Properties

Label 44.0.220...625.1
Degree $44$
Signature $[0, 22]$
Discriminant $2.204\times 10^{80}$
Root discriminant \(66.99\)
Ramified primes $3,5,23$
Class number not computed
Class group not computed
Galois group $C_2\times C_{22}$ (as 44T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 - x^43 + 32*x^42 - 21*x^41 + 608*x^40 - 302*x^39 + 7441*x^38 - 2334*x^37 + 66293*x^36 - 12644*x^35 + 432280*x^34 - 28161*x^33 + 2141316*x^32 + 33285*x^31 + 8001765*x^30 + 693375*x^29 + 23077394*x^28 + 2792911*x^27 + 51292654*x^26 + 7229970*x^25 + 89635724*x^24 + 12947356*x^23 + 123068074*x^22 + 17592284*x^21 + 133953248*x^20 + 17840654*x^19 + 114270996*x^18 + 13803865*x^17 + 76240835*x^16 + 7576064*x^15 + 38619954*x^14 + 2830121*x^13 + 14698338*x^12 + 489978*x^11 + 3921204*x^10 - 46901*x^9 + 736586*x^8 - 41505*x^7 + 78482*x^6 - 8849*x^5 + 5761*x^4 - 386*x^3 + 117*x^2 + 6*x + 1)
 
gp: K = bnfinit(y^44 - y^43 + 32*y^42 - 21*y^41 + 608*y^40 - 302*y^39 + 7441*y^38 - 2334*y^37 + 66293*y^36 - 12644*y^35 + 432280*y^34 - 28161*y^33 + 2141316*y^32 + 33285*y^31 + 8001765*y^30 + 693375*y^29 + 23077394*y^28 + 2792911*y^27 + 51292654*y^26 + 7229970*y^25 + 89635724*y^24 + 12947356*y^23 + 123068074*y^22 + 17592284*y^21 + 133953248*y^20 + 17840654*y^19 + 114270996*y^18 + 13803865*y^17 + 76240835*y^16 + 7576064*y^15 + 38619954*y^14 + 2830121*y^13 + 14698338*y^12 + 489978*y^11 + 3921204*y^10 - 46901*y^9 + 736586*y^8 - 41505*y^7 + 78482*y^6 - 8849*y^5 + 5761*y^4 - 386*y^3 + 117*y^2 + 6*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^44 - x^43 + 32*x^42 - 21*x^41 + 608*x^40 - 302*x^39 + 7441*x^38 - 2334*x^37 + 66293*x^36 - 12644*x^35 + 432280*x^34 - 28161*x^33 + 2141316*x^32 + 33285*x^31 + 8001765*x^30 + 693375*x^29 + 23077394*x^28 + 2792911*x^27 + 51292654*x^26 + 7229970*x^25 + 89635724*x^24 + 12947356*x^23 + 123068074*x^22 + 17592284*x^21 + 133953248*x^20 + 17840654*x^19 + 114270996*x^18 + 13803865*x^17 + 76240835*x^16 + 7576064*x^15 + 38619954*x^14 + 2830121*x^13 + 14698338*x^12 + 489978*x^11 + 3921204*x^10 - 46901*x^9 + 736586*x^8 - 41505*x^7 + 78482*x^6 - 8849*x^5 + 5761*x^4 - 386*x^3 + 117*x^2 + 6*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^44 - x^43 + 32*x^42 - 21*x^41 + 608*x^40 - 302*x^39 + 7441*x^38 - 2334*x^37 + 66293*x^36 - 12644*x^35 + 432280*x^34 - 28161*x^33 + 2141316*x^32 + 33285*x^31 + 8001765*x^30 + 693375*x^29 + 23077394*x^28 + 2792911*x^27 + 51292654*x^26 + 7229970*x^25 + 89635724*x^24 + 12947356*x^23 + 123068074*x^22 + 17592284*x^21 + 133953248*x^20 + 17840654*x^19 + 114270996*x^18 + 13803865*x^17 + 76240835*x^16 + 7576064*x^15 + 38619954*x^14 + 2830121*x^13 + 14698338*x^12 + 489978*x^11 + 3921204*x^10 - 46901*x^9 + 736586*x^8 - 41505*x^7 + 78482*x^6 - 8849*x^5 + 5761*x^4 - 386*x^3 + 117*x^2 + 6*x + 1)
 

\( x^{44} - x^{43} + 32 x^{42} - 21 x^{41} + 608 x^{40} - 302 x^{39} + 7441 x^{38} - 2334 x^{37} + 66293 x^{36} - 12644 x^{35} + 432280 x^{34} - 28161 x^{33} + 2141316 x^{32} + 33285 x^{31} + \cdots + 1 \) 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:   \(220\!\cdots\!625\) \(\medspace = 3^{22}\cdot 5^{22}\cdot 23^{40}\) 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:  \(66.99\)
