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Properties

Label 44.0.182...881.1
Degree $44$
Signature $[0, 22]$
Discriminant $1.830\times 10^{84}$
Root discriminant \(82.23\)
Ramified primes $17,23$
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 + 5*x^42 - 9*x^41 + 29*x^40 - 65*x^39 + 181*x^38 - 441*x^37 + 1165*x^36 - 2929*x^35 + 7589*x^34 - 19305*x^33 + 49661*x^32 - 126881*x^31 + 325525*x^30 - 833049*x^29 + 2135149*x^28 - 5467345*x^27 + 14007941*x^26 - 35877321*x^25 + 91909085*x^24 - 235418369*x^23 + 603054709*x^22 + 941673476*x^21 + 1470545360*x^20 + 2296148544*x^19 + 3586032896*x^18 + 5598561280*x^17 + 8745570304*x^16 + 13648674816*x^15 + 21333606400*x^14 + 33261092864*x^13 + 52073332736*x^12 + 80971038720*x^11 + 127322292224*x^10 + 196561862656*x^9 + 312727306240*x^8 + 473520144384*x^7 + 777389080576*x^6 + 1116691496960*x^5 + 1992864825344*x^4 + 2473901162496*x^3 + 5497558138880*x^2 + 4398046511104*x + 17592186044416)
 
gp: K = bnfinit(y^44 - y^43 + 5*y^42 - 9*y^41 + 29*y^40 - 65*y^39 + 181*y^38 - 441*y^37 + 1165*y^36 - 2929*y^35 + 7589*y^34 - 19305*y^33 + 49661*y^32 - 126881*y^31 + 325525*y^30 - 833049*y^29 + 2135149*y^28 - 5467345*y^27 + 14007941*y^26 - 35877321*y^25 + 91909085*y^24 - 235418369*y^23 + 603054709*y^22 + 941673476*y^21 + 1470545360*y^20 + 2296148544*y^19 + 3586032896*y^18 + 5598561280*y^17 + 8745570304*y^16 + 13648674816*y^15 + 21333606400*y^14 + 33261092864*y^13 + 52073332736*y^12 + 80971038720*y^11 + 127322292224*y^10 + 196561862656*y^9 + 312727306240*y^8 + 473520144384*y^7 + 777389080576*y^6 + 1116691496960*y^5 + 1992864825344*y^4 + 2473901162496*y^3 + 5497558138880*y^2 + 4398046511104*y + 17592186044416, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^44 - x^43 + 5*x^42 - 9*x^41 + 29*x^40 - 65*x^39 + 181*x^38 - 441*x^37 + 1165*x^36 - 2929*x^35 + 7589*x^34 - 19305*x^33 + 49661*x^32 - 126881*x^31 + 325525*x^30 - 833049*x^29 + 2135149*x^28 - 5467345*x^27 + 14007941*x^26 - 35877321*x^25 + 91909085*x^24 - 235418369*x^23 + 603054709*x^22 + 941673476*x^21 + 1470545360*x^20 + 2296148544*x^19 + 3586032896*x^18 + 5598561280*x^17 + 8745570304*x^16 + 13648674816*x^15 + 21333606400*x^14 + 33261092864*x^13 + 52073332736*x^12 + 80971038720*x^11 + 127322292224*x^10 + 196561862656*x^9 + 312727306240*x^8 + 473520144384*x^7 + 777389080576*x^6 + 1116691496960*x^5 + 1992864825344*x^4 + 2473901162496*x^3 + 5497558138880*x^2 + 4398046511104*x + 17592186044416);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^44 - x^43 + 5*x^42 - 9*x^41 + 29*x^40 - 65*x^39 + 181*x^38 - 441*x^37 + 1165*x^36 - 2929*x^35 + 7589*x^34 - 19305*x^33 + 49661*x^32 - 126881*x^31 + 325525*x^30 - 833049*x^29 + 2135149*x^28 - 5467345*x^27 + 14007941*x^26 - 35877321*x^25 + 91909085*x^24 - 235418369*x^23 + 603054709*x^22 + 941673476*x^21 + 1470545360*x^20 + 2296148544*x^19 + 3586032896*x^18 + 5598561280*x^17 + 8745570304*x^16 + 13648674816*x^15 + 21333606400*x^14 + 33261092864*x^13 + 52073332736*x^12 + 80971038720*x^11 + 127322292224*x^10 + 196561862656*x^9 + 312727306240*x^8 + 473520144384*x^7 + 777389080576*x^6 + 1116691496960*x^5 + 1992864825344*x^4 + 2473901162496*x^3 + 5497558138880*x^2 + 4398046511104*x + 17592186044416)
 

\( x^{44} - x^{43} + 5 x^{42} - 9 x^{41} + 29 x^{40} - 65 x^{39} + 181 x^{38} - 441 x^{37} + \cdots + 17592186044416 \) 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:   \(182\!\cdots\!881\) \(\medspace = 17^{22}\cdot 23^{42}\) 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:  \(82.23\)
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:  $17^{1/2}23^{21/22}\approx 82.23478794939922$
