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(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)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17592186044416, 4398046511104, 5497558138880, 2473901162496, 1992864825344, 1116691496960, 777389080576, 473520144384, 312727306240, 196561862656, 127322292224, 80971038720, 52073332736, 33261092864, 21333606400, 13648674816, 8745570304, 5598561280, 3586032896, 2296148544, 1470545360, 941673476, 603054709, -235418369, 91909085, -35877321, 14007941, -5467345, 2135149, -833049, 325525, -126881, 49661, -19305, 7589, -2929, 1165, -441, 181, -65, 29, -9, 5, -1, 1]);
 

\( x^{44} - x^{43} + 5 x^{42} - 9 x^{41} + 29 x^{40} - 65 x^{39} + 181 x^{38} - 441 x^{37} + 1165 x^{36} - 2929 x^{35} + 7589 x^{34} - 19305 x^{33} + 49661 x^{32} - 126881 x^{31} + 325525 x^{30} - 833049 x^{29} + 2135149 x^{28} - 5467345 x^{27} + 14007941 x^{26} - 35877321 x^{25} + 91909085 x^{24} - 235418369 x^{23} + 603054709 x^{22} + 941673476 x^{21} + 1470545360 x^{20} + 2296148544 x^{19} + 3586032896 x^{18} + 5598561280 x^{17} + 8745570304 x^{16} + 13648674816 x^{15} + 21333606400 x^{14} + 33261092864 x^{13} + 52073332736 x^{12} + 80971038720 x^{11} + 127322292224 x^{10} + 196561862656 x^{9} + 312727306240 x^{8} + 473520144384 x^{7} + 777389080576 x^{6} + 1116691496960 x^{5} + 1992864825344 x^{4} + 2473901162496 x^{3} + 5497558138880 x^{2} + 4398046511104 x + 17592186044416 \)

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

Invariants

Degree:  $44$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 22]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(182\!\cdots\!881\)\(\medspace = 17^{22}\cdot 23^{42}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $82.23$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $17, 23$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $44$
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.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{2412218836} a^{23} - \frac{1}{4} a^{22} + \frac{1}{4} a^{21} - \frac{1}{4} a^{20} + \frac{1}{4} a^{19} - \frac{1}{4} a^{18} + \frac{1}{4} a^{17} - \frac{1}{4} a^{16} + \frac{1}{4} a^{15} - \frac{1}{4} a^{14} + \frac{1}{4} a^{13} - \frac{1}{4} a^{12} + \frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{235418369}{603054709}$, $\frac{1}{9648875344} a^{24} - \frac{1}{9648875344} a^{23} + \frac{1}{16} a^{22} - \frac{5}{16} a^{21} - \frac{7}{16} a^{20} + \frac{3}{16} a^{19} + \frac{1}{16} a^{18} - \frac{5}{16} a^{17} - \frac{7}{16} a^{16} + \frac{3}{16} a^{15} + \frac{1}{16} a^{14} - \frac{5}{16} a^{13} - \frac{7}{16} a^{12} + \frac{3}{16} a^{11} + \frac{1}{16} a^{10} - \frac{5}{16} a^{9} - \frac{7}{16} a^{8} + \frac{3}{16} a^{7} + \frac{1}{16} a^{6} - \frac{5}{16} a^{5} - \frac{7}{16} a^{4} + \frac{3}{16} a^{3} + \frac{1}{16} a^{2} + \frac{235418369}{2412218836} a + \frac{91909085}{603054709}$, $\frac{1}{38595501376} a^{25} - \frac{1}{38595501376} a^{24} + \frac{5}{38595501376} a^{23} - \frac{5}{64} a^{22} + \frac{9}{64} a^{21} - \frac{29}{64} a^{20} + \frac{1}{64} a^{19} + \frac{11}{64} a^{18} - \frac{7}{64} a^{17} - \frac{13}{64} a^{16} - \frac{15}{64} a^{15} + \frac{27}{64} a^{14} - \frac{23}{64} a^{13} + \frac{3}{64} a^{12} - \frac{31}{64} a^{11} - \frac{21}{64} a^{10} + \frac{25}{64} a^{9} + \frac{19}{64} a^{8} + \frac{17}{64} a^{7} - \frac{5}{64} a^{6} + \frac{9}{64} a^{5} - \frac{29}{64} a^{4} + \frac{1}{64} a^{3} + \frac{235418369}{9648875344} a^{2} + \frac{91909085}{2412218836} a + \frac{35877321}{603054709}$, $\frac{1}{154382005504} a^{26} - \frac{1}{154382005504} a^{25} + \frac{5}{154382005504} a^{24} - \frac{9}{154382005504} a^{23} + \frac{9}{256} a^{22} - \frac{29}{256} a^{21} + \frac{65}{256} a^{20} + \frac{75}{256} a^{19} - \frac{71}{256} a^{18} + \frac{115}{256} a^{17} + \frac{113}{256} a^{16} + \frac{91}{256} a^{15} + \frac{105}{256} a^{14} + \frac{3}{256} a^{13} - \frac{95}{256} a^{12} + \frac{107}{256} a^{11} + \frac{25}{256} a^{10} - \frac{109}{256} a^{9} - \frac{47}{256} a^{8} + \frac{123}{256} a^{7} - \frac{55}{256} a^{6} + \frac{35}{256} a^{5} + \frac{1}{256} a^{4} + \frac{235418369}{38595501376} a^{3} + \frac{91909085}{9648875344} a^{2} + \frac{35877321}{2412218836} a + \frac{14007941}{603054709}$, $\frac{1}{617528022016} a^{27} - \frac{1}{617528022016} a^{26} + \frac{5}{617528022016} a^{25} - \frac{9}{617528022016} a^{24} + \frac{29}{617528022016} a^{23} - \frac{285}{1024} a^{22} + \frac{321}{1024} a^{21} - \frac{437}{1024} a^{20} - \frac{327}{1024} a^{19} - \frac{397}{1024} a^{18} + \frac{113}{1024} a^{17} + \frac{347}{1024} a^{16} + \frac{105}{1024} a^{15} + \frac{259}{1024} a^{14} + \frac{161}{1024} a^{13} - \frac{149}{1024} a^{12} - \frac{231}{1024} a^{11} - \frac{365}{1024} a^{10} + \frac{465}{1024} a^{9} + \frac{123}{1024} a^{8} - \frac{311}{1024} a^{7} - \frac{221}{1024} a^{6} + \frac{1}{1024} a^{5} + \frac{235418369}{154382005504} a^{4} + \frac{91909085}{38595501376} a^{3} + \frac{35877321}{9648875344} a^{2} + \frac{14007941}{2412218836} a + \frac{5467345}{603054709}$, $\frac{1}{2470112088064} a^{28} - \frac{1}{2470112088064} a^{27} + \frac{5}{2470112088064} a^{26} - \frac{9}{2470112088064} a^{25} + \frac{29}{2470112088064} a^{24} - \frac{65}{2470112088064} a^{23} - \frac{703}{4096} a^{22} - \frac{437}{4096} a^{21} + \frac{1721}{4096} a^{20} + \frac{627}{4096} a^{19} - \frac{1935}{4096} a^{18} + \frac{347}{4096} a^{17} + \frac{105}{4096} a^{16} + \frac{1283}{4096} a^{15} - \frac{863}{4096} a^{14} + \frac{1899}{4096} a^{13} - \frac{1255}{4096} a^{12} + \frac{659}{4096} a^{11} - \frac{1583}{4096} a^{10} + \frac{123}{4096} a^{9} + \frac{1737}{4096} a^{8} - \frac{1245}{4096} a^{7} + \frac{1}{4096} a^{6} + \frac{235418369}{617528022016} a^{5} + \frac{91909085}{154382005504} a^{4} + \frac{35877321}{38595501376} a^{3} + \frac{14007941}{9648875344} a^{2} + \frac{5467345}{2412218836} a + \frac{2135149}{603054709}$, $\frac{1}{9880448352256} a^{29} - \frac{1}{9880448352256} a^{28} + \frac{5}{9880448352256} a^{27} - \frac{9}{9880448352256} a^{26} + \frac{29}{9880448352256} a^{25} - \frac{65}{9880448352256} a^{24} + \frac{181}{9880448352256} a^{23} + \frac{3659}{16384} a^{22} - \frac{6471}{16384} a^{21} + \frac{4723}{16384} a^{20} + \frac{2161}{16384} a^{19} + \frac{347}{16384} a^{18} - \frac{8087}{16384} a^{17} - \frac{6909}{16384} a^{16} + \frac{7329}{16384} a^{15} - \frac{2197}{16384} a^{14} - \frac{1255}{16384} a^{13} - \frac{7533}{16384} a^{12} + \frac{2513}{16384} a^{11} + \frac{123}{16384} a^{10} - \frac{6455}{16384} a^{9} + \frac{6947}{16384} a^{8} + \frac{1}{16384} a^{7} + \frac{235418369}{2470112088064} a^{6} + \frac{91909085}{617528022016} a^{5} + \frac{35877321}{154382005504} a^{4} + \frac{14007941}{38595501376} a^{3} + \frac{5467345}{9648875344} a^{2} + \frac{2135149}{2412218836} a + \frac{833049}{603054709}$, $\frac{1}{39521793409024} a^{30} - \frac{1}{39521793409024} a^{29} + \frac{5}{39521793409024} a^{28} - \frac{9}{39521793409024} a^{27} + \frac{29}{39521793409024} a^{26} - \frac{65}{39521793409024} a^{25} + \frac{181}{39521793409024} a^{24} - \frac{441}{39521793409024} a^{23} + \frac{9913}{65536} a^{22} + \frac{4723}{65536} a^{21} - \frac{30607}{65536} a^{20} - \frac{16037}{65536} a^{19} + \frac{24681}{65536} a^{18} - \frac{23293}{65536} a^{17} - \frac{9055}{65536} a^{16} - \frac{18581}{65536} a^{15} - \frac{17639}{65536} a^{14} + \frac{8851}{65536} a^{13} - \frac{13871}{65536} a^{12} - \frac{16261}{65536} a^{11} + \frac{26313}{65536} a^{10} - \frac{25821}{65536} a^{9} + \frac{1}{65536} a^{8} + \frac{235418369}{9880448352256} a^{7} + \frac{91909085}{2470112088064} a^{6} + \frac{35877321}{617528022016} a^{5} + \frac{14007941}{154382005504} a^{4} + \frac{5467345}{38595501376} a^{3} + \frac{2135149}{9648875344} a^{2} + \frac{833049}{2412218836} a + \frac{325525}{603054709}$, $\frac{1}{158087173636096} a^{31} - \frac{1}{158087173636096} a^{30} + \frac{5}{158087173636096} a^{29} - \frac{9}{158087173636096} a^{28} + \frac{29}{158087173636096} a^{27} - \frac{65}{158087173636096} a^{26} + \frac{181}{158087173636096} a^{25} - \frac{441}{158087173636096} a^{24} + \frac{1165}{158087173636096} a^{23} - \frac{60813}{262144} a^{22} + \frac{100465}{262144} a^{21} - \frac{81573}{262144} a^{20} - \frac{40855}{262144} a^{19} - \frac{23293}{262144} a^{18} + \frac{122017}{262144} a^{17} + \frac{46955}{262144} a^{16} - \frac{83175}{262144} a^{15} + \frac{8851}{262144} a^{14} - \frac{79407}{262144} a^{13} + \frac{114811}{262144} a^{12} + \frac{91849}{262144} a^{11} + \frac{105251}{262144} a^{10} + \frac{1}{262144} a^{9} + \frac{235418369}{39521793409024} a^{8} + \frac{91909085}{9880448352256} a^{7} + \frac{35877321}{2470112088064} a^{6} + \frac{14007941}{617528022016} a^{5} + \frac{5467345}{154382005504} a^{4} + \frac{2135149}{38595501376} a^{3} + \frac{833049}{9648875344} a^{2} + \frac{325525}{2412218836} a + \frac{126881}{603054709}$, $\frac{1}{632348694544384} a^{32} - \frac{1}{632348694544384} a^{31} + \frac{5}{632348694544384} a^{30} - \frac{9}{632348694544384} a^{29} + \frac{29}{632348694544384} a^{28} - \frac{65}{632348694544384} a^{27} + \frac{181}{632348694544384} a^{26} - \frac{441}{632348694544384} a^{25} + \frac{1165}{632348694544384} a^{24} - \frac{2929}{632348694544384} a^{23} - \frac{423823}{1048576} a^{22} + \frac{180571}{1048576} a^{21} + \frac{221289}{1048576} a^{20} + \frac{500995}{1048576} a^{19} + \frac{384161}{1048576} a^{18} - \frac{477333}{1048576} a^{17} - \frac{83175}{1048576} a^{16} + \frac{270995}{1048576} a^{15} + \frac{444881}{1048576} a^{14} - \frac{409477}{1048576} a^{13} + \frac{91849}{1048576} a^{12} + \frac{367395}{1048576} a^{11} + \frac{1}{1048576} a^{10} + \frac{235418369}{158087173636096} a^{9} + \frac{91909085}{39521793409024} a^{8} + \frac{35877321}{9880448352256} a^{7} + \frac{14007941}{2470112088064} a^{6} + \frac{5467345}{617528022016} a^{5} + \frac{2135149}{154382005504} a^{4} + \frac{833049}{38595501376} a^{3} + \frac{325525}{9648875344} a^{2} + \frac{126881}{2412218836} a + \frac{49661}{603054709}$, $\frac{1}{2529394778177536} a^{33} - \frac{1}{2529394778177536} a^{32} + \frac{5}{2529394778177536} a^{31} - \frac{9}{2529394778177536} a^{30} + \frac{29}{2529394778177536} a^{29} - \frac{65}{2529394778177536} a^{28} + \frac{181}{2529394778177536} a^{27} - \frac{441}{2529394778177536} a^{26} + \frac{1165}{2529394778177536} a^{25} - \frac{2929}{2529394778177536} a^{24} + \frac{7589}{2529394778177536} a^{23} - \frac{1916581}{4194304} a^{22} + \frac{221289}{4194304} a^{21} + \frac{500995}{4194304} a^{20} + \frac{384161}{4194304} a^{19} + \frac{1619819}{4194304} a^{18} - \frac{83175}{4194304} a^{17} - \frac{1826157}{4194304} a^{16} + \frac{1493457}{4194304} a^{15} - \frac{409477}{4194304} a^{14} - \frac{2005303}{4194304} a^{13} + \frac{367395}{4194304} a^{12} + \frac{1}{4194304} a^{11} + \frac{235418369}{632348694544384} a^{10} + \frac{91909085}{158087173636096} a^{9} + \frac{35877321}{39521793409024} a^{8} + \frac{14007941}{9880448352256} a^{7} + \frac{5467345}{2470112088064} a^{6} + \frac{2135149}{617528022016} a^{5} + \frac{833049}{154382005504} a^{4} + \frac{325525}{38595501376} a^{3} + \frac{126881}{9648875344} a^{2} + \frac{49661}{2412218836} a + \frac{19305}{603054709}$, $\frac{1}{10117579112710144} a^{34} - \frac{1}{10117579112710144} a^{33} + \frac{5}{10117579112710144} a^{32} - \frac{9}{10117579112710144} a^{31} + \frac{29}{10117579112710144} a^{30} - \frac{65}{10117579112710144} a^{29} + \frac{181}{10117579112710144} a^{28} - \frac{441}{10117579112710144} a^{27} + \frac{1165}{10117579112710144} a^{26} - \frac{2929}{10117579112710144} a^{25} + \frac{7589}{10117579112710144} a^{24} - \frac{19305}{10117579112710144} a^{23} - \frac{8167319}{16777216} a^{22} + \frac{500995}{16777216} a^{21} + \frac{384161}{16777216} a^{20} + \frac{1619819}{16777216} a^{19} - \frac{83175}{16777216} a^{18} + \frac{6562451}{16777216} a^{17} - \frac{6895151}{16777216} a^{16} - \frac{409477}{16777216} a^{15} + \frac{6383305}{16777216} a^{14} - \frac{8021213}{16777216} a^{13} + \frac{1}{16777216} a^{12} + \frac{235418369}{2529394778177536} a^{11} + \frac{91909085}{632348694544384} a^{10} + \frac{35877321}{158087173636096} a^{9} + \frac{14007941}{39521793409024} a^{8} + \frac{5467345}{9880448352256} a^{7} + \frac{2135149}{2470112088064} a^{6} + \frac{833049}{617528022016} a^{5} + \frac{325525}{154382005504} a^{4} + \frac{126881}{38595501376} a^{3} + \frac{49661}{9648875344} a^{2} + \frac{19305}{2412218836} a + \frac{7589}{603054709}$, $\frac{1}{40470316450840576} a^{35} - \frac{1}{40470316450840576} a^{34} + \frac{5}{40470316450840576} a^{33} - \frac{9}{40470316450840576} a^{32} + \frac{29}{40470316450840576} a^{31} - \frac{65}{40470316450840576} a^{30} + \frac{181}{40470316450840576} a^{29} - \frac{441}{40470316450840576} a^{28} + \frac{1165}{40470316450840576} a^{27} - \frac{2929}{40470316450840576} a^{26} + \frac{7589}{40470316450840576} a^{25} - \frac{19305}{40470316450840576} a^{24} + \frac{49661}{40470316450840576} a^{23} + \frac{500995}{67108864} a^{22} - \frac{33170271}{67108864} a^{21} - \frac{31934613}{67108864} a^{20} + \frac{33471257}{67108864} a^{19} - \frac{26991981}{67108864} a^{18} + \frac{26659281}{67108864} a^{17} - \frac{409477}{67108864} a^{16} - \frac{27171127}{67108864} a^{15} + \frac{25533219}{67108864} a^{14} + \frac{1}{67108864} a^{13} + \frac{235418369}{10117579112710144} a^{12} + \frac{91909085}{2529394778177536} a^{11} + \frac{35877321}{632348694544384} a^{10} + \frac{14007941}{158087173636096} a^{9} + \frac{5467345}{39521793409024} a^{8} + \frac{2135149}{9880448352256} a^{7} + \frac{833049}{2470112088064} a^{6} + \frac{325525}{617528022016} a^{5} + \frac{126881}{154382005504} a^{4} + \frac{49661}{38595501376} a^{3} + \frac{19305}{9648875344} a^{2} + \frac{7589}{2412218836} a + \frac{2929}{603054709}$, $\frac{1}{161881265803362304} a^{36} - \frac{1}{161881265803362304} a^{35} + \frac{5}{161881265803362304} a^{34} - \frac{9}{161881265803362304} a^{33} + \frac{29}{161881265803362304} a^{32} - \frac{65}{161881265803362304} a^{31} + \frac{181}{161881265803362304} a^{30} - \frac{441}{161881265803362304} a^{29} + \frac{1165}{161881265803362304} a^{28} - \frac{2929}{161881265803362304} a^{27} + \frac{7589}{161881265803362304} a^{26} - \frac{19305}{161881265803362304} a^{25} + \frac{49661}{161881265803362304} a^{24} - \frac{126881}{161881265803362304} a^{23} + \frac{33938593}{268435456} a^{22} - \frac{31934613}{268435456} a^{21} - \frac{100746471}{268435456} a^{20} - \frac{26991981}{268435456} a^{19} - \frac{107558447}{268435456} a^{18} - \frac{409477}{268435456} a^{17} + \frac{107046601}{268435456} a^{16} - \frac{108684509}{268435456} a^{15} + \frac{1}{268435456} a^{14} + \frac{235418369}{40470316450840576} a^{13} + \frac{91909085}{10117579112710144} a^{12} + \frac{35877321}{2529394778177536} a^{11} + \frac{14007941}{632348694544384} a^{10} + \frac{5467345}{158087173636096} a^{9} + \frac{2135149}{39521793409024} a^{8} + \frac{833049}{9880448352256} a^{7} + \frac{325525}{2470112088064} a^{6} + \frac{126881}{617528022016} a^{5} + \frac{49661}{154382005504} a^{4} + \frac{19305}{38595501376} a^{3} + \frac{7589}{9648875344} a^{2} + \frac{2929}{2412218836} a + \frac{1165}{603054709}$, $\frac{1}{647525063213449216} a^{37} - \frac{1}{647525063213449216} a^{36} + \frac{5}{647525063213449216} a^{35} - \frac{9}{647525063213449216} a^{34} + \frac{29}{647525063213449216} a^{33} - \frac{65}{647525063213449216} a^{32} + \frac{181}{647525063213449216} a^{31} - \frac{441}{647525063213449216} a^{30} + \frac{1165}{647525063213449216} a^{29} - \frac{2929}{647525063213449216} a^{28} + \frac{7589}{647525063213449216} a^{27} - \frac{19305}{647525063213449216} a^{26} + \frac{49661}{647525063213449216} a^{25} - \frac{126881}{647525063213449216} a^{24} + \frac{325525}{647525063213449216} a^{23} - \frac{31934613}{1073741824} a^{22} + \frac{167688985}{1073741824} a^{21} - \frac{295427437}{1073741824} a^{20} - \frac{107558447}{1073741824} a^{19} - \frac{409477}{1073741824} a^{18} - \frac{429824311}{1073741824} a^{17} + \frac{428186403}{1073741824} a^{16} + \frac{1}{1073741824} a^{15} + \frac{235418369}{161881265803362304} a^{14} + \frac{91909085}{40470316450840576} a^{13} + \frac{35877321}{10117579112710144} a^{12} + \frac{14007941}{2529394778177536} a^{11} + \frac{5467345}{632348694544384} a^{10} + \frac{2135149}{158087173636096} a^{9} + \frac{833049}{39521793409024} a^{8} + \frac{325525}{9880448352256} a^{7} + \frac{126881}{2470112088064} a^{6} + \frac{49661}{617528022016} a^{5} + \frac{19305}{154382005504} a^{4} + \frac{7589}{38595501376} a^{3} + \frac{2929}{9648875344} a^{2} + \frac{1165}{2412218836} a + \frac{441}{603054709}$, $\frac{1}{2590100252853796864} a^{38} - \frac{1}{2590100252853796864} a^{37} + \frac{5}{2590100252853796864} a^{36} - \frac{9}{2590100252853796864} a^{35} + \frac{29}{2590100252853796864} a^{34} - \frac{65}{2590100252853796864} a^{33} + \frac{181}{2590100252853796864} a^{32} - \frac{441}{2590100252853796864} a^{31} + \frac{1165}{2590100252853796864} a^{30} - \frac{2929}{2590100252853796864} a^{29} + \frac{7589}{2590100252853796864} a^{28} - \frac{19305}{2590100252853796864} a^{27} + \frac{49661}{2590100252853796864} a^{26} - \frac{126881}{2590100252853796864} a^{25} + \frac{325525}{2590100252