↑

Properties

Label 44.0.860...376.2
Degree $44$
Signature $[0, 22]$
Discriminant $8.601\times 10^{80}$
Root discriminant \(69.09\)
Ramified primes $2,3,23$
Class number not computed
Class group not computed
Galois group $C_2\times C_{22}$ (as 44T2)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 + 23*x^42 + 299*x^40 + 2668*x^38 + 18055*x^36 + 96646*x^34 + 421245*x^32 + 1516689*x^30 + 4557519*x^28 + 11467961*x^26 + 24199128*x^24 + 42662286*x^22 + 62532561*x^20 + 75392022*x^18 + 73935156*x^16 + 57768387*x^14 + 35301228*x^12 + 16195335*x^10 + 5446584*x^8 + 1210352*x^6 + 180389*x^4 + 11638*x^2 + 529)
 
gp: K = bnfinit(y^44 + 23*y^42 + 299*y^40 + 2668*y^38 + 18055*y^36 + 96646*y^34 + 421245*y^32 + 1516689*y^30 + 4557519*y^28 + 11467961*y^26 + 24199128*y^24 + 42662286*y^22 + 62532561*y^20 + 75392022*y^18 + 73935156*y^16 + 57768387*y^14 + 35301228*y^12 + 16195335*y^10 + 5446584*y^8 + 1210352*y^6 + 180389*y^4 + 11638*y^2 + 529, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^44 + 23*x^42 + 299*x^40 + 2668*x^38 + 18055*x^36 + 96646*x^34 + 421245*x^32 + 1516689*x^30 + 4557519*x^28 + 11467961*x^26 + 24199128*x^24 + 42662286*x^22 + 62532561*x^20 + 75392022*x^18 + 73935156*x^16 + 57768387*x^14 + 35301228*x^12 + 16195335*x^10 + 5446584*x^8 + 1210352*x^6 + 180389*x^4 + 11638*x^2 + 529);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^44 + 23*x^42 + 299*x^40 + 2668*x^38 + 18055*x^36 + 96646*x^34 + 421245*x^32 + 1516689*x^30 + 4557519*x^28 + 11467961*x^26 + 24199128*x^24 + 42662286*x^22 + 62532561*x^20 + 75392022*x^18 + 73935156*x^16 + 57768387*x^14 + 35301228*x^12 + 16195335*x^10 + 5446584*x^8 + 1210352*x^6 + 180389*x^4 + 11638*x^2 + 529)
 

\( x^{44} + 23 x^{42} + 299 x^{40} + 2668 x^{38} + 18055 x^{36} + 96646 x^{34} + 421245 x^{32} + 1516689 x^{30} + \cdots + 529 \) Copy content Toggle raw display