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:  $3^{1/2}5^{1/2}23^{10/11}\approx 66.98538007600956$
Ramified primes:   \(3\), \(5\), \(23\) 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:  \(345=3\cdot 5\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{345}(256,·)$, $\chi_{345}(1,·)$, $\chi_{345}(259,·)$, $\chi_{345}(4,·)$, $\chi_{345}(266,·)$, $\chi_{345}(139,·)$, $\chi_{345}(269,·)$, $\chi_{345}(271,·)$, $\chi_{345}(16,·)$, $\chi_{345}(146,·)$, $\chi_{345}(131,·)$, $\chi_{345}(151,·)$, $\chi_{345}(154,·)$, $\chi_{345}(284,·)$, $\chi_{345}(26,·)$, $\chi_{345}(31,·)$, $\chi_{345}(289,·)$, $\chi_{345}(164,·)$, $\chi_{345}(41,·)$, $\chi_{345}(301,·)$, $\chi_{345}(29,·)$, $\chi_{345}(49,·)$, $\chi_{345}(179,·)$, $\chi_{345}(311,·)$, $\chi_{345}(59,·)$, $\chi_{345}(64,·)$, $\chi_{345}(196,·)$, $\chi_{345}(326,·)$, $\chi_{345}(71,·)$, $\chi_{345}(331,·)$, $\chi_{345}(334,·)$, $\chi_{345}(209,·)$, $\chi_{345}(211,·)$, $\chi_{345}(94,·)$, $\chi_{345}(101,·)$, $\chi_{345}(104,·)$, $\chi_{345}(236,·)$, $\chi_{345}(239,·)$, $\chi_{345}(116,·)$, $\chi_{345}(169,·)$, $\chi_{345}(121,·)$, $\chi_{345}(119,·)$, $\chi_{345}(124,·)$, $\chi_{345}(254,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{2097152}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $\frac{1}{2}a^{33}-\frac{1}{2}a^{27}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{34}-\frac{1}{2}a^{28}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{35}-\frac{1}{2}a^{29}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{36}-\frac{1}{2}a^{30}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{37}-\frac{1}{2}a^{31}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{38}-\frac{1}{2}a^{32}-\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{39}-\frac{1}{2}a^{27}-\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{278}a^{40}-\frac{29}{278}a^{39}+\frac{11}{278}a^{38}-\frac{15}{139}a^{37}+\frac{27}{278}a^{36}+\frac{26}{139}a^{35}-\frac{8}{139}a^{34}-\frac{51}{278}a^{33}-\frac{99}{278}a^{32}+\frac{41}{139}a^{31}+\frac{39}{278}a^{30}+\frac{64}{139}a^{29}+\frac{45}{278}a^{28}+\frac{67}{139}a^{27}-\frac{64}{139}a^{26}+\frac{57}{139}a^{25}+\frac{39}{139}a^{24}+\frac{21}{139}a^{23}+\frac{30}{139}a^{22}-\frac{55}{139}a^{21}-\frac{61}{139}a^{20}-\frac{52}{139}a^{19}+\frac{7}{139}a^{18}-\frac{35}{139}a^{17}-\frac{40}{139}a^{16}-\frac{55}{139}a^{15}-\frac{20}{139}a^{14}-\frac{27}{278}a^{13}+\frac{61}{278}a^{12}+\frac{131}{278}a^{11}+\frac{77}{278}a^{10}+\frac{32}{139}a^{9}+\frac{17}{278}a^{8}+\frac{50}{139}a^{7}+\frac{5}{139}a^{6}-\frac{43}{278}a^{5}-\frac{127}{278}a^{4}+\frac{31}{278}a^{3}+\frac{20}{139}a^{2}+\frac{35}{278}a+\frac{27}{139}$, $\frac{1}{278}a^{41}+\frac{2}{139}a^{39}+\frac{11}{278}a^{38}-\frac{9}{278}a^{37}+\frac{1}{278}a^{36}-\frac{37