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Properties

Label 44.0.534...389.1
Degree $44$
Signature $[0, 22]$
Discriminant $5.349\times 10^{102}$
Root discriminant \(216.14\)
Ramified primes $23,29$
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 - 7*x^43 + 22*x^42 - 242*x^41 + 1657*x^40 - 7240*x^39 + 42086*x^38 - 221328*x^37 + 1001535*x^36 - 4832011*x^35 + 21458307*x^34 - 90157847*x^33 + 380606142*x^32 - 1508426297*x^31 + 5783312496*x^30 - 21705960413*x^29 + 77541755704*x^28 - 268975788387*x^27 + 902794694673*x^26 - 2900080859207*x^25 + 9012566283473*x^24 - 26903177336836*x^23 + 76787979588976*x^22 - 210428309538158*x^21 + 549139502886158*x^20 - 1359857293394851*x^19 + 3192442755982022*x^18 - 7031729314930893*x^17 + 14441318484021425*x^16 - 27465922309147700*x^15 + 47542803690121335*x^14 - 73794535762603301*x^13 + 100954209324888769*x^12 - 117329395660925510*x^11 + 112228211865505796*x^10 - 90562555536299943*x^9 + 66866683023604529*x^8 - 46551285520550895*x^7 + 28899354676200824*x^6 - 15707692452050658*x^5 + 7816421449406728*x^4 - 3447228254556642*x^3 + 1348443431557763*x^2 - 386163463261299*x + 114988389055037)
 
gp: K = bnfinit(y^44 - 7*y^43 + 22*y^42 - 242*y^41 + 1657*y^40 - 7240*y^39 + 42086*y^38 - 221328*y^37 + 1001535*y^36 - 4832011*y^35 + 21458307*y^34 - 90157847*y^33 + 380606142*y^32 - 1508426297*y^31 + 5783312496*y^30 - 21705960413*y^29 + 77541755704*y^28 - 268975788387*y^27 + 902794694673*y^26 - 2900080859207*y^25 + 9012566283473*y^24 - 26903177336836*y^23 + 76787979588976*y^22 - 210428309538158*y^21 + 549139502886158*y^20 - 1359857293394851*y^19 + 3192442755982022*y^18 - 7031729314930893*y^17 + 14441318484021425*y^16 - 27465922309147700*y^15 + 47542803690121335*y^14 - 73794535762603301*y^13 + 100954209324888769*y^12 - 117329395660925510*y^11 + 112228211865505796*y^10 - 90562555536299943*y^9 + 66866683023604529*y^8 - 46551285520550895*y^7 + 28899354676200824*y^6 - 15707692452050658*y^5 + 7816421449406728*y^4 - 3447228254556642*y^3 + 1348443431557763*y^2 - 386163463261299*y + 114988389055037, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^44 - 7*x^43 + 22*x^42 - 242*x^41 + 1657*x^40 - 7240*x^39 + 42086*x^38 - 221328*x^37 + 1001535*x^36 - 4832011*x^35 + 21458307*x^34 - 90157847*x^33 + 380606142*x^32 - 1508426297*x^31 + 5783312496*x^30 - 21705960413*x^29 + 77541755704*x^28 - 268975788387*x^27 + 902794694673*x^26 - 2900080859207*x^25 + 9012566283473*x^24 - 26903177336836*x^23 + 76787979588976*x^22 - 210428309538158*x^21 + 549139502886158*x^20 - 1359857293394851*x^19 + 3192442755982022*x^18 - 7031729314930893*x^17 + 14441318484021425*x^16 - 27465922309147700*x^15 + 47542803690121335*x^14 - 