853796864} a^{24} - \frac{833049}{2590100252853796864} a^{23} + \frac{167688985}{4294967296} a^{22} - \frac{295427437}{4294967296} a^{21} + \frac{966183377}{4294967296} a^{20} + \frac{2147074171}{4294967296} a^{19} + \frac{1717659337}{4294967296} a^{18} - \frac{1719297245}{4294967296} a^{17} + \frac{1}{4294967296} a^{16} + \frac{235418369}{647525063213449216} a^{15} + \frac{91909085}{161881265803362304} a^{14} + \frac{35877321}{40470316450840576} a^{13} + \frac{14007941}{10117579112710144} a^{12} + \frac{5467345}{2529394778177536} a^{11} + \frac{2135149}{632348694544384} a^{10} + \frac{833049}{158087173636096} a^{9} + \frac{325525}{39521793409024} a^{8} + \frac{126881}{9880448352256} a^{7} + \frac{49661}{2470112088064} a^{6} + \frac{19305}{617528022016} a^{5} + \frac{7589}{154382005504} a^{4} + \frac{2929}{38595501376} a^{3} + \frac{1165}{9648875344} a^{2} + \frac{441}{2412218836} a + \frac{181}{603054709}$, $\frac{1}{10360401011415187456} a^{39} - \frac{1}{10360401011415187456} a^{38} + \frac{5}{10360401011415187456} a^{37} - \frac{9}{10360401011415187456} a^{36} + \frac{29}{10360401011415187456} a^{35} - \frac{65}{10360401011415187456} a^{34} + \frac{181}{10360401011415187456} a^{33} - \frac{441}{10360401011415187456} a^{32} + \frac{1165}{10360401011415187456} a^{31} - \frac{2929}{10360401011415187456} a^{30} + \frac{7589}{10360401011415187456} a^{29} - \frac{19305}{10360401011415187456} a^{28} + \frac{49661}{10360401011415187456} a^{27} - \frac{126881}{10360401011415187456} a^{26} + \frac{325525}{10360401011415187456} a^{25} - \frac{833049}{10360401011415187456} a^{24} + \frac{2135149}{10360401011415187456} a^{23} + \frac{3999539859}{17179869184} a^{22} - \frac{3328783919}{17179869184} a^{21} + \frac{2147074171}{17179869184} a^{20} + \frac{1717659337}{17179869184} a^{19} + \frac{6870637347}{17179869184} a^{18} + \frac{1}{17179869184} a^{17} + \frac{235418369}{2590100252853796864} a^{16} + \frac{91909085}{647525063213449216} a^{15} + \frac{35877321}{161881265803362304} a^{14} + \frac{14007941}{40470316450840576} a^{13} + \frac{5467345}{10117579112710144} a^{12} + \frac{2135149}{2529394778177536} a^{11} + \frac{833049}{632348694544384} a^{10} + \frac{325525}{158087173636096} a^{9} + \frac{126881}{39521793409024} a^{8} + \frac{49661}{9880448352256} a^{7} + \frac{19305}{2470112088064} a^{6} + \frac{7589}{617528022016} a^{5} + \frac{2929}{154382005504} a^{4} + \frac{1165}{38595501376} a^{3} + \frac{441}{9648875344} a^{2} + \frac{181}{2412218836} a + \frac{65}{603054709}$, $\frac{1}{41441604045660749824} a^{40} - \frac{1}{41441604045660749824} a^{39} + \frac{5}{41441604045660749824} a^{38} - \frac{9}{41441604045660749824} a^{37} + \frac{29}{41441604045660749824} a^{36} - \frac{65}{41441604045660749824} a^{35} + \frac{181}{41441604045660749824} a^{34} - \frac{441}{41441604045660749824} a^{33} + \frac{1165}{41441604045660749824} a^{32} - \frac{2929}{41441604045660749824} a^{31} + \frac{7589}{41441604045660749824} a^{30} - \frac{19305}{41441604045660749824} a^{29} + \frac{49661}{41441604045660749824} a^{28} - \frac{126881}{41441604045660749824} a^{27} + \frac{325525}{41441604045660749824} a^{26} - \frac{833049}{41441604045660749824} a^{25} + \frac{2135149}{41441604045660749824} a^{24} - \frac{5467345}{41441604045660749824} a^{23} + \frac{31030954449}{68719476736} a^{22} - \frac{15032795013}{68719476736} a^{21} + \frac{1717659337}{68719476736} a^{20} + \frac{6870637347}{68719476736} a^{19} + \frac{1}{68719476736} a^{18} + \frac{235418369}{10360401011415187456} a^{17} + \frac{91909085}{2590100252853796864} a^{16} + \frac{35877321}{647525063213449216} a^{15} + \frac{14007941}{161881265803362304} a^{14} + \frac{5467345}{40470316450840576} a^{13} + \frac{2135149}{10117579112710144} a^{12} + \frac{833049}{2529394778177536} a^{11} + \frac{325525}{632348694544384} a^{10} + \frac{126881}{158087173636096} a^{9} + \frac{49661}{39521793409024} a^{8} + \frac{19305}{9880448352256} a^{7} + \frac{7589}{2470112088064} a^{6} + \frac{2929}{617528022016} a^{5} + \frac{1165}{154382005504} a^{4} + \frac{441}{38595501376} a^{3} + \frac{181}{9648875344} a^{2} + \frac{65}{2412218836} a + \frac{29}{603054709}$, $\frac{1}{165766416182642999296} a^{41} - \frac{1}{165766416182642999296} a^{40} + \frac{5}{165766416182642999296} a^{39} - \frac{9}{165766416182642999296} a^{38} + \frac{29}{165766416182642999296} a^{37} - \frac{65}{165766416182642999296} a^{36} + \frac{181}{165766416182642999296} a^{35} - \frac{441}{165766416182642999296} a^{34} + \frac{1165}{165766416182642999296} a^{33} - \frac{2929}{165766416182642999296} a^{32} + \frac{7589}{165766416182642999296} a^{31} - \frac{19305}{165766416182642999296} a^{30} + \frac{49661}{165766416182642999296} a^{29} - \frac{126881}{165766416182642999296} a^{28} + \frac{325525}{165766416182642999296} a^{27} - \frac{833049}{165766416182642999296} a^{26} + \frac{2135149}{165766416182642999296} a^{25} - \frac{5467345}{165766416182642999296} a^{24} + \frac{14007941}{165766416182642999296} a^{23} - \frac{83752271749}{274877906944} a^{22} - \frac{67001817399}{274877906944} a^{21} + \frac{6870637347}{274877906944} a^{20} + \frac{1}{274877906944} a^{19} + \frac{235418369}{41441604045660749824} a^{18} + \frac{91909085}{10360401011415187456} a^{17} + \frac{35877321}{2590100252853796864} a^{16} + \frac{14007941}{647525063213449216} a^{15} + \frac{5467345}{161881265803362304} a^{14} + \frac{2135149}{40470316450840576} a^{13} + \frac{833049}{10117579112710144} a^{12} + \frac{325525}{2529394778177536} a^{11} + \frac{126881}{632348694544384} a^{10} + \frac{49661}{158087173636096} a^{9} + \frac{19305}{39521793409024} a^{8} + \frac{7589}{9880448352256} a^{7} + \frac{2929}{2470112088064} a^{6} + \frac{1165}{617528022016} a^{5} + \frac{441}{154382005504} a^{4} + \frac{181}{38595501376} a^{3} + \frac{65}{9648875344} a^{2} + \frac{29}{2412218836} a + \frac{9}{603054709}$, $\frac{1}{663065664730571997184} a^{42} - \frac{1}{663065664730571997184} a^{41} + \frac{5}{663065664730571997184} a^{40} - \frac{9}{663065664730571997184} a^{39} + \frac{29}{663065664730571997184} a^{38} - \frac{65}{663065664730571997184} a^{37} + \frac{181}{663065664730571997184} a^{36} - \frac{441}{663065664730571997184} a^{35} + \frac{1165}{663065664730571997184} a^{34} - \frac{2929}{663065664730571997184} a^{33} + \frac{7589}{663065664730571997184} a^{32} - \frac{19305}{663065664730571997184} a^{31} + \frac{49661}{663065664730571997184} a^{30} - \frac{126881}{663065664730571997184} a^{29} + \frac{325525}{663065664730571997184} a^{28} - \frac{833049}{663065664730571997184} a^{27} + \frac{2135149}{663065664730571997184} a^{26} - \frac{5467345}{663065664730571997184} a^{25} + \frac{14007941}{663065664730571997184} a^{24} - \frac{35877321}{663065664730571997184} a^{23} - \frac{67001817399}{1099511627776} a^{22} - \frac{268007269597}{1099511627776} a^{21} + \frac{1}{1099511627776} a^{20} + \frac