}{278}a^{35}+\frac{41}{278}a^{34}-\frac{49}{278}a^{33}-\frac{9}{278}a^{32}-\frac{85}{278}a^{31}-\frac{131}{278}a^{30}+\frac{2}{139}a^{29}+\frac{49}{278}a^{28}+\frac{5}{278}a^{27}+\frac{8}{139}a^{26}+\frac{24}{139}a^{25}+\frac{40}{139}a^{24}-\frac{56}{139}a^{23}-\frac{19}{139}a^{22}+\frac{12}{139}a^{21}-\frac{14}{139}a^{20}+\frac{28}{139}a^{19}+\frac{29}{139}a^{18}+\frac{57}{139}a^{17}+\frac{36}{139}a^{16}+\frac{53}{139}a^{15}-\frac{75}{278}a^{14}+\frac{56}{139}a^{13}-\frac{23}{139}a^{12}-\frac{8}{139}a^{11}+\frac{73}{278}a^{10}-\frac{73}{278}a^{9}-\frac{51}{139}a^{8}+\frac{65}{139}a^{7}+\frac{54}{139}a^{6}-\frac{123}{278}a^{5}-\frac{19}{139}a^{4}-\frac{17}{139}a^{3}-\frac{28}{139}a^{2}-\frac{43}{278}a+\frac{37}{278}$, $\frac{1}{27\!\cdots\!86}a^{42}-\frac{12\!\cdots\!91}{13\!\cdots\!43}a^{41}-\frac{33\!\cdots\!05}{13\!\cdots\!43}a^{40}+\frac{11\!\cdots\!64}{13\!\cdots\!43}a^{39}+\frac{39\!\cdots\!47}{27\!\cdots\!86}a^{38}+\frac{33\!\cdots\!39}{13\!\cdots\!43}a^{37}+\frac{18\!\cdots\!90}{13\!\cdots\!43}a^{36}-\frac{40\!\cdots\!29}{27\!\cdots\!86}a^{35}-\frac{18\!\cdots\!99}{13\!\cdots\!43}a^{34}-\frac{15\!\cdots\!75}{27\!\cdots\!86}a^{33}+\frac{10\!\cdots\!65}{27\!\cdots\!86}a^{32}+\frac{47\!\cdots\!93}{13\!\cdots\!43}a^{31}+\frac{34\!\cdots\!35}{27\!\cdots\!86}a^{30}-\frac{12\!\cdots\!53}{27\!\cdots\!86}a^{29}+\frac{26\!\cdots\!28}{13\!\cdots\!43}a^{28}-\frac{10\!\cdots\!15}{27\!\cdots\!86}a^{27}+\frac{65\!\cdots\!17}{13\!\cdots\!43}a^{26}-\frac{33\!\cdots\!64}{13\!\cdots\!43}a^{25}-\frac{72\!\cdots\!17}{13\!\cdots\!43}a^{24}+\frac{57\!\cdots\!93}{13\!\cdots\!43}a^{23}+\frac{38\!\cdots\!35}{13\!\cdots\!43}a^{22}+\frac{57\!\cdots\!98}{13\!\cdots\!43}a^{21}+\frac{47\!\cdots\!52}{13\!\cdots\!43}a^{20}+\frac{16\!\cdots\!68}{13\!\cdots\!43}a^{19}-\frac{51\!\cdots\!67}{13\!\cdots\!43}a^{18}-\frac{29\!\cdots\!62}{13\!\cdots\!43}a^{17}+\frac{59\!\cdots\!36}{13\!\cdots\!43}a^{16}+\frac{77\!\cdots\!29}{27\!\cdots\!86}a^{15}-\frac{54\!\cdots\!13}{13\!\cdots\!43}a^{14}-\frac{59\!\cdots\!51}{13\!\cdots\!43}a^{13}+\frac{58\!\cdots\!47}{27\!\cdots\!86}a^{12}-\frac{76\!\cdots\!73}{27\!\cdots\!86}a^{11}+\frac{12\!\cdots\!53}{13\!\cdots\!43}a^{10}-\frac{61\!\cdots\!05}{13\!\cdots\!43}a^{9}-\frac{12\!\cdots\!47}{13\!\cdots\!43}a^{8}+\frac{58\!\cdots\!17}{13\!\cdots\!43}a^{7}-\frac{42\!\cdots\!26}{13\!\cdots\!43}a^{6}+\frac{61\!\cdots\!81}{13\!\cdots\!43}a^{5}-\frac{45\!\cdots\!69}{13\!\cdots\!43}a^{4}-\frac{35\!\cdots\!82}{13\!\cdots\!43}a^{3}-\frac{26\!\cdots\!73}{27\!\cdots\!86}a^{2}-\frac{19\!\cdots\!90}{13\!\cdots\!43}a-\frac{48\!\cdots\!19}{27\!\cdots\!86}$, $\frac{1}{40\!\cdots\!34}a^{43}+\frac{26\!\cdots\!59}{40\!\cdots\!34}a^{42}-\frac{49\!\cdots\!21}{40\!\cdots\!34}a^{41}-\frac{20\!\cdots\!11}{40\!\cdots\!34}a^{40}-\frac{18\!\cdots\!31}{40\!\cdots\!34}a^{39}+\frac{52\!\cdots\!93}{40\!\cdots\!34}a^{38}-\frac{20\!\cdots\!18}{20\!\cdots\!17}a^{37}-\frac{35\!\cdots\!31}{40\!\cdots\!34}a^{36}-\frac{31\!\cdots\!13}{40\!