73794535762603301*x^13 + 100954209324888769*x^12 - 117329395660925510*x^11 + 112228211865505796*x^10 - 90562555536299943*x^9 + 66866683023604529*x^8 - 46551285520550895*x^7 + 28899354676200824*x^6 - 15707692452050658*x^5 + 7816421449406728*x^4 - 3447228254556642*x^3 + 1348443431557763*x^2 - 386163463261299*x + 114988389055037);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^44 - 7*x^43 + 22*x^42 - 242*x^41 + 1657*x^40 - 7240*x^39 + 42086*x^38 - 221328*x^37 + 1001535*x^36 - 4832011*x^35 + 21458307*x^34 - 90157847*x^33 + 380606142*x^32 - 1508426297*x^31 + 5783312496*x^30 - 21705960413*x^29 + 77541755704*x^28 - 268975788387*x^27 + 902794694673*x^26 - 2900080859207*x^25 + 9012566283473*x^24 - 26903177336836*x^23 + 76787979588976*x^22 - 210428309538158*x^21 + 549139502886158*x^20 - 1359857293394851*x^19 + 3192442755982022*x^18 - 7031729314930893*x^17 + 14441318484021425*x^16 - 27465922309147700*x^15 + 47542803690121335*x^14 - 73794535762603301*x^13 + 100954209324888769*x^12 - 117329395660925510*x^11 + 112228211865505796*x^10 - 90562555536299943*x^9 + 66866683023604529*x^8 - 46551285520550895*x^7 + 28899354676200824*x^6 - 15707692452050658*x^5 + 7816421449406728*x^4 - 3447228254556642*x^3 + 1348443431557763*x^2 - 386163463261299*x + 114988389055037)
 

\( x^{44} - 7 x^{43} + 22 x^{42} - 242 x^{41} + 1657 x^{40} - 7240 x^{39} + 42086 x^{38} + \cdots + 114988389055037 \) 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:   \(534\!\cdots\!389\) \(\medspace = 23^{40}\cdot 29^{33}\) 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:  \(216.14\)
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:  $23^{10/11}29^{3/4}\approx 216.13887906264316$
Ramified primes:   \(23\), \(29\) 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:  \(\Q(\sqrt{29}) \)
$\card{ \Gal(K/\Q) }$:  $44$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(667=23\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{667}(128,·)$, $\chi_{667}(1,·)$, $\chi_{667}(133,·)$, $\chi_{667}(262,·)$, $\chi_{667}(394,·)$, $\chi_{667}(12,·)$, $\chi_{667}(144,·)$, $\chi_{667}(146,·)$, $\chi_{667}(278,·)$, $\chi_{667}(407,·)$, $\chi_{667}(289,·)$, $\chi_{667}(162,·)$, $\chi_{667}(423,·)$, $\chi_{667}(41,·)$, $\chi_{667}(173,·)$, $\chi_{667}(302,·)$, $\chi_{667}(307,·)$, $\chi_{667}(568,·)$, $\chi_{667}(186,·)$, $\chi_{667}(59,·)$, $\chi_{667}(650,·)$, $\chi_{667}(579,·)$, $\chi_{667}(581,·)$, $\chi_{667}(70,·)$, $\chi_{667}(202,·)$, $\chi_{667}(331,·)$, $\chi_{667}(418,·)$, $\chi_{667}(463,·)$, $\chi_{667}(75,·)$, $\chi_{667}(220,·)$, $\chi_{667}(215,·)$, $\chi_{667}(347,·)$, $\chi_{667}(476,·)$, $\chi_{667}(349,·)$, $\chi_{667}(610,·)$, $\chi_{667}(231,·)$, $\chi_{667