Properties

Label 44.0.202...984.1
Degree $44$
Signature $[0, 22]$
Discriminant $2.026\times 10^{86}$
Root discriminant $91.52$
Ramified primes $2, 7, 23$
Class number not computed
Class group not computed
Galois group $C_2\times C_{22}$ (as 44T2)

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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 - 63*x^42 + 1954*x^40 - 39159*x^38 + 564136*x^36 - 6171204*x^34 + 52905976*x^32 - 362220741*x^30 + 2002105655*x^28 - 8980281297*x^26 + 32710198942*x^24 - 96439370415*x^22 + 228524812948*x^20 - 430395212448*x^18 + 634098522688*x^16 - 714985491456*x^14 + 598979244032*x^12 - 357650202624*x^10 + 143735308288*x^8 - 35340484608*x^6 + 4755030016*x^4 - 207618048*x^2 + 4194304)
 
gp: K = bnfinit(x^44 - 63*x^42 + 1954*x^40 - 39159*x^38 + 564136*x^36 - 6171204*x^34 + 52905976*x^32 - 362220741*x^30 + 2002105655*x^28 - 8980281297*x^26 + 32710198942*x^24 - 96439370415*x^22 + 228524812948*x^20 - 430395212448*x^18 + 634098522688*x^16 - 714985491456*x^14 + 598979244032*x^12 - 357650202624*x^10 + 143735308288*x^8 - 35340484608*x^6 + 4755030016*x^4 - 207618048*x^2 + 4194304, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4194304, 0, -207618048, 0, 4755030016, 0, -35340484608, 0, 143735308288, 0, -357650202624, 0, 598979244032, 0, -714985491456, 0, 634098522688, 0, -430395212448, 0, 228524812948, 0, -96439370415, 0, 32710198942, 0, -8980281297, 0, 2002105655, 0, -362220741, 0, 52905976, 0, -6171204, 0, 564136, 0, -39159, 0, 1954, 0, -63, 0, 1]);
 

\( x^{44} - 63 x^{42} + 1954 x^{40} - 39159 x^{38} + 564136 x^{36} - 6171204 x^{34} + 52905976 x^{32} - 362220741 x^{30} + 2002105655 x^{28} - 8980281297 x^{26} + 32710198942 x^{24} - 96439370415 x^{22} + 228524812948 x^{20} - 430395212448 x^{18} + 634098522688 x^{16} - 714985491456 x^{14} + 598979244032 x^{12} - 357650202624 x^{10} + 143735308288 x^{8} - 35340484608 x^{6} + 4755030016 x^{4} - 207618048 x^{2} + 4194304 \)

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:  \(202\!\cdots\!984\)\(\medspace = 2^{44}\cdot 7^{22}\cdot 23^{40}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $91.52$
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:  $2, 7, 23$
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:  \(644=2^{2}\cdot 7\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{644}(1,·)$, $\chi_{644}(363,·)$, $\chi_{644}(519,·)$, $\chi_{644}(265,·)$, $\chi_{644}(139,·)$, $\chi_{644}(13,·)$, $\chi_{644}(531,·)$, $\chi_{644}(533,·)$, $\chi_{644}(407,·)$, $\chi_{644}(27,·)$, $\chi_{644}(29,·)$, $\chi_{644}(545,·)$, $\chi_{644}(547,·)$, $\chi_{644}(167,·)$, $\chi_{644}(41,·)$, $\chi_{644}(561,·)$, $\chi_{644}(307,·)$, $\chi_{644}(393,·)$, $\chi_{644}(223,·)$, $\chi_{644}(279,·)$, $\chi_{644}(449,·)$, $\chi_{644}(323,·)$, $\chi_{644}(197,·)$, $\chi_{644}(71,·)$, $\chi_{644}(55,·)$, $\chi_{644}(461,·)$, $\chi_{644}(141,·)$, $\chi_{644}(209,·)$, $\chi_{644}(335,·)$, $\chi_{644}(211,·)$, $\chi_{644}(85,·)$, $\chi_{644}(601,·)$, $\chi_{644}(463,·)$, $\chi_{644}(349,·)$, $\chi_{644}(351,·)$, $\chi_{644}(225,·)$, $\chi_{644}(587,·)$, $\chi_{644}(489,·)$, $\chi_{644}(491,·)$, $\chi_{644}(239,·)$, $\chi_{644}(629,·)$, $\chi_{644}(169,·)$, $\chi_{644}(377,·)$, $\chi_{644}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{21} - \frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{24} + \frac{1}{4} a^{22} - \frac{1}{2} a^{20} + \frac{1}{4} a^{18} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{25} + \frac{1}{8} a^{23} + \frac{1}{4} a^{21} + \frac{1}{8} a^{19} - \frac{1}{2} a^{15} + \frac{3}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{4} a^{5} + \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{26} + \frac{1}{16} a^{24} + \frac{1}{8} a^{22} - \frac{7}{16} a^{20} - \frac{1}{2} a^{18} - \frac{1}{4} a^{16} - \frac{1}{2} a^{14} - \frac{5}{16} a^{12} + \frac{7}{16} a^{10} - \frac{1}{16} a^{8} - \frac{1}{8} a^{6} + \frac{1}{16} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{32} a^{27} + \frac{1}{32} a^{25} + \frac{1}{16} a^{23} + \frac{9}{32} a^{21} + \frac{1}{4} a^{19} - \frac{1}{8} a^{17} - \frac{1}{4} a^{15} - \frac{5}{32} a^{13} - \frac{9}{32} a^{11} + \frac{15}{32} a^{9} - \frac{1}{16} a^{7} - \frac{15}{32} a^{5} - \frac{3}{8} a^{3}$, $\frac{1}{64} a^{28} + \frac{1}{64} a^{26} + \frac{1}{32} a^{24} - \frac{23}{64} a^{22} - \frac{3}{8} a^{20} + \frac{7}{16} a^{18} - \frac{1}{8} a^{16} - \frac{5}{64} a^{14} - \frac{9}{64} a^{12} + \frac{15}{64} a^{10} - \frac{1}{32} a^{8} - \frac{15}{64} a^{6} + \frac{5}{16} a^{4}$, $\frac{1}{128} a^{29} + \frac{1}{128} a^{27} + \frac{1}{64} a^{25} - \frac{23}{128} a^{23} - \frac{3}{16} a^{21} - \frac{9}{32} a^{19} + \frac{7}{16} a^{17} + \frac{59}{128} a^{15} - \frac{9}{128} a^{13} - \frac{49}{128} a^{11} - \frac{1}{64} a^{9} - \frac{15}{128} a^{7} - \frac{11}{32} a^{5}$, $\frac{1}{256} a^{30} + \frac{1}{256} a^{28} + \frac{1}{128} a^{26} - \frac{23}{256} a^{24} + \frac{13}{32} a^{22} + \frac{23}{64} a^{20} + \frac{7}{32} a^{18} - \frac{69}{256} a^{16} - \frac{9}{256} a^{14} + \frac{79}{256} a^{12} - \frac{1}{128} a^{10} + \frac{113}{256} a^{8} + \frac{21}{64} a^{6}$, $\frac{1}{512} a^{31} + \frac{1}{512} a^{29} + \frac{1}{256} a^{27} - \frac{23}{512} a^{25} + \frac{13}{64} a^{23} - \frac{41}{128} a^{21} + \frac{7}{64} a^{19} + \frac{187}{512} a^{17} - \frac{9}{512} a^{15} - \frac{177}{512} a^{13} - \frac{1}{256} a^{11} + \frac{113}{512} a^{9} + \frac{21}{128} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{1024} a^{32} + \frac{1}{1024} a^{30} + \frac{1}{512} a^{28} - \frac{23}{1024} a^{26} + \frac{13}{128} a^{24} - \frac{41}{256} a^{22} - \frac{57}{128} a^{20} + \frac{187}{1024} a^{18} + \frac{503}{1024} a^{16} - \frac{177}{1024} a^{14} - \frac{1}{512} a^{12} - \frac{399}{1024} a^{10} + \frac{21}{256} a^{8} - \frac{1}{2} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{2048} a^{33} + \frac{1}{2048} a^{31} + \frac{1}{1024} a^{29} - \frac{23}{2048} a^{27} + \frac{13}{256} a^{25} - \frac{41}{512} a^{23} + \frac{71}{256} a^{21} + \frac{187}{2048} a^{19} + \frac{503}{2048} a^{17} + \frac{847}{2048} a^{15} - \frac{1}{1024} a^{13} + \frac{625}{2048} a^{11} + \frac{21}{512} a^{9} - \frac{1}{4} a^{7} - \frac{3}{8} a^{5} + \frac{3}{8} a^{3}$, $\frac{1}{4096} a^{34} + \frac{1}{4096} a^{32} + \frac{1}{2048} a^{30} - \frac{23}{4096} a^{28} + \frac{13}{512} a^{26} - \frac{41}{1024} a^{24} - \frac{185}{512} a^{22} + \frac{187}{4096} a^{20} + \frac{503}{4096} a^{18} - \frac{1201}{4096} a^{16} - \frac{1}{2048} a^{14} - \frac{1423}{4096} a^{12} + \frac{21}{1024} a^{10} - \frac{1}{8} a^{8} - \frac{3}{16} a^{6} + \frac{3}{16} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8192} a^{35} + \frac{1}{8192} a^{33} + \frac{1}{4096} a^{31} - \frac{23}{8192} a^{29} + \frac{13}{1024} a^{27} - \frac{41}{2048} a^{25} - \frac{185}{1024} a^{23} - \frac{3909}{8192} a^{21} - \frac{3593}{8192} a^{19} + \frac{2895}{8192} a^{17} - \frac{1}{4096} a^{15} + \frac{2673}{8192} a^{13} + \frac{21}{2048} a^{11} - \frac{1}{16} a^{9} + \frac{13}{32} a^{7} + \frac{3}{32} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{16384} a^{36} + \frac{1}{16384} a^{34} + \frac{1}{8192} a^{32} - \frac{23}{16384} a^{30} + \frac{13}{2048} a^{28} - \frac{41}{4096} a^{26} - \frac{185}{2048} a^{24} + \frac{4283}{16384} a^{22} + \frac{4599}{16384} a^{20} - \frac{5297}{16384} a^{18} + \frac{4095}{8192} a^{16} + \frac{2673}{16384} a^{14} - \frac{2027}{4096} a^{12} + \frac{15}{32} a^{10} + \frac{13}{64} a^{8} + \frac{3}{64} a^{6} + \frac{3}{8} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{32768} a^{37} + \frac{1}{32768} a^{35} + \frac{1}{16384} a^{33} - \frac{23}{32768} a^{31} + \frac{13}{4096} a^{29} - \frac{41}{8192} a^{27} - \frac{185}{4096} a^{25} + \frac{4283}{32768} a^{23} + \frac{4599}{32768} a^{21} - \frac{5297}{32768} a^{19} + \frac{4095}{16384} a^{17} + \frac{2673}{32768} a^{15} - \frac{2027}{8192} a^{13} - \frac{17}{64} a^{11} - \frac{51}{128} a^{9} + \frac{3}{128} a^{7} + \frac{3}{16} a^{5} + \frac{1}{4} a^{3}$, $\frac{1}{65536} a^{38} + \frac{1}{65536} a^{36} + \frac{1}{32768} a^{34} - \frac{23}{65536} a^{32} + \frac{13}{8192} a^{30} - \frac{41}{16384} a^{28} - \frac{185}{8192} a^{26} + \frac{4283}{65536} a^{24} - \frac{28169}{65536} a^{22} - \frac{5297}{65536} a^{20} + \frac{4095}{32768} a^{18} + \frac{2673}{65536} a^{16} - \frac{2027}{16384} a^{14} + \frac{47}{128} a^{12} + \frac{77}{256} a^{10} + \frac{3}{256} a^{8} - \frac{13}{32} a^{6} + \frac{1}{8} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{131072} a^{39} + \frac{1}{131072} a^{37} + \frac{1}{65536} a^{35} - \frac{23}{131072} a^{33} + \frac{13}{16384} a^{31} - \frac{41}{32768} a^{29} - \frac{185}{16384} a^{27} + \frac{4283}{131072} a^{25} - \frac{28169}{131072} a^{23} - \frac{5297}{131072} a^{21} - \frac{28673}{65536} a^{19} - \frac{62863}{131072} a^{17} + \frac{14357}{32768} a^{15} - \frac{81}{256} a^{13} - \frac{179}{512} a^{11} - \frac{253}{512} a^{9} - \frac{13}{64} a^{7} - \frac{7}{16} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{157024256} a^{40} + \frac{485}{157024256} a^{38} - \frac{1173}{78512128} a^{36} - \frac{16031}{157024256} a^{34} + \frac{2915}{39256064} a^{32} + \frac{31611}{39256064} a^{30} - \frac{91251}{19628032} a^{28} + \frac{1254619}{157024256} a^{26} + \frac{8769795}{157024256} a^{24} - \frac{56729701}{157024256} a^{22} - \frac{32836459}{78512128} a^{20} - \frac{21168135}{157024256} a^{18} - \frac{4634197}{19628032} a^{16} + \frac{575677}{2453504} a^{14} + \frac{536641}{1226752} a^{12} + \frac{44517}{153344} a^{10} + \frac{9443}{19168} a^{8} + \frac{679}{4792} a^{6} + \frac{1005}{9584} a^{4} - \frac{473}{2396} a^{2} - \frac{61}{599}$, $\frac{1}{314048512} a^{41} + \frac{485}{314048512} a^{39} - \frac{1173}{157024256} a^{37} - \frac{16031}{314048512} a^{35} + \frac{2915}{78512128} a^{33} + \frac{31611}{78512128} a^{31} - \frac{91251}{39256064} a^{29} + \frac{1254619}{314048512} a^{27} + \frac{8769795}{314048512} a^{25} - \frac{56729701}{314048512} a^{23} + \frac{45675669}{157024256} a^{21} - \frac{21168135}{314048512} a^{19} + \frac{14993835}{39256064} a^{17} + \frac{575677}{4907008} a^{15} - \frac{690111}{2453504} a^{13} - \frac{108827}{306688} a^{11} + \frac{9443}{38336} a^{9} - \frac{4113}{9584} a^{7} + \frac{1005}{19168} a^{5} + \frac{1923}{4792} a^{3} - \frac{61}{1198} a$, $\frac{1}{516819457040926749080791774386788456262479293526432090363584451786333605833211904} a^{42} + \frac{859175077027477963117331256570430034262199856696075117147979037733866429}{516819457040926749080791774386788456262479293526432090363584451