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

Invariants

Degree:  $44$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 22]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(860\!\cdots\!376\) \(\medspace = 2^{44}\cdot 3^{22}\cdot 23^{42}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(69.09\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2\cdot 3^{1/2}23^{21/22}\approx 69.09104146788619$
Ramified primes:   \(2\), \(3\), \(23\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $44$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(276=2^{2}\cdot 3\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{276}(1,·)$, $\chi_{276}(107,·)$, $\chi_{276}(133,·)$, $\chi_{276}(7,·)$, $\chi_{276}(265,·)$, $\chi_{276}(11,·)$, $\chi_{276}(13,·)$, $\chi_{276}(143,·)$, $\chi_{276}(19,·)$, $\chi_{276}(269,·)$, $\chi_{276}(25,·)$, $\chi_{276}(155,·)$, $\chi_{276}(29,·)$, $\chi_{276}(41,·)$, $\chi_{276}(43,·)$, $\chi_{276}(257,·)$, $\chi_{276}(173,·)$, $\chi_{276}(175,·)$, $\chi_{276}(49,·)$, $\chi_{276}(185,·)$, $\chi_{276}(191,·)$, $\chi_{276}(193,·)$, $\chi_{276}(67,·)$, $\chi_{276}(197,·)$, $\chi_{276}(199,·)$, $\chi_{276}(73,·)$, $\chi_{276}(203,·)$, $\chi_{276}(77,·)$, $\chi_{276}(79,·)$, $\chi_{276}(209,·)$, $\chi_{276}(83,·)$, $\chi_{276}(85,·)$, $\chi_{276}(91,·)$, $\chi_{276}(263,·)$, $\chi_{276}(227,·)$, $\chi_{276}(101,·)$, $\chi_{276}(103,·)$, $\chi_{276}(233,·)$, $\chi_{276}(235,·)$, $\chi_{276}(275,·)$, $\chi_{276}(169,·)$, $\chi_{276}(121,·)$, $\chi_{276}(251,·)$, $\chi_{276}(247,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{2097152}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{23}a^{22}$, $\frac{1}{23}a^{23}$, $\frac{1}{23}a^{24}$, $\frac{1}{23}a^{25}$, $\frac{1}{23}a^{26}$, $\frac{1}{23}a^{27}$, $\frac{1}{23}a^{28}$, $\frac{1}{23}a^{29}$, $\frac{1}{23}a^{30}$, $\frac{1}{23}a^{31}$, $\frac{1}{23}a^{32}$, $\frac{1}{23}a^{33}$, $\frac{1}{23}a^{34}$, $\frac{1}{23}a^{35}$, $\frac{1}{23}a^{36}$, $\frac{1}{23}a^{37}$, $\frac{1}{23}a^{38}$, $\frac{1}{23}a^{39}$, $\frac{1}{24556157}a^{40}+\frac{305889}{24556157}a^{38}-\frac{203316}{24556157}a^{36}-\frac{15232}{24556157}a^{34}+\frac{423632}{24556157}a^{32}-\frac{263735}{24556157}a^{30}-\frac{17058}{1067659}a^{28}+\frac{79693}{24556157}a^{26}+\frac{384404}{24556157}a^{24}+\frac{210138}{24556157}a^{22}+\frac{392320}{1067659}a^{20}-\frac{491477}{1067659}a^{18}+\frac{460291}{1067659}a^{16}+\frac{409417}{1067659}a^{14}-\frac{360327}{1067659}a^{12}+\frac{372646}{1067659}a^{10}-\frac{195531}{1067659}a^{8}+\frac{255097}{1067659}a^{6}+\frac{12929}{1067659}a^{4}+\frac{446170