}(104,·)$, $\chi_{667}(233,·)$, $\chi_{667}(492,·)$, $\chi_{667}(117,·)$, $\chi_{667}(376,·)$, $\chi_{667}(637,·)$, $\chi_{667}(510,·)$, $\chi_{667}(639,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{2097152}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $\frac{1}{7}a^{33}+\frac{2}{7}a^{32}-\frac{1}{7}a^{31}-\frac{3}{7}a^{30}+\frac{2}{7}a^{29}+\frac{2}{7}a^{28}+\frac{3}{7}a^{27}+\frac{1}{7}a^{26}-\frac{3}{7}a^{24}+\frac{3}{7}a^{23}-\frac{2}{7}a^{22}+\frac{2}{7}a^{21}-\frac{2}{7}a^{20}+\frac{3}{7}a^{19}-\frac{2}{7}a^{18}-\frac{2}{7}a^{16}-\frac{1}{7}a^{15}-\frac{1}{7}a^{14}+\frac{3}{7}a^{13}-\frac{3}{7}a^{12}+\frac{1}{7}a^{11}-\frac{3}{7}a^{10}+\frac{1}{7}a^{9}-\frac{1}{7}a^{8}-\frac{1}{7}a^{7}-\frac{1}{7}a^{6}+\frac{3}{7}a^{5}+\frac{3}{7}a^{4}-\frac{2}{7}a^{3}+\frac{1}{7}a-\frac{1}{7}$, $\frac{1}{7}a^{34}+\frac{2}{7}a^{32}-\frac{1}{7}a^{31}+\frac{1}{7}a^{30}-\frac{2}{7}a^{29}-\frac{1}{7}a^{28}+\frac{2}{7}a^{27}-\frac{2}{7}a^{26}-\frac{3}{7}a^{25}+\frac{2}{7}a^{24}-\frac{1}{7}a^{23}-\frac{1}{7}a^{22}+\frac{1}{7}a^{21}-\frac{1}{7}a^{19}-\frac{3}{7}a^{18}-\frac{2}{7}a^{17}+\frac{3}{7}a^{16}+\frac{1}{7}a^{15}-\frac{2}{7}a^{14}-\frac{2}{7}a^{13}+\frac{2}{7}a^{11}-\frac{3}{7}a^{9}+\frac{1}{7}a^{8}+\frac{1}{7}a^{7}-\frac{2}{7}a^{6}-\frac{3}{7}a^{5}-\frac{1}{7}a^{4}-\frac{3}{7}a^{3}+\frac{1}{7}a^{2}-\frac{3}{7}a+\frac{2}{7}$, $\frac{1}{7}a^{35}+\frac{2}{7}a^{32}+\frac{3}{7}a^{31}-\frac{3}{7}a^{30}+\frac{2}{7}a^{29}-\frac{2}{7}a^{28}-\frac{1}{7}a^{27}+\frac{2}{7}a^{26}+\frac{2}{7}a^{25}-\frac{2}{7}a^{24}-\frac{2}{7}a^{22}+\frac{3}{7}a^{21}+\frac{3}{7}a^{20}-\frac{2}{7}a^{19}+\frac{2}{7}a^{18}+\frac{3}{7}a^{17}-\frac{2}{7}a^{16}+\frac{1}{7}a^{13}+\frac{1}{7}a^{12}-\frac{2}{7}a^{11}+\frac{3}{7}a^{10}-\frac{1}{7}a^{9}+\frac{3}{7}a^{8}-\frac{1}{7}a^{6}-\frac{2}{7}a^{4}-\frac{2}{7}a^{3}-\frac{3}{7}a^{2}+\frac{2}{7}$, $\frac{1}{7}a^{36}-\frac{1}{7}a^{32}-\frac{1}{7}a^{31}+\frac{1}{7}a^{30}+\frac{1}{7}a^{29}+\frac{2}{7}a^{28}+\frac{3}{7}a^{27}-\frac{2}{7}a^{25}-\frac{1}{7}a^{24}-\frac{1}{7}a^{23}-\frac{1}{7}a^{21}+\frac{2}{7}a^{20}+\frac{3}{7}a^{19}-\frac{2}{7}a^{17}-\frac{3}{7}a^{16}+\frac{2}{7}a^{15}+\frac{3}{7}a^{14}+\frac{2}{7}a^{13}-\frac{3}{7}a^{12}+\frac{1}{7}a^{11}-\frac{2}{7}a^{10}+\frac{1}{7}a^{9}+\frac{2}{7}a^{8}+\frac{1}{7}a^{7}+\frac{2}{7}a^{6}-\frac{1}{7}a^{5}-\frac{1}{7}a^{4}+\frac{1}{7}a^{3}+\frac{2}{7}$, $\frac{1}{7}a^{37}+\frac{1}{7}a^{32}-\frac{2}{7}a^{30}-\frac{3}{7}a^{29}-\frac{2}{7}a^{28}+\frac{3}{7}a^{27}-\frac{1}{7}a^{26}-\frac{1}{7}a^{25}+\frac{3}{7}a^{24}+\frac{3}{7}a^{23}-\frac{3}{7}a^{22}-\frac{3}{7}a^{21}+\frac{1}{7}a^{20}+\frac{3}{7}a^{19}+\frac{3}{7}a^{18}-\frac{3}{7}a^{17}+\frac{2}{7}a^{15}+\frac