Properties

Label 44.44.576...125.1
Degree $44$
Signature $[44, 0]$
Discriminant $5.770\times 10^{99}$
Root discriminant \(185.05\)
Ramified primes $5,67$
Class number not computed
Class group not computed
Galois group $C_{44}$ (as 44T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 - x^43 - 129*x^42 + 162*x^41 + 7330*x^40 - 10994*x^39 - 242871*x^38 + 421787*x^37 + 5231229*x^36 - 10324428*x^35 - 77259269*x^34 + 171714508*x^33 + 803790504*x^32 - 2012223025*x^31 - 5953080558*x^30 + 16977674434*x^29 + 31267083742*x^28 - 104442409979*x^27 - 113803293756*x^26 + 471438587020*x^25 + 268829517530*x^24 - 1564104309645*x^23 - 324159144950*x^22 + 3807787546527*x^21 - 169117650533*x^20 - 6777424820140*x^19 + 1431182547858*x^18 + 8784959795618*x^17 - 2701422052577*x^16 - 8267166900667*x^15 + 2815756227137*x^14 + 5623685086734*x^13 - 1774390295890*x^12 - 2724687389461*x^11 + 659704565348*x^10 + 902043228936*x^9 - 127476458594*x^8 - 186420518356*x^7 + 8576764156*x^6 + 20461360797*x^5 + 54760783*x^4 - 1031359974*x^3 - 10114401*x^2 + 16422117*x - 215069)
 
gp: K = bnfinit(y^44 - y^43 - 129*y^42 + 162*y^41 + 7330*y^40 - 10994*y^39 - 242871*y^38 + 421787*y^37 + 5231229*y^36 - 10324428*y^35 - 77259269*y^34 + 171714508*y^33 + 803790504*y^32 - 2012223025*y^31 - 5953080558*y^30 + 16977674434*y^29 + 31267083742*y^28 - 104442409979*y^27 - 113803293756*y^26 + 471438587020*y^25 + 268829517530*y^24 - 1564104309645*y^23 - 324159144950*y^22 + 3807787546527*y^21 - 169117650533*y^20 - 6777424820140*y^19 + 1431182547858*y^18 + 8784959795618*y^17 - 2701422052577*y^16 - 8267166900667*y^15 + 2815756227137*y^14 + 5623685086734*y^13 - 1774390295890*y^12 - 2724687389461*y^11 + 659704565348*y^10 + 902043228936*y^9 - 127476458594*y^8 - 186420518356*y^7 + 8576764156*y^6 + 20461360797*y^5 + 54760783*y^4 - 1031359974*y^3 - 10114401*y^2 + 16422117*y - 215069, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^44 - x^43 - 129*x^42 + 162*x^41 + 7330*x^40 - 10994*x^39 - 242871*x^38 + 421787*x^37 + 5231229*x^36 - 10324428*x^35 - 77259269*x^34 + 171714508*x^33 + 803790504*x^32 - 2012223025*x^31 - 5953080558*x^30 + 16977674434*x^29 + 31267083742*x^28 - 104442409979*x^27 - 113803293756*x^26 + 471438587020*x^25 + 268829517530*x^24 - 1564104309645*x^23 - 324159144950*x^22 + 3807787546527*x^21 - 169117650533*x^20 - 6777424820140*x^19 + 1431182547858*x^18 + 8784959795618*x^17 - 2701422052577*x^16 - 8267166900667*x^15 + 2815756227137*x^14 + 5623685086734*x^13 - 1774390295890*x^12 - 2724687389461*x^11 + 659704565348*x^10 + 902043228936*x^9 - 127476458594*x^8 - 186420518356*x^7 + 8576764156*x^6 + 20461360797*x^5 + 54760783*x^4 - 1031359974*x^3 - 10114401*x^2 + 16422117*x - 215069);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^44 - x^43 - 129*x^42 + 162*x^41 + 7330*x^40 - 10994*x^39 - 242871*x^38 + 421787*x^37 + 5231229*x^36 - 10324428*x^35 - 77259269*x^34 + 171714508*x^33 + 803790504*x^32 - 2012223025*x^31 - 5953080558*x^30 + 16977674434*x^29 + 31267083742*x^28 - 104442409979*x^27 - 113803293756*x^26 + 471438587020*x^25 + 268829517530*x^24 - 1564104309645*x^23 - 324159144950*x^22 + 3807787546527*x^21 - 169117650533*x^20 - 6777424820140*x^19 + 1431182547858*x^18 + 8784959795618*x^17 - 2701422052577*x^16 - 8267166900667*x^15 + 2815756227137*x^14 + 5623685086734*x^13 - 1774390295890*x^12 - 2724687389461*x^11 + 659704565348*x^10 + 902043228936*x^9 - 127476458594*x^8 - 186420518356*x^7 + 8576764156*x^6 + 20461360797*x^5 + 54760783*x^4 - 1031359974*x^3 - 10114401*x^2 + 16422117*x - 215069)
 