Ramified primes:   \(17\), \(23\) 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:  \(391=17\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{391}(256,·)$, $\chi_{391}(1,·)$, $\chi_{391}(390,·)$, $\chi_{391}(135,·)$, $\chi_{391}(137,·)$, $\chi_{391}(271,·)$, $\chi_{391}(16,·)$, $\chi_{391}(273,·)$, $\chi_{391}(18,·)$, $\chi_{391}(152,·)$, $\chi_{391}(154,·)$, $\chi_{391}(288,·)$, $\chi_{391}(33,·)$, $\chi_{391}(290,·)$, $\chi_{391}(35,·)$, $\chi_{391}(169,·)$, $\chi_{391}(171,·)$, $\chi_{391}(305,·)$, $\chi_{391}(50,·)$, $\chi_{391}(307,·)$, $\chi_{391}(52,·)$, $\chi_{391}(186,·)$, $\chi_{391}(188,·)$, $\chi_{391}(67,·)$, $\chi_{391}(324,·)$, $\chi_{391}(203,·)$, $\chi_{391}(205,·)$, $\chi_{391}(339,·)$, $\chi_{391}(84,·)$, $\chi_{391}(341,·)$, $\chi_{391}(86,·)$, $\chi_{391}(220,·)$, $\chi_{391}(222,·)$, $\chi_{391}(356,·)$, $\chi_{391}(101,·)$, $\chi_{391}(358,·)$, $\chi_{391}(103,·)$, $\chi_{391}(237,·)$, $\chi_{391}(239,·)$, $\chi_{391}(373,·)$, $\chi_{391}(118,·)$, $\chi_{391}(375,·)$, $\chi_{391}(120,·)$, $\chi_{391}(254,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{2097152}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{2412218836}a^{23}-\frac{1}{4}a^{22}+\frac{1}{4}a^{21}-\frac{1}{4}a^{20}+\frac{1}{4}a^{19}-\frac{1}{4}a^{18}+\frac{1}{4}a^{17}-\frac{1}{4}a^{16}+\frac{1}{4}a^{15}-\frac{1}{4}a^{14}+\frac{1}{4}a^{13}-\frac{1}{4}a^{12}+\frac{1}{4}a^{11}-\frac{1}{4}a^{10}+\frac{1}{4}a^{9}-\frac{1}{4}a^{8}+\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a+\frac{235418369}{603054709}$, $\frac{1}{9648875344}a^{24}-\frac{1}{9648875344}a^{23}+\frac{1}{16}a^{22}-\frac{5}{16}a^{21}-\frac{7}{16}a^{20}+\frac{3}{16}a^{19}+\frac{1}{16}a^{18}-\frac{5}{16}a^{17}-\frac{7}{16}a^{16}+\frac{3}{16}a^{15}+\frac{1}{16}a^{14}-\frac{5}{16}a^{13}-\frac{7}{16}a^{12}+\frac{3}{16}a^{11}+\frac{1}{16}a^{10}-\frac{5}{16}a^{9}-\frac{7}{16}a^{8}+\frac{3}{16}a^{7}+\frac{1}{16}a^{6}-\frac{5}{16}a^{5}-\frac{7}{16}a^{4}+\frac{3}{16}a^{3}+\frac{1}{16}a^{2}+\frac{235418369}{2412218836}a+\frac{91909085}{603054709}$, $\frac{1}{38595501376}a^{25}-\frac{1}{38595501376}a^{24}+\frac{5}{38595501376}a^{23}-\frac{5}{64}a^{22}+\frac{9}{64}a^{21}-\frac{29}{64}a^{20}+\frac{1}{64}a^{19}+\frac{11}{64}a^{18}-\frac{7}{64}a^{17}-\frac{13}{64}a^{16}-\frac{15}{64}a^{15}+\frac{27}{64}a^{14}-\frac{23}{64}a^{13}+\frac{3}{64}a^{12}-\frac{31}{64}a^{11}-\frac{21}{64}a^{10}+\frac{25}{64}a^{9}+\frac{19}{64}a^{8}+\frac{17}{64}a^{7}-\frac{5}{64}a^{6}+\frac{9}{64}a^{5}-\frac{29}{64}a^{4}+\frac{1}{64}a^{3}+\frac{235418369}{9648875344}a^{2}+\frac{91909085}{2412218836}a+\frac{35877321}{603054709}$, $\frac{1}{154382005504}a^{26}-\frac{1}{154382005504}a^{25}+\frac{5}{154382005504}a^{24}-\frac{9}{154382005504}a^{23}+\frac{9}{256}a^{22}-\frac{29}{256}a^{21}+\frac{65}{256}a^{20}+\frac{75}{256}a^{19}-\frac{71}{256}a^{18}+\frac{115}{256}a^{17}+\frac{113}{256}a^{16}+\frac{91}{256}a^{15}+\frac{105}{256}a^{14}+\frac{3}{256}a^{13}-\frac{95}{256}a^{12}+\frac{107}{256}a^{11}+\frac{25}{256}a^{10}-\frac{109}{256}a^{9}-\frac{47}{256}a^{8}+\frac{123}{256}a^{7}-\frac{55}{256}a^{6}+\frac{35}{256}a^{5}+\frac{1}{256}a^{4}+\frac{235418369}{38595501376}a^{3}+\frac{91909085}{9648875344}a^{2}+\frac{35877321}{2412218836}a+\frac{14007941}{603054709}$, $\frac{1}{617528022016}a^{27}-\frac{1}{617528022016}a^{26}+\frac{5}{617528022016}a^{25}-\frac{9}{617528022016}a^{24}+\frac{29}{617528022016}a^{23}-\frac{285}{1024}a^{22}+\frac{321}{1024}a^{21}-\frac{437}{1024}a^{20}-\frac{327}{1024}a^{19}-\frac{397}{1024}a^{18}+\frac{113}{1024}a^{17}+\frac{347}{1024}a^{16}+\frac{105}{1024}a^{15}+\frac{259}{1024}a^{14}+\frac{161}{1024}a^{13}-\frac{149}{1024}a^{12}-\frac{231}{1024}a^{11}-\frac{365}{1024}a^{10}+\frac{465}{1024}a^{9}+\frac{123}{1024}a^{8}-\frac{311}{1024}a^{7}-\frac{221}{1024}a^{6}+\frac{1}{1024}a^{5}+\frac{235418369}{154382005504}a^{4}+\frac{91909085}{38595501376}a^{3}+\frac{35877321}{9648875344}a^{2}+\frac{14007941}{2412218836}a+\frac{5467345}{603054709}$, $\frac{1}{2470112088064}a^{28}-\frac{1