\cdots\!34}a^{35}-\frac{20\!\cdots\!41}{20\!\cdots\!17}a^{34}-\frac{57\!\cdots\!77}{40\!\cdots\!34}a^{33}-\frac{13\!\cdots\!37}{40\!\cdots\!34}a^{32}-\frac{11\!\cdots\!69}{40\!\cdots\!34}a^{31}-\frac{82\!\cdots\!79}{20\!\cdots\!17}a^{30}+\frac{11\!\cdots\!29}{20\!\cdots\!17}a^{29}-\frac{98\!\cdots\!45}{40\!\cdots\!34}a^{28}+\frac{98\!\cdots\!52}{20\!\cdots\!17}a^{27}-\frac{70\!\cdots\!18}{20\!\cdots\!17}a^{26}+\frac{54\!\cdots\!27}{20\!\cdots\!17}a^{25}-\frac{56\!\cdots\!63}{20\!\cdots\!17}a^{24}+\frac{20\!\cdots\!87}{20\!\cdots\!17}a^{23}+\frac{73\!\cdots\!03}{20\!\cdots\!17}a^{22}+\frac{29\!\cdots\!18}{20\!\cdots\!17}a^{21}-\frac{87\!\cdots\!40}{20\!\cdots\!17}a^{20}-\frac{38\!\cdots\!53}{20\!\cdots\!17}a^{19}-\frac{67\!\cdots\!09}{20\!\cdots\!17}a^{18}+\frac{36\!\cdots\!78}{20\!\cdots\!17}a^{17}-\frac{52\!\cdots\!47}{40\!\cdots\!34}a^{16}-\frac{90\!\cdots\!91}{40\!\cdots\!34}a^{15}+\frac{72\!\cdots\!73}{40\!\cdots\!34}a^{14}+\frac{77\!\cdots\!80}{20\!\cdots\!17}a^{13}+\frac{81\!\cdots\!53}{20\!\cdots\!17}a^{12}+\frac{81\!\cdots\!64}{20\!\cdots\!17}a^{11}-\frac{20\!\cdots\!01}{40\!\cdots\!34}a^{10}+\frac{54\!\cdots\!75}{20\!\cdots\!17}a^{9}+\frac{43\!\cdots\!41}{20\!\cdots\!17}a^{8}-\frac{14\!\cdots\!65}{40\!\cdots\!34}a^{7}+\frac{10\!\cdots\!87}{40\!\cdots\!34}a^{6}-\frac{11\!\cdots\!97}{40\!\cdots\!34}a^{5}+\frac{13\!\cdots\!75}{20\!\cdots\!17}a^{4}-\frac{29\!\cdots\!33}{20\!\cdots\!17}a^{3}-\frac{40\!\cdots\!05}{20\!\cdots\!17}a^{2}-\frac{15\!\cdots\!57}{40\!\cdots\!34}a-\frac{43\!\cdots\!11}{20\!\cdots\!17}$ 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{6792537412232801002582422064039516108762981549823466733020436687789928976142436901339171}{98738597904054451933088240413608426834088629326608151319907309022236666515400190351436434} a^{43} + \frac{7862287986351534617270734220958010146390452828161249737916451883067457486006366617813055}{98738597904054451933088240413608426834088629326608151319907309022236666515400190351436434} a^{42} - \frac{109171396861104011903136110132700282623844166962443384810696586294855644431390760778404037}{49369298952027225966544120206804213417044314663304075659953654511118333257700095175718217} a^{41} + \frac{88377696366719351285027004520954977694878599726611594995810437454470061566879699174571232}{49369298952027225966544120206804213417044314663304075659953654511118333257700095175718217} a^{40} - \frac{2074737833413453857497870010842034870596341325623721905541019992640211483070535452696431041}{49369298952027225966544120206804213417044314663304075659953654511118333257700095175718217} a^{39} + \frac{2698892377114999965934202642720473295492445261665539208493079506239452805918450144263706537}{98738597904054451933088240413608426834088629326608151319907309022236666515400190351436434} a^{38} - \frac{50812102483984794737244574375454751669030671216058350535478016482970660062502082468285945033}{98738597904054451933088240413608426834088629326608151319907309022236666515400190351436434} a^{37} + \frac{11