}{1067659}a^{2}+\frac{55697}{1067659}$, $\frac{1}{24556157}a^{41}+\frac{305889}{24556157}a^{39}-\frac{203316}{24556157}a^{37}-\frac{15232}{24556157}a^{35}+\frac{423632}{24556157}a^{33}-\frac{263735}{24556157}a^{31}-\frac{17058}{1067659}a^{29}+\frac{79693}{24556157}a^{27}+\frac{384404}{24556157}a^{25}+\frac{210138}{24556157}a^{23}+\frac{392320}{1067659}a^{21}-\frac{491477}{1067659}a^{19}+\frac{460291}{1067659}a^{17}+\frac{409417}{1067659}a^{15}-\frac{360327}{1067659}a^{13}+\frac{372646}{1067659}a^{11}-\frac{195531}{1067659}a^{9}+\frac{255097}{1067659}a^{7}+\frac{12929}{1067659}a^{5}+\frac{446170}{1067659}a^{3}+\frac{55697}{1067659}a$, $\frac{1}{63\!\cdots\!23}a^{42}+\frac{57\!\cdots\!43}{63\!\cdots\!23}a^{40}-\frac{27\!\cdots\!26}{63\!\cdots\!23}a^{38}+\frac{84\!\cdots\!16}{63\!\cdots\!23}a^{36}+\frac{53\!\cdots\!37}{63\!\cdots\!23}a^{34}-\frac{50\!\cdots\!79}{63\!\cdots\!23}a^{32}+\frac{64\!\cdots\!90}{63\!\cdots\!23}a^{30}+\frac{13\!\cdots\!99}{63\!\cdots\!23}a^{28}+\frac{13\!\cdots\!86}{63\!\cdots\!23}a^{26}+\frac{91\!\cdots\!01}{63\!\cdots\!23}a^{24}+\frac{11\!\cdots\!02}{63\!\cdots\!23}a^{22}-\frac{13\!\cdots\!71}{27\!\cdots\!01}a^{20}+\frac{66\!\cdots\!14}{27\!\cdots\!01}a^{18}-\frac{68\!\cdots\!46}{27\!\cdots\!01}a^{16}+\frac{10\!\cdots\!00}{27\!\cdots\!01}a^{14}-\frac{11\!\cdots\!04}{27\!\cdots\!01}a^{12}-\frac{91\!\cdots\!05}{27\!\cdots\!01}a^{10}-\frac{12\!\cdots\!60}{27\!\cdots\!01}a^{8}+\frac{40\!\cdots\!76}{27\!\cdots\!01}a^{6}-\frac{36\!\cdots\!39}{27\!\cdots\!01}a^{4}-\frac{16\!\cdots\!51}{27\!\cdots\!01}a^{2}-\frac{21\!\cdots\!83}{27\!\cdots\!01}$, $\frac{1}{63\!\cdots\!23}a^{43}+\frac{57\!\cdots\!43}{63\!\cdots\!23}a^{41}-\frac{27\!\cdots\!26}{63\!\cdots\!23}a^{39}+\frac{84\!\cdots\!16}{63\!\cdots\!23}a^{37}+\frac{53\!\cdots\!37}{63\!\cdots\!23}a^{35}-\frac{50\!\cdots\!79}{63\!\cdots\!23}a^{33}+\frac{64\!\cdots\!90}{63\!\cdots\!23}a^{31}+\frac{13\!\cdots\!99}{63\!\cdots\!23}a^{29}+\frac{13\!\cdots\!86}{63\!\cdots\!23}a^{27}+\frac{91\!\cdots\!01}{63\!\cdots\!23}a^{25}+\frac{11\!\cdots\!02}{63\!\cdots\!23}a^{23}-\frac{13\!\cdots\!71}{27\!\cdots\!01}a^{21}+\frac{66\!\cdots\!14}{27\!\cdots\!01}a^{19}-\frac{68\!\cdots\!46}{27\!\cdots\!01}a^{17}+\frac{10\!\cdots\!00}{27\!\cdots\!01}a^{15}-\frac{11\!\cdots\!04}{27\!\cdots\!01}a^{13}-\frac{91\!\cdots\!05}{27\!\cdots\!01}a^{11}-\frac{12\!\cdots\!60}{27\!\cdots\!01}a^{9}+\frac{40\!\cdots\!76}{27\!\cdots\!01}a^{7}-\frac{36\!\cdots\!39}{27\!\cdots\!01}a^{5}-\frac{16\!\cdots\!51}{27\!\cdots\!01}a^{3}-\frac{21\!\cdots\!83}{27\!\cdots\!01}a$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