}{2470112088064}a^{27}+\frac{5}{2470112088064}a^{26}-\frac{9}{2470112088064}a^{25}+\frac{29}{2470112088064}a^{24}-\frac{65}{2470112088064}a^{23}-\frac{703}{4096}a^{22}-\frac{437}{4096}a^{21}+\frac{1721}{4096}a^{20}+\frac{627}{4096}a^{19}-\frac{1935}{4096}a^{18}+\frac{347}{4096}a^{17}+\frac{105}{4096}a^{16}+\frac{1283}{4096}a^{15}-\frac{863}{4096}a^{14}+\frac{1899}{4096}a^{13}-\frac{1255}{4096}a^{12}+\frac{659}{4096}a^{11}-\frac{1583}{4096}a^{10}+\frac{123}{4096}a^{9}+\frac{1737}{4096}a^{8}-\frac{1245}{4096}a^{7}+\frac{1}{4096}a^{6}+\frac{235418369}{617528022016}a^{5}+\frac{91909085}{154382005504}a^{4}+\frac{35877321}{38595501376}a^{3}+\frac{14007941}{9648875344}a^{2}+\frac{5467345}{2412218836}a+\frac{2135149}{603054709}$, $\frac{1}{9880448352256}a^{29}-\frac{1}{9880448352256}a^{28}+\frac{5}{9880448352256}a^{27}-\frac{9}{9880448352256}a^{26}+\frac{29}{9880448352256}a^{25}-\frac{65}{9880448352256}a^{24}+\frac{181}{9880448352256}a^{23}+\frac{3659}{16384}a^{22}-\frac{6471}{16384}a^{21}+\frac{4723}{16384}a^{20}+\frac{2161}{16384}a^{19}+\frac{347}{16384}a^{18}-\frac{8087}{16384}a^{17}-\frac{6909}{16384}a^{16}+\frac{7329}{16384}a^{15}-\frac{2197}{16384}a^{14}-\frac{1255}{16384}a^{13}-\frac{7533}{16384}a^{12}+\frac{2513}{16384}a^{11}+\frac{123}{16384}a^{10}-\frac{6455}{16384}a^{9}+\frac{6947}{16384}a^{8}+\frac{1}{16384}a^{7}+\frac{235418369}{2470112088064}a^{6}+\frac{91909085}{617528022016}a^{5}+\frac{35877321}{154382005504}a^{4}+\frac{14007941}{38595501376}a^{3}+\frac{5467345}{9648875344}a^{2}+\frac{2135149}{2412218836}a+\frac{833049}{603054709}$, $\frac{1}{39521793409024}a^{30}-\frac{1}{39521793409024}a^{29}+\frac{5}{39521793409024}a^{28}-\frac{9}{39521793409024}a^{27}+\frac{29}{39521793409024}a^{26}-\frac{65}{39521793409024}a^{25}+\frac{181}{39521793409024}a^{24}-\frac{441}{39521793409024}a^{23}+\frac{9913}{65536}a^{22}+\frac{4723}{65536}a^{21}-\frac{30607}{65536}a^{20}-\frac{16037}{65536}a^{19}+\frac{24681}{65536}a^{18}-\frac{23293}{65536}a^{17}-\frac{9055}{65536}a^{16}-\frac{18581}{65536}a^{15}-\frac{17639}{65536}a^{14}+\frac{8851}{65536}a^{13}-\frac{13871}{65536}a^{12}-\frac{16261}{65536}a^{11}+\frac{26313}{65536}a^{10}-\frac{25821}{65536}a^{9}+\frac{1}{65536}a^{8}+\frac{235418369}{9880448352256}a^{7}+\frac{91909085}{2470112088064}a^{6}+\frac{35877321}{617528022016}a^{5}+\frac{14007941}{154382005504}a^{4}+\frac{5467345}{38595501376}a^{3}+\frac{2135149}{9648875344}a^{2}+\frac{833049}{2412218836}a+\frac{325525}{603054709}$, $\frac{1}{158087173636096}a^{31}-\frac{1}{158087173636096}a^{30}+\frac{5}{158087173636096}a^{29}-\frac{9}{158087173636096}a^{28}+\frac{29}{158087173636096}a^{27}-\frac{65}{158087173636096}a^{26}+\frac{181}{158087173636096}a^{25}-\frac{441}{158087173636096}a^{24}+\frac{1165}{158087173636096}a^{23}-\frac{60813}{262144}a^{22}+\frac{100465}{262144}a^{21}-\frac{81573}{262144}a^{20}-\frac{40855}{262144}a^{19}-\frac{23293}{262144}a^{18}+\frac{122017}{262144}a^{17}+\frac{46955}{262144}a^{16}-\frac{83175}{262144}a^{15}+\frac{8851}{262144}a^{14}-\frac{79407}{262144}a^{13}+\frac{114811}{262144}a^{12}+\frac{91849}{262144}a^{11}+\frac{105251}{262144}a^{10}+\frac{1}{262144}a^{9}+\frac{235418369}{39521793409024}a^{8}+\frac{91909085}{9880448352256}a^{7}+\frac{35877321}{2470112088064}a^{6}+\frac{14007941}{617528022016}a^{5}+\frac{5467345}{154382005504}a^{4}+\frac{2135149}{38595501376}a^{3}+\frac{833049}{9648875344}a^{2}+\frac{325525}{2412218836}a+\frac{126881}{603054709}$, $\frac{1}{632348694544384}a^{32}-\frac{1}{632348694544384}a^{31}+\frac{5}{632348694544384}a^{30}-\frac{9}{632348694544384}a^{29}+\frac{29}{632348694544384}a^{28}-\frac{65}{632348694544384}a^{27}+\frac{181}{632348694544384}a^{26}-\frac{441}{632348694544384}a^{25}+\frac{1165}{632348694544384}a^{24}-\frac{2929}{632348694544384}a^{23}-\frac{423823}{1048576}a^