884010867066869156132457460931025607122610084745401001080017362503247432227338789110665101}{49369298952027225966544120206804213417044314663304075659953654511118333257700095175718217} a^{36} - \frac{452129543082053920475876424755606692357315365725838645508537202412884158181241044688352792001}{98738597904054451933088240413608426834088629326608151319907309022236666515400190351436434} a^{35} + \frac{156362639120498672083203133610923934580146316221562493779969934068377156865138102243488748067}{98738597904054451933088240413608426834088629326608151319907309022236666515400190351436434} a^{34} - \frac{2943893527479993671004722122854516348643034913321253742069536768133355921724931991909933536537}{98738597904054451933088240413608426834088629326608151319907309022236666515400190351436434} a^{33} + \frac{650530440242827947891460030127296486867072816190906942175510842965173206895647550340567117049}{98738597904054451933088240413608426834088629326608151319907309022236666515400190351436434} a^{32} - \frac{7268326102570836808275091637009056707859352982703123228007913408502505473596540770382740750136}{49369298952027225966544120206804213417044314663304075659953654511118333257700095175718217} a^{31} + \frac{2048667653533177086174343560380945338572128206056369226281585747344258296365671865695953883065}{98738597904054451933088240413608426834088629326608151319907309022236666515400190351436434} a^{30} - \frac{54127446678058260660366163219124973275918596483952446631147897405507176447842215422936974057273}{98738597904054451933088240413608426834088629326608151319907309022236666515400190351436434} a^{29} + \frac{3786905442078774953476028414534071839585288097110518214560317080112558212297580272951826927321}{98738597904054451933088240413608426834088629326608151319907309022236666515400190351436434} a^{28} - \frac{155309958738887427320295918479949958516564902956521824874628165655447923893525568468143768796513}{98738597904054451933088240413608426834088629326608151319907309022236666515400190351436434} a^{27} + \frac{2766042641865626289738720362165756051142887813362467586800295929835652993452157583780118930843}{49369298952027225966544120206804213417044314663304075659953654511118333257700095175718217} a^{26} - \frac{171707008983546036794994132755831113050659321377966435322700304817457996437195690957641373200895}{49369298952027225966544120206804213417044314663304075659953654511118333257700095175718217} a^{25} + \frac{2653251156095528940697405141551432281748093118032476883827877630173547528812767723252179819600}{49369298952027225966544120206804213417044314663304075659953654511118333257700095175718217} a^{24} - \frac{298352045488683764644476593016163741561077853133871603655084945209969895003777035366987494204437}{49369298952027225966544120206804213417044314663304075659953654511118333257700095175718217} a^{23} + \frac{3521579295410616217973245589959850943219621107149842499962709401451411871979664304591934518767}{49369298952027225966544120206804213417044314