\( x^{44} - x^{43} - 129 x^{42} + 162 x^{41} + 7330 x^{40} - 10994 x^{39} - 242871 x^{38} + 421787 x^{37} + 5231229 x^{36} - 10324428 x^{35} - 77259269 x^{34} + \cdots - 215069 \) 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:  $[44, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(576\!\cdots\!125\) \(\medspace = 5^{33}\cdot 67^{42}\) 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:  \(185.05\)
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:  $5^{3/4}67^{21/22}\approx 185.05418274979348$
Ramified primes:   \(5\), \(67\) 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:  \(\Q(\sqrt{5}) \)
$\card{ \Gal(K/\Q) }$:  $44$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(335=5\cdot 67\)
Dirichlet character group:    $\lbrace$$\chi_{335}(1,·)$, $\chi_{335}(3,·)$, $\chi_{335}(133,·)$, $\chi_{335}(129,·)$, $\chi_{335}(8,·)$, $\chi_{335}(9,·)$, $\chi_{335}(267,·)$, $\chi_{335}(269,·)$, $\chi_{335}(14,·)$, $\chi_{335}(273,·)$, $\chi_{335}(131,·)$, $\chi_{335}(149,·)$, $\chi_{335}(24,·)$, $\chi_{335}(27,·)$, $\chi_{335}(156,·)$, $\chi_{335}(159,·)$, $\chi_{335}(64,·)$, $\chi_{335}(42,·)$, $\chi_{335}(43,·)$, $\chi_{335}(174,·)$, $\chi_{335}(177,·)$, $\chi_{335}(52,·)$, $\chi_{335}(53,·)$, $\chi_{335}(137,·)$, $\chi_{335}(313,·)$, $\chi_{335}(58,·)$, $\chi_{335}(59,·)$, $\chi_{335}(192,·)$, $\chi_{335}(196,·)$, $\chi_{335}(72,·)$, $\chi_{335}(76,·)$, $\chi_{335}(81,·)$, $\chi_{335}(142,·)$, $\chi_{335}(216,·)$, $\chi_{335}(89,·)$, $\chi_{335}(91,·)$, $\chi_{335}(226,·)$, $\chi_{335}(187,·)$, $\chi_{335}(228,·)$, $\chi_{335}(112,·)$, $\chi_{335}(241,·)$, $\chi_{335}(243,·)$, $\chi_{335}(253,·)$, $\chi_{335}(126,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $\frac{1}{75\!\cdots\!91}a^{42}-\frac{30\!\cdots\!27}{75\!\cdots\!91}a^{41}-\frac{35\!\cdots\!02}{75\!\cdots\!91}a^{40}-\frac{12\!\cdots\!61}{75\!\cdots\!91}a^{39}-\frac{33\!\cdots\!38}{75\!\cdots\!91}a^{38}+\frac{32\!\cdots\!49}{75\!\cdots\!91}a^{37}-\frac{10\!\cdots\!02}{75\!\cdots\!91}a^{36}-\frac{12\!\cdots\!35}{75\!\cdots\!91}a^{35}+\frac{41\!\cdots\!77}{75\!\cdots\!91}a^{34}-\frac{90\!\cdots\!56}{75\!\cdots\!91}a^{33}-\frac{52\!\cdots\!80}{75\!\cdots\!91}a^{32}-\frac{25\!\cdots\!89}{75\!\cdots\!91}a^{31}+\frac{28\!\cdots\!74}{75\!\cdots\!91}a^{30}-\frac{30\!\cdots\!35}{75\!\cdots\!91}a^{29}+\frac{50\!\cdots\!40}{75\!\cdots\!91}a^{28}+\frac{10\!\cdots\!27}{75\!\cdots\!91}a^{27}+\frac{37\!\cdots\!59}{75\!\cdots\!91}a^{26}-\frac{27\!\cdots\!05}{75\!\cdots\!91}a^{25}-\frac{15\!\cdots\!12}{75\!\cdots\!91}a^{24}-\frac{32\!\cdots\!41}{75\!\cdots\!91}a^{23}-\frac{32\!\cdots\!68}{75\!\cdots\!91}a^{22}-\frac{35\!\cdots\!06}{75\!\cdots\!91}a^{21}+\frac{11\!\cdots\!09}{75\!\cdots\!91}a^{20}+\frac{26\!\cdots\!41}{75\!\cdots\!91}a^{19}+\frac{28\!\cdots\!42}{75\!\cdots\!91}a^{18}-\frac{35\!\cdots\!22}{75\!\cdots\!91}a^{17}-\frac{21\!\cdots\!88}{75\!\cdots\!91}a^{16}-\frac{33\!\cdots\!47}{75\!\cdots\!91}a^{15}+\frac{16\!\cdots\!03}{75\!\cdots\!91}a^{14}-\frac{14\!\cdots\!78}{75\!\cdots\!91}a^{13}+\frac{10\!\cdots\!14}{75\!\cdots\!91}a^{12}+\frac{16\!\cdots\!14}{75\!\cdots\!91}a^{11}+\frac{28\!\cdots\!67}{75\!\cdots\!91}a^{10}-\frac{13\!\cdots\!14}{75\!\cdots\!91}a^{9}-\frac{11\!