786333605833211904} a^{40} + \frac{297206390731727073728983793093562770062893349814407368903661746785592438631}{258409728520463374540395887193394228131239646763216045181792225893166802916605952} a^{38} + \frac{14725007884739143725246463221014743789013326067180541354071271363411234830705}{516819457040926749080791774386788456262479293526432090363584451786333605833211904} a^{36} + \frac{3240111073245903510470166227641530299207705462093469937327994430651715343529}{129204864260231687270197943596697114065619823381608022590896112946583401458302976} a^{34} - \frac{26015000466506741518241900098632371552772353445856243721544796000350189800045}{129204864260231687270197943596697114065619823381608022590896112946583401458302976} a^{32} + \frac{60963636205548151122433501595387120639649471889459215976439625492697894321705}{64602432130115843635098971798348557032809911690804011295448056473291700729151488} a^{30} + \frac{3614621327787403063127779412301630939499947987938544793573449550899162956643995}{516819457040926749080791774386788456262479293526432090363584451786333605833211904} a^{28} + \frac{8824629591024193059995669079218725224231422506296043707348040710226840813652939}{516819457040926749080791774386788456262479293526432090363584451786333605833211904} a^{26} + \frac{48258367097277841693893653570785630166883030559928781437151497123987293996720739}{516819457040926749080791774386788456262479293526432090363584451786333605833211904} a^{24} + \frac{9855714617307085939779724212058600496189392050516352520946464804792927736353337}{258409728520463374540395887193394228131239646763216045181792225893166802916605952} a^{22} - \frac{166194439282747636098212664164303221511842424532625114925975320353760799107176727}{516819457040926749080791774386788456262479293526432090363584451786333605833211904} a^{20} - \frac{6685122622710136712280184291269447048327164108821570848774895785573782403204637}{32301216065057921817549485899174278516404955845402005647724028236645850364575744} a^{18} - \frac{3413257171401905270887286343934543769975817602149067388003155800196447746058845}{8075304016264480454387371474793569629101238961350501411931007059161462591143936} a^{16} + \frac{3165840080402967907722570111563777983196010624187059048037016510785001503150973}{8075304016264480454387371474793569629101238961350501411931007059161462591143936} a^{14} - \frac{106950218185059682703179892304439526379167197120772238722175740483589652448405}{1009413002033060056798421434349196203637654870168812676491375882395182823892992} a^{12} - \frac{29114572419418816530183476301274721498326588062148272046356887974287392582505}{252353250508265014199605358587299050909413717542203169122843970598795705973248} a^{10} - \frac{32398411192728840554637670090797170559596246538686982613040991462773002907785}{126176625254132507099802679293649525454706858771101584561421985299397852986624} a^{8} + \frac{7886004923044513323192486317327575521578107709386730464537366078222094916771}{31544156313533126774950669823412381363676714692775396140355496324849463246656} a^{6} + \frac{1789111783261032652088998666587261264803912214969432625506665763966363734161}{3943019539191640846868833727926547670459589336596924517544437040606182905832} a^{4} + \frac{255713400844338524425340000652027183749998478194074132612224671903318028967}{1971509769595820423434416863963273835229794668298462258772218520303091452916} a^{2} - \frac{150334567084334752917939062303618938510229300594273652784607345512995539221}{492877442398955105858604215990818458807448667074615