663304075659953654511118333257700095175718217} a^{22} - \frac{407227944274716510138075460892987442029866387420365119973289519224941154368122115978120529641409}{49369298952027225966544120206804213417044314663304075659953654511118333257700095175718217} a^{21} + \frac{5325647941403645676318037066739116579556454920890011603179457414966382693926186475813580864253}{49369298952027225966544120206804213417044314663304075659953654511118333257700095175718217} a^{20} - \frac{440338284837480420571208828802953621725951048932460565060820394044660505662327020115600968140329}{49369298952027225966544120206804213417044314663304075659953654511118333257700095175718217} a^{19} + \frac{10049597194458510232919951553651935759418407248730746590787897524844396468905730632681290061086}{49369298952027225966544120206804213417044314663304075659953654511118333257700095175718217} a^{18} - \frac{372958155943165037373520606583179812471292708411050694656306316745415844276964000052334166419841}{49369298952027225966544120206804213417044314663304075659953654511118333257700095175718217} a^{17} + \frac{26257124406217657000601742236210262673112914776931808514175620984917317044574625873914533387451}{98738597904054451933088240413608426834088629326608151319907309022236666515400190351436434} a^{16} - \frac{493651764123536908188622087513321756717318767361330054253950349494350767510757973466025366388013}{98738597904054451933088240413608426834088629326608151319907309022236666515400190351436434} a^{15} + \frac{14086971435991475501980149817873036438665362353029044205158570115067537362132910726213815466525}{49369298952027225966544120206804213417044314663304075659953654511118333257700095175718217} a^{14} - \frac{247982499515803253894864255949532867111013763293409661501289912226209378141279312488956960116825}{98738597904054451933088240413608426834088629326608151319907309022236666515400190351436434} a^{13} + \frac{20739404099681994415433082496450452372239266799245985525964304014579018549912445337146968917435}{98738597904054451933088240413608426834088629326608151319907309022236666515400190351436434} a^{12} - \frac{93672318487042511772242679009357568650499876740282782839632521238421297416041969353124968240145}{98738597904054451933088240413608426834088629326608151319907309022236666515400190351436434} a^{11} + \frac{11677301291614736173565342975030254902364131137917579714407394509264744533269317727953757422397}{98738597904054451933088240413608426834088629326608151319907309022236666515400190351436434} a^{10} - \frac{12459185032534679325612269119511998160934557294134506836368389503733772580932051195932895261572}{49369298952027225966544120206804213417044314663304075659953654511118333257700095175718217} a^{9} + \frac{2107113531750466663180543587545159079570086294827497964512981810976202619222337089372537220185}{49369298952027225966544120206804213417044314663304075659953654511118333257700095175718217} a^{8} - \frac{4729953063596148730147699366911205728975516874871970440402045341