{22}+\frac{180571}{1048576}a^{21}+\frac{221289}{1048576}a^{20}+\frac{500995}{1048576}a^{19}+\frac{384161}{1048576}a^{18}-\frac{477333}{1048576}a^{17}-\frac{83175}{1048576}a^{16}+\frac{270995}{1048576}a^{15}+\frac{444881}{1048576}a^{14}-\frac{409477}{1048576}a^{13}+\frac{91849}{1048576}a^{12}+\frac{367395}{1048576}a^{11}+\frac{1}{1048576}a^{10}+\frac{235418369}{158087173636096}a^{9}+\frac{91909085}{39521793409024}a^{8}+\frac{35877321}{9880448352256}a^{7}+\frac{14007941}{2470112088064}a^{6}+\frac{5467345}{617528022016}a^{5}+\frac{2135149}{154382005504}a^{4}+\frac{833049}{38595501376}a^{3}+\frac{325525}{9648875344}a^{2}+\frac{126881}{2412218836}a+\frac{49661}{603054709}$, $\frac{1}{25\!\cdots\!36}a^{33}-\frac{1}{25\!\cdots\!36}a^{32}+\frac{5}{25\!\cdots\!36}a^{31}-\frac{9}{25\!\cdots\!36}a^{30}+\frac{29}{25\!\cdots\!36}a^{29}-\frac{65}{25\!\cdots\!36}a^{28}+\frac{181}{25\!\cdots\!36}a^{27}-\frac{441}{25\!\cdots\!36}a^{26}+\frac{1165}{25\!\cdots\!36}a^{25}-\frac{2929}{25\!\cdots\!36}a^{24}+\frac{7589}{25\!\cdots\!36}a^{23}-\frac{1916581}{4194304}a^{22}+\frac{221289}{4194304}a^{21}+\frac{500995}{4194304}a^{20}+\frac{384161}{4194304}a^{19}+\frac{1619819}{4194304}a^{18}-\frac{83175}{4194304}a^{17}-\frac{1826157}{4194304}a^{16}+\frac{1493457}{4194304}a^{15}-\frac{409477}{4194304}a^{14}-\frac{2005303}{4194304}a^{13}+\frac{367395}{4194304}a^{12}+\frac{1}{4194304}a^{11}+\frac{235418369}{632348694544384}a^{10}+\frac{91909085}{158087173636096}a^{9}+\frac{35877321}{39521793409024}a^{8}+\frac{14007941}{9880448352256}a^{7}+\frac{5467345}{2470112088064}a^{6}+\frac{2135149}{617528022016}a^{5}+\frac{833049}{154382005504}a^{4}+\frac{325525}{38595501376}a^{3}+\frac{126881}{9648875344}a^{2}+\frac{49661}{2412218836}a+\frac{19305}{603054709}$, $\frac{1}{10\!\cdots\!44}a^{34}-\frac{1}{10\!\cdots\!44}a^{33}+\frac{5}{10\!\cdots\!44}a^{32}-\frac{9}{10\!\cdots\!44}a^{31}+\frac{29}{10\!\cdots\!44}a^{30}-\frac{65}{10\!\cdots\!44}a^{29}+\frac{181}{10\!\cdots\!44}a^{28}-\frac{441}{10\!\cdots\!44}a^{27}+\frac{1165}{10\!\cdots\!44}a^{26}-\frac{2929}{10\!\cdots\!44}a^{25}+\frac{7589}{10\!\cdots\!44}a^{24}-\frac{19305}{10\!\cdots\!44}a^{23}-\frac{8167319}{16777216}a^{22}+\frac{500995}{16777216}a^{21}+\frac{384161}{16777216}a^{20}+\frac{1619819}{16777216}a^{19}-\frac{83175}{16777216}a^{18}+\frac{6562451}{16777216}a^{17}-\frac{6895151}{16777216}a^{16}-\frac{409477}{16777216}a^{15}+\frac{6383305}{16777216}a^{14}-\frac{8021213}{16777216}a^{13}+\frac{1}{16777216}a^{12}+\frac{235418369}{25\!\cdots\!36}a^{11}+\frac{91909085}{632348694544384}a^{10}+\frac{35877321}{158087173636096}a^{9}+\frac{14007941}{39521793409024}a^{8}+\frac{5467345}{9880448352256}a^{7}+\frac{2135149}{2470112088064}a^{6}+\frac{833049}{617528022016}a^{5}+\frac{325525}{154382005504}a^{4}+\frac{126881}{38595501376}a^{3}+\frac{49661}{9648875344}a^{2}+\frac{19305}{2412218836}a+\frac{7589}{603054709}$, $\frac{1}{40\!\cdots\!76}a^{35}-\frac{1}{40\!\cdots\!76}a^{34}+\frac{5}{40\!\cdots\!76}a^{33}-\frac{9}{40\!\cdots\!76}a^{32}+\frac{29}{40\!\cdots\!76}a^{31}-\frac{65}{40\!\cdots\!76}a^{30}+\frac{181}{40\!\cdots\!76}a^{29}-\frac{441}{40\!\cdots\!76}a^{28}+\frac{1165}{40\!\cdots\!76}a^{27}-\frac{2929}{40\!\cdots\!76}a^{26}+\frac{7589}{40\!\cdots\!76}a^{25}-\frac{19305}{40\!\cdots\!76}a^{24}+\frac{49661}{40\!\cdots\!76}a^{23}+\frac{500995}{67108864}a^{22}-\frac{33170271}{67108864}a^{21}-\frac{31934613}{67108864}a^{20}+\frac{33471257}{67108864}a^{19}-\frac{26991981}{67108864}a^{18}+\frac{26659281}{67108864}a^{17}-\frac{409477}{67108864}a^{16}-\frac{27171127}{67108864}a^{15}+\frac{25533219}{67108864}a^{14}+\frac{1}{67108864}a^{13}+\frac{235418369}{10\!\cdots\!44}a^{12}+\frac{91909085}{25\!