not computed

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $21$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{122886834586606027611739775648793658}{636968065704100522440858389810697591023} a^{42} - \frac{2811939192352668411885229675135639801}{636968065704100522440858389810697591023} a^{40} - \frac{36414697296856284759939776538503707259}{636968065704100522440858389810697591023} a^{38} - \frac{323631039610506013057573865960345462816}{636968065704100522440858389810697591023} a^{36} - \frac{94840309042035377828763732789936159253}{27694263726265240106124277817856417001} a^{34} - \frac{11625862554251792319858035429465402898456}{636968065704100522440858389810697591023} a^{32} - \frac{50437469985103885077225421117261836876234}{636968065704100522440858389810697591023} a^{30} - \frac{180656467729518231288054018432112012295220}{636968065704100522440858389810697591023} a^{28} - \frac{539700580015441526193324798291357128560367}{636968065704100522440858389810697591023} a^{26} - \frac{1348925463646387817733501972052698954948762}{636968065704100522440858389810697591023} a^{24} - \frac{2824335798154840894937367872769089973737085}{636968065704100522440858389810697591023} a^{22} - \frac{214483121678187214802353819142117544539397}{27694263726265240106124277817856417001} a^{20} - \frac{310899250664862701002677708578524574741901}{27694263726265240106124277817856417001} a^{18} - \frac{369687660510746256879893799985492355417343}{27694263726265240106124277817856417001} a^{16} - \frac{356393319410208792238368353696847232360374}{27694263726265240106124277817856417001} a^{14} - \frac{272294467684043844839338109120170952735511}{27694263726265240106124277817856417001} a^{12} - \frac{161725619685950643130152670201485133448169}{27694263726265240106124277817856417001} a^{10} - \frac{71231131240740897059214689752582901822260}{27694263726265240106124277817856417001} a^{8} - \frac{22806403126614556303212626774792213774456}{27694263726265240106124277817856417001} a^{6} - \frac{4627967503254677839370131558107172115801}{27694263726265240106124277817856417001} a^{4} - \frac{692427001424194557692742582610301257789}{27694263726265240106124277817856417001} a^{2} - \frac{16470264746805560920516187670108347434}{27694263726265240106124277817856417001} \)  (order $6$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^44 + 23*x^42 + 299*x^40 + 2668*x^38 + 18055*x^36 + 96646*x^34 + 421245*x^32 + 1516689*x^30 + 4557519*x^28 + 11467961*x^26 + 24199128*x^24 + 42662286*x^22 + 62532561*x^20 + 75392022*x^18 + 73935156*x^16 + 57768387*x^14 + 35301228*x^12 + 16195335*x^10 + 5446584*x^8 + 1210352*x^6 + 180389*x^4 + 11638*x^2 + 529)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^44 + 23*x^42 + 299*x^40 + 2668*x^38 + 18055*x^36 + 96646*x^34 + 421245*x^32 + 1516689*x^30 + 4557519*x^28 + 11467961*x^26 + 24199128*x^24 + 42662286*x^22 + 62532561*x^20 + 75392022*x^18 + 73935156*x^16 + 57768387*x^14 + 35301228*x^12 + 16195335*x^10 + 5446584*x^8 + 1210352*x^6 + 180389*x^4 + 11638*x^2 + 529, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^44 + 23*x^42 + 299*x^40 + 2668*x^38 + 18055*x^36 + 96646*x^34 + 421245*x^32 + 1516689*x^30 + 4557519*x^28 + 11467961*x^26 + 24199128*x^24 + 42662286*x^22 + 62532561*x^20 + 75392022*x^18 + 73935156*x^16 + 57768387*x^14 + 35301228*x^12 + 16195335*x^10 + 5446584*x^8 + 1210352*x^6 + 180389*x^4 + 11638*x^2 + 529);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^44 + 23*x^42 + 299*x^40 + 2668*x^38 + 18055*x^36 + 96646*x^34 + 421245*x^32 + 1516689*x^30 + 4557519*x^28 + 11467961*x^26 + 24199128*x^24 + 42662286*x^22 + 62532561*x^20 + 75392022*x^18 + 73935156*x^16 + 57768387*x^14 + 35301228*x^12 + 16195335*x^10 + 5446584*x^8 + 1210352*x^6 + 180389*x^4 + 11638*x^2 + 529);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
An abelian group of order 44
The 44 conjugacy class representatives for $C_2\times C_{22}$
Character table for $C_2\times C_{22}$

Intermediate fields

\(\Q(\sqrt{-69}) \), \(\Q(\sqrt{23}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{23})\), \(\Q(\zeta_{23})^+\), 22.0.29327717405992286110481815659381862170624.1, \(\Q(\zeta_{92})^+\), 22.0.304011857053427966889939263171547.1

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

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 $22^{2}$ ${\href{/padicField/7.11.0.1}{11} }^{4}$ $22^{2}$ ${\href{/padicField/13.11.0.1}{11} }^{4}$ $22^{2}$ ${\href{/padicField/19.11.0.1}{11} }^{4}$ R $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ ${\href{/padicField/43.11.0.1}{11} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{22}$ $22^{2}$ $22^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display Deg $44$$2$$22$$44$
\(3\) Copy content Toggle raw display Deg $44$$2$$22$$22$
\(23\) Copy content Toggle raw display Deg $44$$22$$2$$42$