548321207543318670276676336937}{98738597904054451933088240413608426834088629326608151319907309022236666515400190351436434} a^{7} + \frac{493059912339677665081317578480511271500361584853201602471937715235206192146266315148488403522}{49369298952027225966544120206804213417044314663304075659953654511118333257700095175718217} a^{6} - \frac{257162925933588249305324405735788761565898297631047619127132354446062206716039947554442339120}{49369298952027225966544120206804213417044314663304075659953654511118333257700095175718217} a^{5} + \frac{127891062645071797144848197347299000675561705289038962183282348698023361701451363191549706923}{98738597904054451933088240413608426834088629326608151319907309022236666515400190351436434} a^{4} - \frac{20713889971870654145296537508736805254303916619756916725443082538014323953372985186938622803}{49369298952027225966544120206804213417044314663304075659953654511118333257700095175718217} a^{3} + \frac{7554030412780138297658967978307955078043046087218370633160616699623320805938948656832700333}{98738597904054451933088240413608426834088629326608151319907309022236666515400190351436434} a^{2} - \frac{634321657111916783033107771539107247088364790519949395901363365242274126840905014941805953}{98738597904054451933088240413608426834088629326608151319907309022236666515400190351436434} a + \frac{76078203866007790348609058080077629739289365921417811032630628455505768442701243858751599}{98738597904054451933088240413608426834088629326608151319907309022236666515400190351436434} \)  (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 - x^43 + 32*x^42 - 21*x^41 + 608*x^40 - 302*x^39 + 7441*x^38 - 2334*x^37 + 66293*x^36 - 12644*x^35 + 432280*x^34 - 28161*x^33 + 2141316*x^32 + 33285*x^31 + 8001765*x^30 + 693375*x^29 + 23077394*x^28 + 2792911*x^27 + 51292654*x^26 + 7229970*x^25 + 89635724*x^24 + 12947356*x^23 + 123068074*x^22 + 17592284*x^21 + 133953248*x^20 + 17840654*x^19 + 114270996*x^18 + 13803865*x^17 + 76240835*x^16 + 7576064*x^15 + 38619954*x^14 + 2830121*x^13 + 14698338*x^12 + 489978*x^11 + 3921204*x^10 - 46901*x^9 + 736586*x^8 - 41505*x^7 + 78482*x^6 - 8849*x^5 + 5761*x^4 - 386*x^3 + 117*x^2 + 6*x + 1)
 
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 + 32*x^42 - 21*x^41 + 608*x^40 - 302*x^39 + 7441*x^38 - 2334*x^37 + 66293*x^36 - 12644*x^35 + 432280*x^34 - 28161*x^33 + 2141316*x^32 + 33285*x^31 + 8001765*x^30 + 693375*x^29 + 23077394*x^28 + 2792911*x^27 + 51292654*x^26 + 7229970*x^25 + 89635724*x^24 + 12947356*x^23 + 123068074*x^22 + 17592284*x^21 + 133953248*x^20 + 17840654*x^19 + 114270996*x^18 + 13803865*x^17 + 76240835*x^16 + 7576064*x^15 + 38619954*x^14 + 2830121*x^13 + 14698338*x^12 + 489978*x^11 + 3921204*x^10 - 46901*x^9 + 736586*x^8 - 41505*x^7 + 78482*x^6 - 8849*x^5 + 5761*x^4 - 386*x^3 + 117*x^2 + 6*x + 1, 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 + 32*x^42 - 21*x^41 + 608*x^40 - 302*x^39 + 7441*x^38 - 2334*x^37 + 66293*x^36 - 12644*x^35 + 432280*x^34 - 28161*x^33 + 