\cdots\!36}a^{11}+\frac{35877321}{632348694544384}a^{10}+\frac{14007941}{158087173636096}a^{9}+\frac{5467345}{39521793409024}a^{8}+\frac{2135149}{9880448352256}a^{7}+\frac{833049}{2470112088064}a^{6}+\frac{325525}{617528022016}a^{5}+\frac{126881}{154382005504}a^{4}+\frac{49661}{38595501376}a^{3}+\frac{19305}{9648875344}a^{2}+\frac{7589}{2412218836}a+\frac{2929}{603054709}$, $\frac{1}{16\!\cdots\!04}a^{36}-\frac{1}{16\!\cdots\!04}a^{35}+\frac{5}{16\!\cdots\!04}a^{34}-\frac{9}{16\!\cdots\!04}a^{33}+\frac{29}{16\!\cdots\!04}a^{32}-\frac{65}{16\!\cdots\!04}a^{31}+\frac{181}{16\!\cdots\!04}a^{30}-\frac{441}{16\!\cdots\!04}a^{29}+\frac{1165}{16\!\cdots\!04}a^{28}-\frac{2929}{16\!\cdots\!04}a^{27}+\frac{7589}{16\!\cdots\!04}a^{26}-\frac{19305}{16\!\cdots\!04}a^{25}+\frac{49661}{16\!\cdots\!04}a^{24}-\frac{126881}{16\!\cdots\!04}a^{23}+\frac{33938593}{268435456}a^{22}-\frac{31934613}{268435456}a^{21}-\frac{100746471}{268435456}a^{20}-\frac{26991981}{268435456}a^{19}-\frac{107558447}{268435456}a^{18}-\frac{409477}{268435456}a^{17}+\frac{107046601}{268435456}a^{16}-\frac{108684509}{268435456}a^{15}+\frac{1}{268435456}a^{14}+\frac{235418369}{40\!\cdots\!76}a^{13}+\frac{91909085}{10\!\cdots\!44}a^{12}+\frac{35877321}{25\!\cdots\!36}a^{11}+\frac{14007941}{632348694544384}a^{10}+\frac{5467345}{158087173636096}a^{9}+\frac{2135149}{39521793409024}a^{8}+\frac{833049}{9880448352256}a^{7}+\frac{325525}{2470112088064}a^{6}+\frac{126881}{617528022016}a^{5}+\frac{49661}{154382005504}a^{4}+\frac{19305}{38595501376}a^{3}+\frac{7589}{9648875344}a^{2}+\frac{2929}{2412218836}a+\frac{1165}{603054709}$, $\frac{1}{64\!\cdots\!16}a^{37}-\frac{1}{64\!\cdots\!16}a^{36}+\frac{5}{64\!\cdots\!16}a^{35}-\frac{9}{64\!\cdots\!16}a^{34}+\frac{29}{64\!\cdots\!16}a^{33}-\frac{65}{64\!\cdots\!16}a^{32}+\frac{181}{64\!\cdots\!16}a^{31}-\frac{441}{64\!\cdots\!16}a^{30}+\frac{1165}{64\!\cdots\!16}a^{29}-\frac{2929}{64\!\cdots\!16}a^{28}+\frac{7589}{64\!\cdots\!16}a^{27}-\frac{19305}{64\!\cdots\!16}a^{26}+\frac{49661}{64\!\cdots\!16}a^{25}-\frac{126881}{64\!\cdots\!16}a^{24}+\frac{325525}{64\!\cdots\!16}a^{23}-\frac{31934613}{1073741824}a^{22}+\frac{167688985}{1073741824}a^{21}-\frac{295427437}{1073741824}a^{20}-\frac{107558447}{1073741824}a^{19}-\frac{409477}{1073741824}a^{18}-\frac{429824311}{1073741824}a^{17}+\frac{428186403}{1073741824}a^{16}+\frac{1}{1073741824}a^{15}+\frac{235418369}{16\!\cdots\!04}a^{14}+\frac{91909085}{40\!\cdots\!76}a^{13}+\frac{35877321}{10\!\cdots\!44}a^{12}+\frac{14007941}{25\!\cdots\!36}a^{11}+\frac{5467345}{632348694544384}a^{10}+\frac{2135149}{158087173636096}a^{9}+\frac{833049}{39521793409024}a^{8}+\frac{325525}{9880448352256}a^{7}+\frac{126881}{2470112088064}a^{6}+\frac{49661}{617528022016}a^{5}+\frac{19305}{154382005504}a^{4}+\frac{7589}{38595501376}a^{3}+\frac{2929}{9648875344}a^{2}+\frac{1165}{2412218836}a+\frac{441}{603054709}$, $\frac{1}{25\!\cdots\!64}a^{38}-\frac{1}{25\!\cdots\!64}a^{37}+\frac{5}{25\!\cdots\!64}a^{36}-\frac{9}{25\!\cdots\!64}a^{35}+\frac{29}{25\!\cdots\!64}a^{34}-\frac{65}{25\!\cdots\!64}a^{33}+\frac{181}{25\!\cdots\!64}a^{32}-\frac{441}{25\!\cdots\!64}a^{31}+\frac{1165}{25\!\cdots\!64}a^{30}-\frac{2929}{25\!\cdots\!64}a^{29}+\frac{7589}{25\!\cdots\!64}a^{28}-\frac{19305}{25\!\cdots\!64}a^{27}+\frac{49661}{25\!\cdots\!64}a^{26}-\frac{126881}{25\!\cdots\!64}a^{25}+\frac{325525}{25\!\cdots\!64}a^{24}-\frac{833049}{25\!\cdots\!64}a^{23}+\frac{167688985}{4294967296}a^{22}-\frac{295427437}{4294967296}a^{21}+\frac{966183377}{4294967296}a^{20}+\frac{2147074171}{4294967296}a^{19}+\frac{1717659337}{4294967296}a^{18}-\frac{1719297245}{4294967296}a^{17}+\frac{1}{4294967296}a^{16}+\frac{235418369}{64\!\cdots\!16}a^{15}+\frac{91909085}{16\!\cdots\!04}a^{14}+\frac{35877321}{40\!\cdots\!76}a^{13}+\frac{14007941}{10\!\cdots\!44}a^{12}+\frac{5467345}{25\!