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Properties

Label 32.0.868...000.1
Degree $32$
Signature $[0, 16]$
Discriminant $8.690\times 10^{55}$
Root discriminant \(55.99\)
Ramified primes $2,5,17$
Class number $320$ (GRH)
Class group [4, 4, 20] (GRH)
Galois group $C_2\times C_4^2$ (as 32T36)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 13*x^30 + 129*x^28 - 1170*x^26 + 10218*x^24 - 43485*x^22 + 171666*x^20 - 623922*x^18 + 1799433*x^16 - 623922*x^14 + 171666*x^12 - 43485*x^10 + 10218*x^8 - 1170*x^6 + 129*x^4 - 13*x^2 + 1)
 
gp: K = bnfinit(y^32 - 13*y^30 + 129*y^28 - 1170*y^26 + 10218*y^24 - 43485*y^22 + 171666*y^20 - 623922*y^18 + 1799433*y^16 - 623922*y^14 + 171666*y^12 - 43485*y^10 + 10218*y^8 - 1170*y^6 + 129*y^4 - 13*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 13*x^30 + 129*x^28 - 1170*x^26 + 10218*x^24 - 43485*x^22 + 171666*x^20 - 623922*x^18 + 1799433*x^16 - 623922*x^14 + 171666*x^12 - 43485*x^10 + 10218*x^8 - 1170*x^6 + 129*x^4 - 13*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 13*x^30 + 129*x^28 - 1170*x^26 + 10218*x^24 - 43485*x^22 + 171666*x^20 - 623922*x^18 + 1799433*x^16 - 623922*x^14 + 171666*x^12 - 43485*x^10 + 10218*x^8 - 1170*x^6 + 129*x^4 - 13*x^2 + 1)
 