2141316*x^32 + 33285*x^31 + 8001765*x^30 + 693375*x^29 + 23077394*x^28 + 2792911*x^27 + 51292654*x^26 + 7229970*x^25 + 89635724*x^24 + 12947356*x^23 + 123068074*x^22 + 17592284*x^21 + 133953248*x^20 + 17840654*x^19 + 114270996*x^18 + 13803865*x^17 + 76240835*x^16 + 7576064*x^15 + 38619954*x^14 + 2830121*x^13 + 14698338*x^12 + 489978*x^11 + 3921204*x^10 - 46901*x^9 + 736586*x^8 - 41505*x^7 + 78482*x^6 - 8849*x^5 + 5761*x^4 - 386*x^3 + 117*x^2 + 6*x + 1);
 
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 + 32*x^42 - 21*x^41 + 608*x^40 - 302*x^39 + 7441*x^38 - 2334*x^37 + 66293*x^36 - 12644*x^35 + 432280*x^34 - 28161*x^33 + 2141316*x^32 + 33285*x^31 + 8001765*x^30 + 693375*x^29 + 23077394*x^28 + 2792911*x^27 + 51292654*x^26 + 7229970*x^25 + 89635724*x^24 + 12947356*x^23 + 123068074*x^22 + 17592284*x^21 + 133953248*x^20 + 17840654*x^19 + 114270996*x^18 + 13803865*x^17 + 76240835*x^16 + 7576064*x^15 + 38619954*x^14 + 2830121*x^13 + 14698338*x^12 + 489978*x^11 + 3921204*x^10 - 46901*x^9 + 736586*x^8 - 41505*x^7 + 78482*x^6 - 8849*x^5 + 5761*x^4 - 386*x^3 + 117*x^2 + 6*x + 1);
 
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 $C_2\times C_{22}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\zeta_{23})^+\), 22.22.83796671451884098775580820361328125.1, 22.0.304011857053427966889939263171547.1, 22.0.14844328957686912445797815584548193359375.1

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 $22^{2}$ R R $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ ${\href{/padicField/19.11.0.1}{11} }^{4}$ R $22^{2}$ ${\href{/padicField/31.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ $22^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{22}$ $22^{2}$ $22^{2}$

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
\(3\) Copy content Toggle raw display Deg $44$$2$$22$$22$
\(5\) Copy content Toggle raw display Deg $44$$2$$22$$22$
\(23\) Copy content Toggle raw display 23.22.20.1$x^{22} + 231 x^{21} + 24310 x^{20} + 1539615 x^{19} + 65271580 x^{18} + 1948237137 x^{17} + 41892053577 x^{16} + 652037968890 x^{15} + 7265319209115 x^{14} + 56290884204555 x^{13} + 287278948122936 x^{12} + 881069580352567 x^{11} + 1436394740619993 x^{10} + 1407272105659090 x^{9} + 908164935089445 x^{8} + 407525140102740 x^{7} + 130953632793951 x^{6} + 31291611122541 x^{5} + 17709125163290 x^{4} + 131485551467820 x^{3} + 905728303498040 x^{2} + 3760233666129578 x + 7096289029978197$$11$$2$$20$22T1$[\ ]_{11}^{2}$
23.22.20.1$x^{22} + 231 x^{21} + 24310 x^{20} + 1539615 x^{19} + 65271580 x^{18} + 1948237137 x^{17} + 41892053577 x^{16} + 652037968890 x^{15} + 7265319209115 x^{14} + 56290884204555 x^{13} + 287278948122936 x^{12} + 881069580352567 x^{11} + 1436394740619993 x^{10} + 1407272105659090 x^{9} + 908164935089445 x^{8} + 407525140102740 x^{7} + 130953632793951 x^{6} + 31291611122541 x^{5} + 17709125163290 x^{4} + 131485551467820 x^{3} + 905728303498040 x^{2} + 3760233666129578 x + 7096289029978197$$11$$2$$20$22T1$[\ ]_{11}^{2}$