{1}{7}a^{14}-\frac{2}{7}a^{12}-\frac{1}{7}a^{11}-\frac{2}{7}a^{10}+\frac{3}{7}a^{9}+\frac{1}{7}a^{7}-\frac{2}{7}a^{6}+\frac{2}{7}a^{5}-\frac{3}{7}a^{4}-\frac{2}{7}a^{3}+\frac{3}{7}a-\frac{1}{7}$, $\frac{1}{7}a^{38}-\frac{2}{7}a^{32}-\frac{1}{7}a^{31}+\frac{3}{7}a^{29}+\frac{1}{7}a^{28}+\frac{3}{7}a^{27}-\frac{2}{7}a^{26}+\frac{3}{7}a^{25}-\frac{1}{7}a^{24}+\frac{1}{7}a^{23}-\frac{1}{7}a^{22}-\frac{1}{7}a^{21}-\frac{2}{7}a^{20}-\frac{1}{7}a^{18}-\frac{3}{7}a^{16}+\frac{2}{7}a^{15}+\frac{1}{7}a^{14}+\frac{2}{7}a^{13}+\frac{2}{7}a^{12}-\frac{3}{7}a^{11}-\frac{1}{7}a^{10}-\frac{1}{7}a^{9}+\frac{2}{7}a^{8}-\frac{1}{7}a^{7}+\frac{3}{7}a^{6}+\frac{1}{7}a^{5}+\frac{2}{7}a^{4}+\frac{2}{7}a^{3}+\frac{3}{7}a^{2}-\frac{2}{7}a+\frac{1}{7}$, $\frac{1}{973}a^{39}+\frac{3}{139}a^{38}-\frac{6}{973}a^{37}-\frac{4}{973}a^{36}-\frac{69}{973}a^{35}+\frac{40}{973}a^{34}+\frac{45}{973}a^{33}-\frac{135}{973}a^{32}-\frac{311}{973}a^{31}-\frac{485}{973}a^{30}+\frac{437}{973}a^{29}+\frac{444}{973}a^{28}-\frac{344}{973}a^{27}-\frac{449}{973}a^{26}+\frac{18}{139}a^{25}-\frac{391}{973}a^{24}+\frac{79}{973}a^{23}+\frac{51}{139}a^{22}+\frac{283}{973}a^{21}+\frac{49}{139}a^{20}+\frac{369}{973}a^{19}-\frac{426}{973}a^{18}+\frac{16}{973}a^{17}+\frac{101}{973}a^{16}-\frac{418}{973}a^{15}+\frac{11}{973}a^{14}+\frac{16}{139}a^{13}+\frac{29}{139}a^{12}+\frac{42}{139}a^{11}-\frac{31}{139}a^{10}-\frac{311}{973}a^{9}+\frac{232}{973}a^{8}+\frac{1}{139}a^{7}-\frac{305}{973}a^{6}+\frac{176}{973}a^{5}-\frac{486}{973}a^{4}+\frac{425}{973}a^{3}-\frac{14}{139}a^{2}-\frac{258}{973}a+\frac{257}{973}$, $\frac{1}{973}a^{40}-\frac{30}{973}a^{38}-\frac{17}{973}a^{37}+\frac{15}{973}a^{36}-\frac{40}{973}a^{35}+\frac{39}{973}a^{34}+\frac{32}{973}a^{33}+\frac{439}{973}a^{32}+\frac{69}{973}a^{31}+\frac{48}{139}a^{30}+\frac{163}{973}a^{29}+\frac{340}{973}a^{28}-\frac{453}{973}a^{27}+\frac{381}{973}a^{26}-\frac{396}{973}a^{25}+\frac{89}{973}a^{24}+\frac{227}{973}a^{23}-\frac{403}{973}a^{22}-\frac{318}{973}a^{21}-\frac{23}{973}a^{20}-\frac{113}{973}a^{19}+\frac{344}{973}a^{18}-\frac{235}{973}a^{17}-\frac{454}{973}a^{16}+\frac{310}{973}a^{15}+\frac{298}{973}a^{14}-\frac{29}{139}a^{13}+\frac{62}{973}a^{12}+\frac{281}{973}a^{11}+\frac{76}{973}a^{10}+\frac{230}{973}a^{9}-\frac{1}{7}a^{8}-\frac{313}{973}a^{7}+\frac{48}{973}a^{6}-\frac{290}{973}a^{5}-\frac{72}{973}a^{4}+\frac{151}{973}a^{3}-\frac{285}{973}a^{2}+\frac{115}{973}a+\frac{63}{139}$, $\frac{1}{973}a^{41}+\frac{57}{973}a^{38}-\frac{26}{973}a^{37}-\frac{3}{139}a^{36}+\frac{54}{973}a^{35}-\frac{19}{973}a^{34}-\frac{18}{973}a^{33}+\frac{50}{973}a^{32}-\frac{237}{973}a^{31}-\frac{10}{139}a^{30}-\frac{33}{973}a^{29}-\frac{60}{973}a^{28}-