564693054630075772863229}$, $\frac{1}{1033638914081853498161583548773576912524958587052864180727168903572667211666423808} a^{43} + \frac{859175077027477963117331256570430034262199856696075117147979037733866429}{1033638914081853498161583548773576912524958587052864180727168903572667211666423808} a^{41} + \frac{297206390731727073728983793093562770062893349814407368903661746785592438631}{516819457040926749080791774386788456262479293526432090363584451786333605833211904} a^{39} + \frac{14725007884739143725246463221014743789013326067180541354071271363411234830705}{1033638914081853498161583548773576912524958587052864180727168903572667211666423808} a^{37} + \frac{3240111073245903510470166227641530299207705462093469937327994430651715343529}{258409728520463374540395887193394228131239646763216045181792225893166802916605952} a^{35} - \frac{26015000466506741518241900098632371552772353445856243721544796000350189800045}{258409728520463374540395887193394228131239646763216045181792225893166802916605952} a^{33} + \frac{60963636205548151122433501595387120639649471889459215976439625492697894321705}{129204864260231687270197943596697114065619823381608022590896112946583401458302976} a^{31} + \frac{3614621327787403063127779412301630939499947987938544793573449550899162956643995}{1033638914081853498161583548773576912524958587052864180727168903572667211666423808} a^{29} + \frac{8824629591024193059995669079218725224231422506296043707348040710226840813652939}{1033638914081853498161583548773576912524958587052864180727168903572667211666423808} a^{27} + \frac{48258367097277841693893653570785630166883030559928781437151497123987293996720739}{1033638914081853498161583548773576912524958587052864180727168903572667211666423808} a^{25} + \frac{9855714617307085939779724212058600496189392050516352520946464804792927736353337}{516819457040926749080791774386788456262479293526432090363584451786333605833211904} a^{23} - \frac{166194439282747636098212664164303221511842424532625114925975320353760799107176727}{1033638914081853498161583548773576912524958587052864180727168903572667211666423808} a^{21} + \frac{25616093442347785105269301607904831468077791736580434798949132451072067961371107}{64602432130115843635098971798348557032809911690804011295448056473291700729151488} a^{19} - \frac{3413257171401905270887286343934543769975817602149067388003155800196447746058845}{16150608032528960908774742949587139258202477922701002823862014118322925182287872} a^{17} - \frac{4909463935861512546664801363229791645905228337163442363893990548376461087992963}{16150608032528960908774742949587139258202477922701002823862014118322925182287872} a^{15} - \frac{106950218185059682703179892304439526379167197120772238722175740483589652448405}{2018826004066120113596842868698392407275309740337625352982751764790365647785984} a^{13} - \frac{29114572419418816530183476301274721498326588062148272046356887974287392582505}{504706501016530028399210717174598101818827435084406338245687941197591411946496} a^{11} + \frac{93778214061403666545165009202852354895110612232414601948380993836624850078839}{252353250508265014199605358587299050909413717542203169122843970598795705973248} a^{9} - \frac{23658151390488613451758183506084805842098606983388665675818130246627368329885}{63088312627066253549901339646824762727353429385550792280710992649698926493312} a^{7} + \frac{1789111783261032652088998666587261264803912214969432625506665763966363734161}{7886039078383281693737667455853095340919178673193849035088874081212365811664} a^{5} - \frac{1715796368751481899009076863311246651479796190104388126159993848399773423949}{3943019539191640846868833727926547670459589336596924517544437040606182905832} a^{3} - \frac{150334567084334752917939062303618938510229300594273652784607345512995539221}{985754884797910211717208431981636917614897334149231129386109260151545726458} a$