\cdots\!36}a^{11}+\frac{2135149}{632348694544384}a^{10}+\frac{833049}{158087173636096}a^{9}+\frac{325525}{39521793409024}a^{8}+\frac{126881}{9880448352256}a^{7}+\frac{49661}{2470112088064}a^{6}+\frac{19305}{617528022016}a^{5}+\frac{7589}{154382005504}a^{4}+\frac{2929}{38595501376}a^{3}+\frac{1165}{9648875344}a^{2}+\frac{441}{2412218836}a+\frac{181}{603054709}$, $\frac{1}{10\!\cdots\!56}a^{39}-\frac{1}{10\!\cdots\!56}a^{38}+\frac{5}{10\!\cdots\!56}a^{37}-\frac{9}{10\!\cdots\!56}a^{36}+\frac{29}{10\!\cdots\!56}a^{35}-\frac{65}{10\!\cdots\!56}a^{34}+\frac{181}{10\!\cdots\!56}a^{33}-\frac{441}{10\!\cdots\!56}a^{32}+\frac{1165}{10\!\cdots\!56}a^{31}-\frac{2929}{10\!\cdots\!56}a^{30}+\frac{7589}{10\!\cdots\!56}a^{29}-\frac{19305}{10\!\cdots\!56}a^{28}+\frac{49661}{10\!\cdots\!56}a^{27}-\frac{126881}{10\!\cdots\!56}a^{26}+\frac{325525}{10\!\cdots\!56}a^{25}-\frac{833049}{10\!\cdots\!56}a^{24}+\frac{2135149}{10\!\cdots\!56}a^{23}+\frac{3999539859}{17179869184}a^{22}-\frac{3328783919}{17179869184}a^{21}+\frac{2147074171}{17179869184}a^{20}+\frac{1717659337}{17179869184}a^{19}+\frac{6870637347}{17179869184}a^{18}+\frac{1}{17179869184}a^{17}+\frac{235418369}{25\!\cdots\!64}a^{16}+\frac{91909085}{64\!\cdots\!16}a^{15}+\frac{35877321}{16\!\cdots\!04}a^{14}+\frac{14007941}{40\!\cdots\!76}a^{13}+\frac{5467345}{10\!\cdots\!44}a^{12}+\frac{2135149}{25\!\cdots\!36}a^{11}+\frac{833049}{632348694544384}a^{10}+\frac{325525}{158087173636096}a^{9}+\frac{126881}{39521793409024}a^{8}+\frac{49661}{9880448352256}a^{7}+\frac{19305}{2470112088064}a^{6}+\frac{7589}{617528022016}a^{5}+\frac{2929}{154382005504}a^{4}+\frac{1165}{38595501376}a^{3}+\frac{441}{9648875344}a^{2}+\frac{181}{2412218836}a+\frac{65}{603054709}$, $\frac{1}{41\!\cdots\!24}a^{40}-\frac{1}{41\!\cdots\!24}a^{39}+\frac{5}{41\!\cdots\!24}a^{38}-\frac{9}{41\!\cdots\!24}a^{37}+\frac{29}{41\!\cdots\!24}a^{36}-\frac{65}{41\!\cdots\!24}a^{35}+\frac{181}{41\!\cdots\!24}a^{34}-\frac{441}{41\!\cdots\!24}a^{33}+\frac{1165}{41\!\cdots\!24}a^{32}-\frac{2929}{41\!\cdots\!24}a^{31}+\frac{7589}{41\!\cdots\!24}a^{30}-\frac{19305}{41\!\cdots\!24}a^{29}+\frac{49661}{41\!\cdots\!24}a^{28}-\frac{126881}{41\!\cdots\!24}a^{27}+\frac{325525}{41\!\cdots\!24}a^{26}-\frac{833049}{41\!\cdots\!24}a^{25}+\frac{2135149}{41\!\cdots\!24}a^{24}-\frac{5467345}{41\!\cdots\!24}a^{23}+\frac{31030954449}{68719476736}a^{22}-\frac{15032795013}{68719476736}a^{21}+\frac{1717659337}{68719476736}a^{20}+\frac{6870637347}{68719476736}a^{19}+\frac{1}{68719476736}a^{18}+\frac{235418369}{10\!\cdots\!56}a^{17}+\frac{91909085}{25\!\cdots\!64}a^{16}+\frac{35877321}{64\!\cdots\!16}a^{15}+\frac{14007941}{16\!\cdots\!04}a^{14}+\frac{5467345}{40\!\cdots\!76}a^{13}+\frac{2135149}{10\!\cdots\!44}a^{12}+\frac{833049}{25\!\cdots\!36}a^{11}+\frac{325525}{632348694544384}a^{10}+\frac{126881}{158087173636096}a^{9}+\frac{49661}{39521793409024}a^{8}+\frac{19305}{9880448352256}a^{7}+\frac{7589}{2470112088064}a^{6}+\frac{2929}{617528022016}a^{5}+\frac{1165}{154382005504}a^{4}+\frac{441}{38595501376}a^{3}+\frac{181}{9648875344}a^{2}+\frac{65}{2412218836}a+\frac{29}{603054709}$, $\frac{1}{16\!\cdots\!96}a^{41}-\frac{1}{16\!\cdots\!96}a^{40}+\frac{5}{16\!\cdots\!96}a^{39}-\frac{9}{16\!\cdots\!96}a^{38}+\frac{29}{16\!\cdots\!96}a^{37}-\frac{65}{16\!\cdots\!96}a^{36}+\frac{181}{16\!\cdots\!96}a^{35}-\frac{441}{16\!\cdots\!96}a^{34}+\frac{1165}{16\!\cdots\!96}a^{33}-\frac{2929}{16\!\cdots\!96}a^{32}+\frac{7589}{16\!\cdots\!96}a^{31}-\frac{19305}{16\!\cdots\!96}a^{30}+\frac{49661}{16\!\cdots\!96}a^{29}-\frac{126881}{16\!\cdots\!96}a^{28}+\frac{325525}{16\!\cdots\!96}a^{27}-\frac{833049}{16\!\cdots\!96}a^{26}+\frac{2135149}{16\!\cdots\!96}a^{25}-\frac{5467345}{16\!\cdots\!96}a^{24}+\frac{14007941}{16\!\cdots\!96}a^{23}-\frac{83752271749}{274877906944}a^{22}-\frac{67001817399}{274877906944}a^{21}+\frac{6870637347}{274877906944}a^{20}+\frac{1}{274877906944}a^{19}+\frac{235418369}{41\!