\frac{348}{973}a^{27}-\frac{244}{973}a^{26}-\frac{23}{973}a^{25}-\frac{244}{973}a^{24}+\frac{438}{973}a^{23}-\frac{450}{973}a^{22}+\frac{127}{973}a^{21}+\frac{169}{973}a^{20}+\frac{16}{973}a^{19}+\frac{468}{973}a^{18}+\frac{304}{973}a^{17}+\frac{282}{973}a^{16}+\frac{407}{973}a^{15}-\frac{429}{973}a^{14}-\frac{192}{973}a^{13}+\frac{394}{973}a^{12}+\frac{1}{7}a^{11}-\frac{442}{973}a^{10}+\frac{261}{973}a^{9}-\frac{25}{973}a^{8}-\frac{298}{973}a^{7}-\frac{127}{973}a^{6}+\frac{204}{973}a^{5}-\frac{16}{139}a^{4}-\frac{184}{973}a^{3}-\frac{323}{973}a^{2}+\frac{68}{973}a+\frac{65}{973}$, $\frac{1}{973}a^{42}+\frac{4}{139}a^{38}+\frac{43}{973}a^{37}+\frac{4}{973}a^{36}+\frac{22}{973}a^{35}+\frac{65}{973}a^{34}-\frac{13}{973}a^{33}+\frac{13}{139}a^{32}+\frac{143}{973}a^{31}+\frac{368}{973}a^{30}+\frac{51}{973}a^{29}-\frac{358}{973}a^{28}-\frac{374}{973}a^{27}-\frac{284}{973}a^{26}-\frac{198}{973}a^{25}-\frac{349}{973}a^{24}-\frac{88}{973}a^{23}+\frac{211}{973}a^{22}+\frac{23}{973}a^{21}+\frac{342}{973}a^{20}+\frac{424}{973}a^{19}-\frac{295}{973}a^{18}-\frac{74}{973}a^{17}-\frac{346}{973}a^{16}+\frac{323}{973}a^{15}-\frac{124}{973}a^{14}-\frac{291}{973}a^{13}-\frac{451}{973}a^{12}-\frac{103}{973}a^{11}+\frac{120}{973}a^{10}+\frac{7}{139}a^{9}-\frac{39}{973}a^{8}+\frac{447}{973}a^{7}-\frac{481}{973}a^{6}+\frac{3}{973}a^{5}+\frac{274}{973}a^{4}+\frac{194}{973}a^{3}+\frac{94}{973}a^{2}+\frac{37}{973}a+\frac{32}{139}$, $\frac{1}{23\!\cdots\!93}a^{43}-\frac{14\!\cdots\!60}{23\!\cdots\!93}a^{42}+\frac{16\!\cdots\!34}{32\!\cdots\!99}a^{41}-\frac{60\!\cdots\!09}{23\!\cdots\!93}a^{40}-\frac{23\!\cdots\!95}{23\!\cdots\!93}a^{39}-\frac{28\!\cdots\!48}{23\!\cdots\!93}a^{38}-\frac{34\!\cdots\!62}{32\!\cdots\!99}a^{37}+\frac{31\!\cdots\!97}{23\!\cdots\!93}a^{36}-\frac{15\!\cdots\!41}{23\!\cdots\!93}a^{35}-\frac{25\!\cdots\!68}{23\!\cdots\!93}a^{34}-\frac{91\!\cdots\!18}{23\!\cdots\!93}a^{33}-\frac{10\!\cdots\!20}{23\!\cdots\!93}a^{32}-\frac{60\!\cdots\!33}{23\!\cdots\!93}a^{31}+\frac{67\!\cdots\!61}{23\!\cdots\!93}a^{30}+\frac{35\!\cdots\!82}{23\!\cdots\!93}a^{29}-\frac{45\!\cdots\!54}{23\!\cdots\!93}a^{28}-\frac{12\!\cdots\!38}{23\!\cdots\!93}a^{27}-\frac{69\!\cdots\!65}{23\!\cdots\!93}a^{26}+\frac{32\!\cdots\!96}{23\!\cdots\!93}a^{25}-\frac{72\!\cdots\!54}{23\!\cdots\!93}a^{24}-\frac{29\!\cdots\!79}{83\!\cdots\!09}a^{23}+\frac{54\!\cdots\!18}{23\!\cdots\!93}a^{22}-\frac{98\!\cdots\!80}{23\!\cdots\!93}a^{21}+\frac{96\!\cdots\!26}{32\!\cdots\!99}a^{20}+\frac{98\!\cdots\!39}{23\!\cdots\!93}a^{19}+\frac{79\!\cdots\!55}{23\!\cdots\!93}a^{18}+\frac{11\!\cdots\!62}{23\!\cdots\!93}a^{17}-\frac{10\!\cdots\!95}{23\!\cdots\!93}a^{16}-\frac{10\!\cdots\!07}{23\!