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{12510636459605472541565569626671330282473}{150992405799960827353876616408260423340252987392} a^{43} + \frac{788155204883906868031788120452544671121787}{150992405799960827353876616408260423340252987392} a^{41} - \frac{12222412535911517858439487774581238522975351}{75496202899980413676938308204130211670126493696} a^{39} + \frac{489873602995256836274423571564526256474629943}{150992405799960827353876616408260423340252987392} a^{37} - \frac{1764268916566608477463675942263307363705141737}{37748101449990206838469154102065105835063246848} a^{35} + \frac{19299108616983551654477233489468805331906499097}{37748101449990206838469154102065105835063246848} a^{33} - \frac{82722860010164010797077671965520911777142553537}{18874050724995103419234577051032552917531623424} a^{31} + \frac{4530681285881169048042403143196267234741261292461}{150992405799960827353876616408260423340252987392} a^{29} - \frac{25040951085494340503193687459240778980623332353731}{150992405799960827353876616408260423340252987392} a^{27} + \frac{112310131461680477894701525764055678298568062198805}{150992405799960827353876616408260423340252987392} a^{25} - \frac{204519381185656251262172760208092959650243567058537}{75496202899980413676938308204130211670126493696} a^{23} + \frac{1205778192527570118676095653861767352138544479378399}{150992405799960827353876616408260423340252987392} a^{21} - \frac{178535194968578488033671148578361950303629091182629}{9437025362497551709617288525516276458765811712} a^{19} + \frac{336119866696916397332529375363045901014608593036917}{9437025362497551709617288525516276458765811712} a^{17} - \frac{7732566688932177300653210328933389763846098481381}{147453521289024245462770133211191819668215808} a^{15} + \frac{8708574449713093653424698475988502986352103631101}{147453521289024245462770133211191819668215808} a^{13} - \frac{227441830194098487198347307634251285861508385891}{4607922540282007670711566662849744364631744} a^{11} + \frac{1080689946012840053506408863981892699816057045763}{36863380322256061365692533302797954917053952} a^{9} - \frac{1674625443307585495222322843759386236156697387}{143997579383812739709736458214054511394742} a^{7} + \frac{6352936899219365866975289920305886325414067353}{2303961270141003835355783331424872182315872} a^{5} - \frac{47692887008831282151574289740134176165856751}{143997579383812739709736458214054511394742} a^{3} + \frac{958711397086014320239456325257015048634325}{143997579383812739709736458214054511394742} a \) (order $4$)
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{7}) \), \(\Q(i, \sqrt{7})\), \(\Q(\zeta_{23})^+\), 22.0.7198079267989980836471065337135104.1, 22.0.3393400820453274956705986794666660343.1, 22.22.14232954634830452964011747236817552143286272.1

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 R $22^{2}$ $22^{2}$ R $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ R ${\href{/LocalNumberField/29.11.0.1}{11} }^{4}$ $22^{2}$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{22}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{4}$ $22^{2}$

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
2Data not computed
7Data not computed
23Data not computed