\cdots\!24}a^{18}+\frac{91909085}{10\!\cdots\!56}a^{17}+\frac{35877321}{25\!\cdots\!64}a^{16}+\frac{14007941}{64\!\cdots\!16}a^{15}+\frac{5467345}{16\!\cdots\!04}a^{14}+\frac{2135149}{40\!\cdots\!76}a^{13}+\frac{833049}{10\!\cdots\!44}a^{12}+\frac{325525}{25\!\cdots\!36}a^{11}+\frac{126881}{632348694544384}a^{10}+\frac{49661}{158087173636096}a^{9}+\frac{19305}{39521793409024}a^{8}+\frac{7589}{9880448352256}a^{7}+\frac{2929}{2470112088064}a^{6}+\frac{1165}{617528022016}a^{5}+\frac{441}{154382005504}a^{4}+\frac{181}{38595501376}a^{3}+\frac{65}{9648875344}a^{2}+\frac{29}{2412218836}a+\frac{9}{603054709}$, $\frac{1}{66\!\cdots\!84}a^{42}-\frac{1}{66\!\cdots\!84}a^{41}+\frac{5}{66\!\cdots\!84}a^{40}-\frac{9}{66\!\cdots\!84}a^{39}+\frac{29}{66\!\cdots\!84}a^{38}-\frac{65}{66\!\cdots\!84}a^{37}+\frac{181}{66\!\cdots\!84}a^{36}-\frac{441}{66\!\cdots\!84}a^{35}+\frac{1165}{66\!\cdots\!84}a^{34}-\frac{2929}{66\!\cdots\!84}a^{33}+\frac{7589}{66\!\cdots\!84}a^{32}-\frac{19305}{66\!\cdots\!84}a^{31}+\frac{49661}{66\!\cdots\!84}a^{30}-\frac{126881}{66\!\cdots\!84}a^{29}+\frac{325525}{66\!\cdots\!84}a^{28}-\frac{833049}{66\!\cdots\!84}a^{27}+\frac{2135149}{66\!\cdots\!84}a^{26}-\frac{5467345}{66\!\cdots\!84}a^{25}+\frac{14007941}{66\!\cdots\!84}a^{24}-\frac{35877321}{66\!\cdots\!84}a^{23}-\frac{67001817399}{1099511627776}a^{22}-\frac{268007269597}{1099511627776}a^{21}+\frac{1}{1099511627776}a^{20}+\frac{235418369}{16\!\cdots\!96}a^{19}+\frac{91909085}{41\!\cdots\!24}a^{18}+\frac{35877321}{10\!\cdots\!56}a^{17}+\frac{14007941}{25\!\cdots\!64}a^{16}+\frac{5467345}{64\!\cdots\!16}a^{15}+\frac{2135149}{16\!\cdots\!04}a^{14}+\frac{833049}{40\!\cdots\!76}a^{13}+\frac{325525}{10\!\cdots\!44}a^{12}+\frac{126881}{25\!\cdots\!36}a^{11}+\frac{49661}{632348694544384}a^{10}+\frac{19305}{158087173636096}a^{9}+\frac{7589}{39521793409024}a^{8}+\frac{2929}{9880448352256}a^{7}+\frac{1165}{2470112088064}a^{6}+\frac{441}{617528022016}a^{5}+\frac{181}{154382005504}a^{4}+\frac{65}{38595501376}a^{3}+\frac{29}{9648875344}a^{2}+\frac{9}{2412218836}a+\frac{5}{603054709}$, $\frac{1}{26\!\cdots\!36}a^{43}-\frac{1}{26\!\cdots\!36}a^{42}+\frac{5}{26\!\cdots\!36}a^{41}-\frac{9}{26\!\cdots\!36}a^{40}+\frac{29}{26\!\cdots\!36}a^{39}-\frac{65}{26\!\cdots\!36}a^{38}+\frac{181}{26\!\cdots\!36}a^{37}-\frac{441}{26\!\cdots\!36}a^{36}+\frac{1165}{26\!\cdots\!36}a^{35}-\frac{2929}{26\!\cdots\!36}a^{34}+\frac{7589}{26\!\cdots\!36}a^{33}-\frac{19305}{26\!\cdots\!36}a^{32}+\frac{49661}{26\!\cdots\!36}a^{31}-\frac{126881}{26\!\cdots\!36}a^{30}+\frac{325525}{26\!\cdots\!36}a^{29}-\frac{833049}{26\!\cdots\!36}a^{28}+\frac{2135149}{26\!\cdots\!36}a^{27}-\frac{5467345}{26\!\cdots\!36}a^{26}+\frac{14007941}{26\!\cdots\!36}a^{25}-\frac{35877321}{26\!\cdots\!36}a^{24}+\frac{91909085}{26\!\cdots\!36}a^{23}-\frac{268007269597}{4398046511104}a^{22}+\frac{1}{4398046511104}a^{21}+\frac{235418369}{66\!\cdots\!84}a^{20}+\frac{91909085}{16\!\cdots\!96}a^{19}+\frac{35877321}{41\!\cdots\!24}a^{18}+\frac{14007941}{10\!\cdots\!56}a^{17}+\frac{5467345}{25\!\cdots\!64}a^{16}+\frac{2135149}{64\!\cdots\!16}a^{15}+\frac{833049}{16\!\cdots\!04}a^{14}+\frac{325525}{40\!\cdots\!76}a^{13}+\frac{126881}{10\!\cdots\!44}a^{12}+\frac{49661}{25\!