\cdots\!93}a^{15}+\frac{52\!\cdots\!83}{23\!\cdots\!93}a^{14}+\frac{67\!\cdots\!80}{23\!\cdots\!93}a^{13}+\frac{92\!\cdots\!66}{23\!\cdots\!93}a^{12}+\frac{28\!\cdots\!35}{23\!\cdots\!93}a^{11}-\frac{39\!\cdots\!09}{23\!\cdots\!93}a^{10}-\frac{73\!\cdots\!96}{23\!\cdots\!93}a^{9}+\frac{61\!\cdots\!01}{23\!\cdots\!93}a^{8}+\frac{60\!\cdots\!32}{23\!\cdots\!93}a^{7}-\frac{98\!\cdots\!47}{23\!\cdots\!93}a^{6}-\frac{16\!\cdots\!17}{23\!\cdots\!93}a^{5}-\frac{10\!\cdots\!63}{23\!\cdots\!93}a^{4}+\frac{32\!\cdots\!72}{32\!\cdots\!99}a^{3}+\frac{95\!\cdots\!51}{23\!\cdots\!93}a^{2}+\frac{26\!\cdots\!04}{16\!\cdots\!87}a+\frac{90\!\cdots\!45}{23\!\cdots\!93}$ 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:   \( -1 \)  (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 - 7*x^43 + 22*x^42 - 242*x^41 + 1657*x^40 - 7240*x^39 + 42086*x^38 - 221328*x^37 + 1001535*x^36 - 4832011*x^35 + 21458307*x^34 - 90157847*x^33 + 380606142*x^32 - 1508426297*x^31 + 5783312496*x^30 - 21705960413*x^29 + 77541755704*x^28 - 268975788387*x^27 + 902794694673*x^26 - 2900080859207*x^25 + 9012566283473*x^24 - 26903177336836*x^23 + 76787979588976*x^22 - 210428309538158*x^21 + 549139502886158*x^20 - 1359857293394851*x^19 + 3192442755982022*x^18 - 7031729314930893*x^17 + 14441318484021425*x^16 - 27465922309147700*x^15 + 47542803690121335*x^14 - 73794535762603301*x^13 + 100954209324888769*x^12 - 117329395660925510*x^11 + 112228211865505796*x^10 - 90562555536299943*x^9 + 66866683023604529*x^8 - 46551285520550895*x^7 + 28899354676200824*x^6 - 15707692452050658*x^5 + 7816421449406728*x^4 - 3447228254556642*x^3 + 1348443431557763*x^2 - 386163463261299*x + 114988389055037)
 
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 - 7*x^43 + 22*x^42 - 242*x^41 + 1657*x^40 - 7240*x^39 + 42086*x^38 - 221328*x^37 + 1001535*x^36 - 4832011*x^35 + 21458307*x^34 - 90157847*x^33 + 380606142*x^32 - 1508426297*x^31 + 5783312496*x^30 - 21705960413*x^29 + 77541755704*x^28 - 268975788387*x^27 + 902794694673*x^26 - 2900080859207*x^25 + 9012566283473*x^24 - 26903177336836*x^23 + 76787979588976*x^22 - 210428309538158*x^21 + 549139502886158*x^20 - 1359857293394851*x^19 + 3192442755982022*x^18 - 7031729314930893*x^17 + 14441318484021425*x^16 - 27465922309147700*x^15 + 47542803690121335*x^14 - 73794535762603301*x^13 + 100954209324888769*x^12 - 117329395660925510*x^11 + 112228211865505796*x^10 - 90562555536299943*x^9 + 66866683023604529*x^8 - 46551285520550895*x^7 + 28899354676200824*x^6 - 15707692452050658*x^5 + 7816421449406728*x^4 - 3447228254556642*x^3 + 1348443431557763*x^2 - 386163463261299*x + 114988389055037, 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 - 7*x^43 + 22*x^42 - 242*x^41 + 1657*x^40 - 7240*x^39 + 42086*x^38 - 221328*x^37 + 1001535*x^36 - 4832011*x^35 + 21458307*x^34 - 90157847*x^33 + 380606142*x^32 - 1508426297*x^31 + 5783312496*x^30 - 21705960413*x^29 + 77541755704*x^28 - 268975788387*x^27 + 902794694673*x^26 - 2900080859207*x^25 + 9012566283473*x^24 - 26903177336836*x^23 + 76787979588976*x^22 - 210428309538158*x^21 + 549139502886158*x^20 - 1359857293394851*x^19 + 3192442755982022*x^18 - 7031729314930893*x^17 + 14441318484021425*x^16 - 27465922309147700*x^15 + 47542803690121335*x^14 - 73794535762603301*x^13 + 100954209324888769*x^12 - 117329395660925510*x^11 + 112228211865505796*x^10 - 90562555536299943*x^9 + 66866683023604529*x^8 - 46551285520550895*x^7 + 28899354676200824*x^6 - 15707692452050658*x^5 + 7816421449406728*x^4 - 3447228254556642*x^3 + 1348443431557763*x^2 - 386163463261299*x + 114988389055037);
 
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 - 7*x^43 + 22*x^42 - 242*x^41 + 1657*x^40 - 7240*x^39 + 42086*x^38 - 221328*x^37 + 1001535*x^36 - 4832011*x^35 + 21458307*x^34 - 90157847*x^33 + 380606142*x^32 - 1508426297*x^31 + 5783312496*x^30 - 21705960413*x^29 + 77541755704*x^28 - 268975788387*x^27 + 902794694673*x^26 - 2900080859207*x^25 + 9012566283473*x^24 - 26903177336836*x^23 + 76787979588976*x^22 - 210428309538158*x^21 + 549139502886158*x^20 - 1359857293394851*x^19 + 3192442755982022*x^18 - 7031729314930893*x^17 + 14441318484021425*x^16 - 27465922309147700*x^15 + 47542803690121335*x^14 - 73794535762603301*x^13 + 100954209324888769*x^12 - 117329395660925510*x^11 + 112228211865505796*x^10 - 90562555536299943*x^9 + 66866683023604529*x^8 - 46551285520550895*x^7 + 28899354676200824*x^6 - 15707692452050658*x^5 + 7816421449406728*x^4 - 3447228254556642*x^3 + 1348443431557763*x^2 - 386163463261299*x + 114988389055037);
 
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 $C_{44}$

Intermediate fields

\(\Q(\sqrt{29}) \), 4.0.24389.1, \(\Q(\zeta_{23})^+\), 22.22.20937975979670626213353681795476767790826629.1

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 $44$ $44$ $22^{2}$ ${\href{/padicField/7.11.0.1}{11} }^{4}$ $44$ $22^{2}$ $44$ $44$ R R $44$ $44$ $44$ $44$ ${\href{/padicField/47.4.0.1}{4} }^{11}$ ${\href{/padicField/53.11.0.1}{11} }^{4}$ ${\href{/padicField/59.11.0.1}{11} }^{4}$

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
\(23\) Copy content Toggle raw display 23.11.10.10$x^{11} + 23$$11$$1$$10$$C_{11}$$[\ ]_{11}$
23.11.10.10$x^{11} + 23$$11$$1$$10$$C_{11}$$[\ ]_{11}$
23.11.10.10$x^{11} + 23$$11$$1$$10$$C_{11}$$[\ ]_{11}$
23.11.10.10$x^{11} + 23$$11$$1$$10$$C_{11}$$[\ ]_{11}$
\(29\) Copy content Toggle raw display Deg $44$$4$$11$$33$