\cdots\!90}{75\!\cdots\!91}a^{8}+\frac{15\!\cdots\!20}{75\!\cdots\!91}a^{7}+\frac{94\!\cdots\!10}{75\!\cdots\!91}a^{6}+\frac{21\!\cdots\!58}{75\!\cdots\!91}a^{5}+\frac{37\!\cdots\!95}{75\!\cdots\!91}a^{4}+\frac{31\!\cdots\!10}{75\!\cdots\!91}a^{3}+\frac{68\!\cdots\!76}{75\!\cdots\!91}a^{2}-\frac{42\!\cdots\!76}{75\!\cdots\!91}a-\frac{25\!\cdots\!39}{75\!\cdots\!91}$, $\frac{1}{10\!\cdots\!51}a^{43}-\frac{23\!\cdots\!40}{10\!\cdots\!51}a^{42}-\frac{32\!\cdots\!74}{10\!\cdots\!51}a^{41}+\frac{23\!\cdots\!65}{10\!\cdots\!51}a^{40}+\frac{28\!\cdots\!92}{10\!\cdots\!51}a^{39}+\frac{20\!\cdots\!92}{10\!\cdots\!51}a^{38}+\frac{43\!\cdots\!42}{10\!\cdots\!51}a^{37}+\frac{32\!\cdots\!15}{10\!\cdots\!51}a^{36}-\frac{23\!\cdots\!81}{10\!\cdots\!51}a^{35}-\frac{37\!\cdots\!06}{10\!\cdots\!51}a^{34}+\frac{52\!\cdots\!59}{10\!\cdots\!51}a^{33}+\frac{51\!\cdots\!07}{10\!\cdots\!51}a^{32}+\frac{37\!\cdots\!72}{10\!\cdots\!51}a^{31}-\frac{10\!\cdots\!66}{10\!\cdots\!51}a^{30}+\frac{28\!\cdots\!23}{10\!\cdots\!51}a^{29}+\frac{36\!\cdots\!85}{10\!\cdots\!51}a^{28}+\frac{39\!\cdots\!43}{10\!\cdots\!51}a^{27}-\frac{41\!\cdots\!37}{10\!\cdots\!51}a^{26}+\frac{18\!\cdots\!79}{10\!\cdots\!51}a^{25}-\frac{51\!\cdots\!25}{10\!\cdots\!51}a^{24}-\frac{13\!\cdots\!93}{10\!\cdots\!51}a^{23}-\frac{18\!\cdots\!81}{10\!\cdots\!51}a^{22}-\frac{34\!\cdots\!40}{10\!\cdots\!51}a^{21}-\frac{28\!\cdots\!72}{10\!\cdots\!51}a^{20}-\frac{21\!\cdots\!64}{10\!\cdots\!51}a^{19}+\frac{11\!\cdots\!69}{10\!\cdots\!51}a^{18}+\frac{26\!\cdots\!25}{10\!\cdots\!51}a^{17}+\frac{63\!\cdots\!18}{10\!\cdots\!51}a^{16}-\frac{35\!\cdots\!45}{10\!\cdots\!51}a^{15}-\frac{18\!\cdots\!12}{10\!\cdots\!51}a^{14}-\frac{32\!\cdots\!50}{10\!\cdots\!51}a^{13}+\frac{34\!\cdots\!53}{10\!\cdots\!51}a^{12}+\frac{63\!\cdots\!16}{10\!\cdots\!51}a^{11}-\frac{61\!\cdots\!48}{10\!\cdots\!51}a^{10}+\frac{30\!\cdots\!33}{10\!\cdots\!51}a^{9}+\frac{12\!\cdots\!87}{10\!\cdots\!51}a^{8}+\frac{16\!\cdots\!61}{10\!\cdots\!51}a^{7}-\frac{31\!\cdots\!31}{10\!\cdots\!51}a^{6}-\frac{14\!\cdots\!77}{10\!\cdots\!51}a^{5}+\frac{50\!\cdots\!81}{10\!\cdots\!51}a^{4}-\frac{34\!\cdots\!07}{10\!\cdots\!51}a^{3}-\frac{98\!\cdots\!81}{10\!\cdots\!51}a^{2}+\frac{50\!\cdots\!48}{10\!\cdots\!51}a-\frac{70\!\cdots\!61}{21\!\cdots\!49}$ 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:  $43$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) 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 - x^43 - 129*x^42 + 162*x^41 + 7330*x^40 - 10994*x^39 - 242871*x^38 + 421787*x^37 + 5231229*x^36 - 10324428*x^35 - 77259269*x^34 + 171714508*x^33 + 803790504*x^32 - 2012223025*x^31 - 5953080558*x^30 + 16977674434*x^29 + 31267083742*x^28 - 104442409979*x^27 - 113803293756*x^26 + 471438587020*x^25 + 268829517530*x^24 - 1564104309645*x^23 - 324159144950*x^22 + 3807787546527*x^21 - 169117650533*x^20 - 6777424820140*x^19 + 1431182547858*x^18 + 8784959795618*x^17 - 2701422052577*x^16 - 8267166900667*x^15 + 2815756227137*x^14 + 5623685086734*x^13 - 1774390295890*x^12 - 2724687389461*x^11 + 659704565348*x^10 + 902043228936*x^9 - 127476458594*x^8 - 186420518356*x^7 + 8576764156*x^6 + 20461360797*x^5 + 54760783*x^4 - 1031359974*x^3 - 10114401*x^2 + 16422117*x - 215069)
 