\cdots\!36}a^{11}+\frac{19305}{632348694544384}a^{10}+\frac{7589}{158087173636096}a^{9}+\frac{2929}{39521793409024}a^{8}+\frac{1165}{9880448352256}a^{7}+\frac{441}{2470112088064}a^{6}+\frac{181}{617528022016}a^{5}+\frac{65}{154382005504}a^{4}+\frac{29}{38595501376}a^{3}+\frac{9}{9648875344}a^{2}+\frac{5}{2412218836}a+\frac{1}{603054709}$ 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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:   \( -\frac{29}{617528022016} a^{28} - \frac{66507086889}{617528022016} a^{5} \)  (order $46$) 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 + 5*x^42 - 9*x^41 + 29*x^40 - 65*x^39 + 181*x^38 - 441*x^37 + 1165*x^36 - 2929*x^35 + 7589*x^34 - 19305*x^33 + 49661*x^32 - 126881*x^31 + 325525*x^30 - 833049*x^29 + 2135149*x^28 - 5467345*x^27 + 14007941*x^26 - 35877321*x^25 + 91909085*x^24 - 235418369*x^23 + 603054709*x^22 + 941673476*x^21 + 1470545360*x^20 + 2296148544*x^19 + 3586032896*x^18 + 5598561280*x^17 + 8745570304*x^16 + 13648674816*x^15 + 21333606400*x^14 + 33261092864*x^13 + 52073332736*x^12 + 80971038720*x^11 + 127322292224*x^10 + 196561862656*x^9 + 312727306240*x^8 + 473520144384*x^7 + 777389080576*x^6 + 1116691496960*x^5 + 1992864825344*x^4 + 2473901162496*x^3 + 5497558138880*x^2 + 4398046511104*x + 17592186044416)
 
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 + 5*x^42 - 9*x^41 + 29*x^40 - 65*x^39 + 181*x^38 - 441*x^37 + 1165*x^36 - 2929*x^35 + 7589*x^34 - 19305*x^33 + 49661*x^32 - 126881*x^31 + 325525*x^30 - 833049*x^29 + 2135149*x^28 - 5467345*x^27 + 14007941*x^26 - 35877321*x^25 + 91909085*x^24 - 235418369*x^23 + 603054709*x^22 + 941673476*x^21 + 1470545360*x^20 + 2296148544*x^19 + 3586032896*x^18 + 5598561280*x^17 + 8745570304*x^16 + 13648674816*x^15 + 21333606400*x^14 + 33261092864*x^13 + 52073332736*x^12 + 80971038720*x^11 + 127322292224*x^10 + 196561862656*x^9 + 312727306240*x^8 + 473520144384*x^7 + 777389080576*x^6 + 1116691496960*x^5 + 1992864825344*x^4 + 2473901162496*x^3 + 5497558138880*x^2 + 4398046511104*x + 17592186044416, 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 + 5*x^42 - 9*x^41 + 29*x^40 - 65*x^39 + 181*x^38 - 441*x^37 + 1165*x^36 - 2929*x^35 + 7589*x^34 - 19305*x^33 + 49661*x^32 - 126881*x^31 + 325525*x^30 - 833049*x^29 + 2135149*x^28 - 5467345*x^27 + 14007941*x^26 - 35877321*x^25 + 91909085*x^24 - 235418369*x^23 + 603054709*x^22 + 941673476*x^21 + 1470545360*x^20 + 2296148544*x^19 + 3586032896*x^18 + 5598561280*x^17 + 8745570304*x^16 + 13648674816*x^15 + 21333606400*x^14 + 33261092864*x^13 + 52073332736*x^12 + 80971038720*x^11 + 127322292224*x^10 + 196561862656*x^9 + 312727306240*x^8 + 473520144384*x^7 + 777389080576*x^6 + 1116691496960*x^5 + 1992864825344*x^4 + 2473901162496*x^3 + 5497558138880*x^2 + 4398046511104*x + 17592186044416);
 
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 + 5*x^42 - 9*x^41 + 29*x^40 - 65*x^39 + 181*x^38 - 441*x^37 + 1165*x^36 - 2929*x^35 + 7589*x^34 - 19305*x^33 + 49661*x^32 - 126881*x^31 + 325525*x^30 - 833049*x^29 + 2135149*x^28 - 5467345*x^27 + 14007941*x^26 - 35877321*x^25 + 91909085*x^24 - 235418369*x^23 + 603054709*x^22 + 941673476*x^21 + 1470545360*x^20 + 2296148544*x^19 + 3586032896*x^18 + 5598561280*x^17 + 8745570304*x^16 + 13648674816*x^15 + 21333606400*x^14 + 33261092864*x^13 + 52073332736*x^12 + 80971038720*x^11 + 127322292224*x^10 + 196561862656*x^9 + 312727306240*x^8 + 473520144384*x^7 + 777389080576*x^6 + 1116691496960*x^5 + 1992864825344*x^4 + 2473901162496*x^3 + 5497558138880*x^2 + 4398046511104*x + 17592186044416);
 
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

\(\Q(\sqrt{17}) \), \(\Q(\sqrt{-391}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{17}, \sqrt{-23})\), \(\Q(\zeta_{23})^+\), 22.22.58815914699238651208660872676277748369233.1, 22.0.1352766038082488977799200071554388212492359.1, \(\Q(\zeta_{23})\)

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 ${\href{/padicField/2.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ ${\href{/padicField/13.11.0.1}{11} }^{4}$ R $22^{2}$ R $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ ${\href{/padicField/47.1.0.1}{1} }^{44}$ $22^{2}$ ${\href{/padicField/59.11.0.1}{11} }^{4}$

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
\(17\) Copy content Toggle raw display Deg $44$$2$$22$$22$
\(23\) Copy content Toggle raw display Deg $44$$22$$2$$42$