]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(488\!\cdots\!561\)\(\medspace = 3^{22}\cdot 23^{42}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $34.55$
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:  $3, 23$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $44$
This field is Galois and abelian over $\Q$.
Conductor:  \(69=3\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{69}(1,·)$, $\chi_{69}(2,·)$, $\chi_{69}(4,·)$, $\chi_{69}(5,·)$, $\chi_{69}(7,·)$, $\chi_{69}(8,·)$, $\chi_{69}(10,·)$, $\chi_{69}(11,·)$, $\chi_{69}(13,·)$, $\chi_{69}(14,·)$, $\chi_{69}(16,·)$, $\chi_{69}(17,·)$, $\chi_{69}(19,·)$, $\chi_{69}(20,·)$, $\chi_{69}(22,·)$, $\chi_{69}(25,·)$, $\chi_{69}(26,·)$, $\chi_{69}(28,·)$, $\chi_{69}(29,·)$, $\chi_{69}(31,·)$, $\chi_{69}(32,·)$, $\chi_{69}(34,·)$, $\chi_{69}(35,·)$, $\chi_{69}(37,·)$, $\chi_{69}(38,·)$, $\chi_{69}(40,·)$, $\chi_{69}(41,·)$, $\chi_{69}(43,·)$, $\chi_{69}(44,·)$, $\chi_{69}(47,·)$, $\chi_{69}(49,·)$, $\chi_{69}(50,·)$, $\chi_{69}(52,·)$, $\chi_{69}(53,·)$, $\chi_{69}(55,·)$, $\chi_{69}(56,·)$, $\chi_{69}(58,·)$, $\chi_{69}(59,·)$, $\chi_{69}(61,·)$, $\chi_{69}(62,·)$, $\chi_{69}(64,·)$, $\chi_{69}(65,·)$, $\chi_{69}(67,·)$, $\chi_{69}(68,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $a^{43}$  Toggle raw display

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

Class group and class number

$C_{69}$, which has order $69$ (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:  $21$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -a \) (order $138$)  Toggle raw display
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:  \( 5770465240890302.0 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{22}\cdot 5770465240890302.0 \cdot 69}{138\sqrt{48891877682180103607391812819535352418736437208892474989920547427561}}\approx 0.149802678697671$ (assuming GRH)

Galois group

$C_2\times C_{22}$ (as 44T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
An abelian group of order 44
The 44 conjugacy class representatives for $C_2\times C_{22}$
Character table for $C_2\times C_{22}$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{69}) \), \(\Q(\sqrt{-3}, \sqrt{-23})\), \(\Q(\zeta_{23})^+\), 22.0.304011857053427966889939263171547.1, \(\Q(\zeta_{23})\), \(\Q(\zeta_{69})^+\)

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 $22^{2}$ R $22^{2}$ $22^{2}$ $22^{2}$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ R $22^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ $22^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{22}$ $22^{2}$ $22^{2}$

In the table, R denotes a ramified prime. 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
3Data not computed
23Data not computed