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 - x^43 - 129*x^42 + 162*x^41 + 7330*x^40 - 10994*x^39 - 242871*x^38 + 421787*x^37 + 5231229*x^36 - 10324428*x^35 - 77259269*x^34 + 171714508*x^33 + 803790504*x^32 - 2012223025*x^31 - 5953080558*x^30 + 16977674434*x^29 + 31267083742*x^28 - 104442409979*x^27 - 113803293756*x^26 + 471438587020*x^25 + 268829517530*x^24 - 1564104309645*x^23 - 324159144950*x^22 + 3807787546527*x^21 - 169117650533*x^20 - 6777424820140*x^19 + 1431182547858*x^18 + 8784959795618*x^17 - 2701422052577*x^16 - 8267166900667*x^15 + 2815756227137*x^14 + 5623685086734*x^13 - 1774390295890*x^12 - 2724687389461*x^11 + 659704565348*x^10 + 902043228936*x^9 - 127476458594*x^8 - 186420518356*x^7 + 8576764156*x^6 + 20461360797*x^5 + 54760783*x^4 - 1031359974*x^3 - 10114401*x^2 + 16422117*x - 215069, 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 - x^43 - 129*x^42 + 162*x^41 + 7330*x^40 - 10994*x^39 - 242871*x^38 + 421787*x^37 + 5231229*x^36 - 10324428*x^35 - 77259269*x^34 + 171714508*x^33 + 803790504*x^32 - 2012223025*x^31 - 5953080558*x^30 + 16977674434*x^29 + 31267083742*x^28 - 104442409979*x^27 - 113803293756*x^26 + 471438587020*x^25 + 268829517530*x^24 - 1564104309645*x^23 - 324159144950*x^22 + 3807787546527*x^21 - 169117650533*x^20 - 6777424820140*x^19 + 1431182547858*x^18 + 8784959795618*x^17 - 2701422052577*x^16 - 8267166900667*x^15 + 2815756227137*x^14 + 5623685086734*x^13 - 1774390295890*x^12 - 2724687389461*x^11 + 659704565348*x^10 + 902043228936*x^9 - 127476458594*x^8 - 186420518356*x^7 + 8576764156*x^6 + 20461360797*x^5 + 54760783*x^4 - 1031359974*x^3 - 10114401*x^2 + 16422117*x - 215069);
 