{235418369}{165766416182642999296} a^{19} + \frac{91909085}{41441604045660749824} a^{18} + \frac{35877321}{10360401011415187456} a^{17} + \frac{14007941}{2590100252853796864} a^{16} + \frac{5467345}{647525063213449216} a^{15} + \frac{2135149}{161881265803362304} a^{14} + \frac{833049}{40470316450840576} a^{13} + \frac{325525}{10117579112710144} a^{12} + \frac{126881}{2529394778177536} a^{11} + \frac{49661}{632348694544384} a^{10} + \frac{19305}{158087173636096} a^{9} + \frac{7589}{39521793409024} a^{8} + \frac{2929}{9880448352256} a^{7} + \frac{1165}{2470112088064} a^{6} + \frac{441}{617528022016} a^{5} + \frac{181}{154382005504} a^{4} + \frac{65}{38595501376} a^{3} + \frac{29}{9648875344} a^{2} + \frac{9}{2412218836} a + \frac{5}{603054709}$, $\frac{1}{2652262658922287988736} a^{43} - \frac{1}{2652262658922287988736} a^{42} + \frac{5}{2652262658922287988736} a^{41} - \frac{9}{2652262658922287988736} a^{40} + \frac{29}{2652262658922287988736} a^{39} - \frac{65}{2652262658922287988736} a^{38} + \frac{181}{2652262658922287988736} a^{37} - \frac{441}{2652262658922287988736} a^{36} + \frac{1165}{2652262658922287988736} a^{35} - \frac{2929}{2652262658922287988736} a^{34} + \frac{7589}{2652262658922287988736} a^{33} - \frac{19305}{2652262658922287988736} a^{32} + \frac{49661}{2652262658922287988736} a^{31} - \frac{126881}{2652262658922287988736} a^{30} + \frac{325525}{2652262658922287988736} a^{29} - \frac{833049}{2652262658922287988736} a^{28} + \frac{2135149}{2652262658922287988736} a^{27} - \frac{5467345}{2652262658922287988736} a^{26} + \frac{14007941}{2652262658922287988736} a^{25} - \frac{35877321}{2652262658922287988736} a^{24} + \frac{91909085}{2652262658922287988736} a^{23} - \frac{268007269597}{4398046511104} a^{22} + \frac{1}{4398046511104} a^{21} + \frac{235418369}{663065664730571997184} a^{20} + \frac{91909085}{165766416182642999296} a^{19} + \frac{35877321}{41441604045660749824} a^{18} + \frac{14007941}{10360401011415187456} a^{17} + \frac{5467345}{2590100252853796864} a^{16} + \frac{2135149}{647525063213449216} a^{15} + \frac{833049}{161881265803362304} a^{14} + \frac{325525}{40470316450840576} a^{13} + \frac{126881}{10117579112710144} a^{12} + \frac{49661}{2529394778177536} a^{11} + \frac{19305}{632348694544384} a^{10} + \frac{7589}{158087173636096} a^{9} + \frac{2929}{39521793409024} a^{8} + \frac{1165}{9880448352256} a^{7} + \frac{441}{2470112088064} a^{6} + \frac{181}{617528022016} a^{5} + \frac{65}{154382005504} a^{4} + \frac{29}{38595501376} a^{3} + \frac{9}{9648875344} a^{2} + \frac{5}{2412218836} a + \frac{1}{603054709}$

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

Class group and class number

not computed

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $21$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{29}{617528022016} a^{28} - \frac{66507086889}{617528022016} a^{5} \) (order $46$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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

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

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{4}$ R $22^{2}$ R $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{44}$ $22^{2}$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{4}$

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

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

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
17Data not computed
23Data not computed