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 - x^43 - 129*x^42 + 162*x^41 + 7330*x^40 - 10994*x^39 - 242871*x^38 + 421787*x^37 + 5231229*x^36 - 10324428*x^35 - 77259269*x^34 + 171714508*x^33 + 803790504*x^32 - 2012223025*x^31 - 5953080558*x^30 + 16977674434*x^29 + 31267083742*x^28 - 104442409979*x^27 - 113803293756*x^26 + 471438587020*x^25 + 268829517530*x^24 - 1564104309645*x^23 - 324159144950*x^22 + 3807787546527*x^21 - 169117650533*x^20 - 6777424820140*x^19 + 1431182547858*x^18 + 8784959795618*x^17 - 2701422052577*x^16 - 8267166900667*x^15 + 2815756227137*x^14 + 5623685086734*x^13 - 1774390295890*x^12 - 2724687389461*x^11 + 659704565348*x^10 + 902043228936*x^9 - 127476458594*x^8 - 186420518356*x^7 + 8576764156*x^6 + 20461360797*x^5 + 54760783*x^4 - 1031359974*x^3 - 10114401*x^2 + 16422117*x - 215069);
 
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

$C_{44}$ (as 44T1):

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)
 
A cyclic group of order 44
The 44 conjugacy class representatives for $C_{44}$
Character table for $C_{44}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.561125.1, 11.11.1822837804551761449.1, 22.22.162243049887845980095628744560672832080078125.1

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 $44$ $44$ R $44$ $22^{2}$ $44$ $44$ $22^{2}$ $44$ ${\href{/padicField/29.2.0.1}{2} }^{22}$ $22^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{11}$ $22^{2}$ $44$ $44$ $44$ $22^{2}$

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
\(5\) Copy content Toggle raw display Deg $44$$4$$11$$33$
\(67\) Copy content Toggle raw display Deg $44$$22$$2$$42$