\( x^{32} - 13 x^{30} + 129 x^{28} - 1170 x^{26} + 10218 x^{24} - 43485 x^{22} + 171666 x^{20} - 623922 x^{18} + \cdots + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $32$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 16]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(86898859856540647588737053982976000000000000000000000000\) \(\medspace = 2^{32}\cdot 5^{24}\cdot 17^{24}\) 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:  \(55.99\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2\cdot 5^{3/4}17^{3/4}\approx 55.98790150987606$
Ramified primes:   \(2\), \(5\), \(17\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $32$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(340=2^{2}\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{340}(1,·)$, $\chi_{340}(259,·)$, $\chi_{340}(137,·)$, $\chi_{340}(13,·)$, $\chi_{340}(271,·)$, $\chi_{340}(273,·)$, $\chi_{340}(149,·)$, $\chi_{340}(89,·)$, $\chi_{340}(157,·)$, $\chi_{340}(33,·)$, $\chi_{340}(293,·)$, $\chi_{340}(169,·)$, $\chi_{340}(171,·)$, $\chi_{340}(47,·)$, $\chi_{340}(307,·)$, $\chi_{340}(183,·)$, $\chi_{340}(191,·)$, $\chi_{340}(67,·)$, $\chi_{340}(69,·)$, $\chi_{340}(327,·)$, $\chi_{340}(203,·)$, $\chi_{340}(81,·)$, $\chi_{340}(339,·)$, $\chi_{340}(217,·)$, $\chi_{340}(123,·)$, $\chi_{340}(101,·)$, $\chi_{340}(103,·)$, $\chi_{340}(237,·)$, $\chi_{340}(239,·)$, $\chi_{340}(319,·)$, $\chi_{340}(251,·)$, $\chi_{340}(21,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{32768}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3}a^{10}-\frac{1}{3}$, $\frac{1}{3}a^{11}-\frac{1}{3}a$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{5}$, $\frac{1}{3}a^{16}-\frac{1}{3}a^{6}$, $\frac{1}{3}a^{17}-\frac{1}{3}a^{7}$, $\frac{1}{3}a^{18}-\frac{1}{3}a^{8}$, $\frac{1}{3}a^{19}-\frac{1}{3}a^{9}$, $\frac{1}{10089}a^{20}+\frac{13}{531}a^{10}-\frac{3362}{10089}$, $\frac{1}{10089}a^{21}+\frac{13}{531}a^{11}-\frac{3362}{10089}a$, $\frac{1}{10089}a^{22}+\frac{13}{531}a^{12}-\frac{3362}{10089}a^{2}$, $\frac{1}{10089}a^{23}+\frac{13}{531}a^{13}-\frac{3362}{10089}a^{3}$, $\frac{1}{80712}a^{24}+\frac{1}{24}a^{18}+\frac{68}{531}a^{14}+\frac{1}{8}a^{12}+\frac{1}{3}a^{8}+\frac{3}{8}a^{6}+\frac{2102}{10089}a^{4}+\frac{1}{8}$, $\frac{1}{80712}a^{25}+\frac{1}{24}a^{19}+\frac{68}{531}a^{15}+\frac{1}{8}a^{13}+\frac{1}{3}a^{9}+\frac{3}{8}a^{7}+\frac{2102}{10089}a^{5}+\frac{1}{8}a$, $\frac{1}{15252227352}a^{26}+\frac{2409}{423672982}a^{24}+\frac{23887}{635509473}a^{22}-\frac{339341}{15252227352}a^{20}-\frac{65069}{1133826}a^{18}+\frac{6627656}{100343601}a^{16}+\frac{5088193}{267582936}a^{14}-\frac{3788539}{66895734}a^{12}+\frac{1244789}{100343601}a^{10}-\frac{1811335}{4535304}a^{8}+\frac{1599840475}{3813056838}a^{6}+\frac{144283600}{635509473}a^{4}+\frac{341390987}{5084075784}a^{2}-\frac{33174233}{3813056838}$, $\frac{1}{15252227352}a^{27}+\frac{2409}{423672982}a^{25}+\frac{23887}{635509473}a^{23}-\frac{339341}{15252227352}a^{21}-\frac{65069}{1133826}a^{19}+\frac{6627656}{100343601}a^{17}+\frac{5088193}{267582936}a^{15}-\frac{3788539}{66895734}a^{13}+\frac{1244789}{100343601}a^{11}-\frac{1811335}{4535304}a^{9}+\frac{1599840475}{3813056838}a^{7}+\frac{144283600}{635509473}a^{5}+\frac{341390987}{5084075784}a^{3}-\frac{33174233}{3813056838}a$, $\frac{1}{15252227352}a^{28}-\frac{1}{15252227352}a^{24}-\frac{217879}{5084075784}a^{22}-\frac{23887}{635509473}a^{20}-\frac{53018921}{802748808}a^{18}-\frac{306637}{4535304}a^{16}-\frac{6627656}{100343601}a^{14}-\frac{18694105}{267582936}a^{12}+\frac{4004867}{267582936}a^{10}-\frac{506650663}{1906528419}a^{8}+\frac{1811335}{4535304}a^{6}+\frac{6946337033}{15252227352}a^{4}-\frac{117827660}{635509473}a^{2}-\frac{678694305}{1694691928}$, $\frac{1}{15252227352}a^{29}-\frac{1}{15252227352}a^{25}-\frac{217879}{5084075784}a^{23}-\frac{23887}{635509473}a^{21}-\frac{53018921}{802748808}a^{19}-\frac{306637}{4535304}a^{17}-\frac{6627656}{100343601}a^{15}-\frac{18694105}{267582936}a^{13}+\frac{4004867}{267582936}a^{11}-\frac{506650663}{1906528419}a^{9}+\frac{1811335}{4535304}a^{7}+\frac{6946337033}{15252227352}a^{5}-\frac{117827660}{635509473}a^{3}-\frac{678694305}{1694691928}a$, $\frac{1}{45756682056}a^{30}+\frac{90091}{1906528419}a^{20}+\frac{275838071}{1906528419}a^{10}-\frac{146703017}{775536984}$, $\frac{1}{45756682056}a^{31}+\frac{90091}{1906528419}a^{21}+\frac{275838071}{1906528419}a^{11}-\frac{146703017}{775536984}a$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

$C_{4}\times C_{4}\times C_{20}$, which has order $320$ (assuming GRH)

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:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{8471948981}{22878341028} a^{31} - \frac{18355965731}{3813056838} a^{29} + \frac{364293806183}{7626113676} a^{27} - \frac{550676683765}{1271018946} a^{25} + \frac{14427729114643}{3813056838} a^{23} - \frac{122800900479595}{7626113676} a^{21} + \frac{12757240003909}{200687202} a^{19} - \frac{46366976773013}{200687202} a^{17} + \frac{89150319127063}{133791468} a^{15} - \frac{46366976773013}{200687202} a^{13} + \frac{26932325810599}{423672982} a^{11} - \frac{131358420785431}{7626113676} a^{9} + \frac{14427729114643}{3813056838} a^{7} - \frac{550676683765}{1271018946} a^{5} + \frac{364293806183}{7626113676} a^{3} - \frac{110135336753}{22878341028} a \)  (order $20$) 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:   $\frac{13824322781}{22878341028}a^{31}-\frac{59325710767}{7626113676}a^{29}+\frac{586952556263}{7626113676}a^{27}-\frac{279856121431}{401374404}a^{25}+\frac{46412983814771}{7626113676}a^{23}-\frac{194510934177235}{7626113676}a^{21}+\frac{40331054970323}{401374404}a^{19}-\frac{146184230215621}{401374404}a^{17}+\frac{417782221323857}{401374404}a^{15}-\frac{97870760491231}{401374404}a^{13}+\frac{56615608209643}{847345964}a^{11}-\frac{137231045318791}{7626113676}a^{9}+\frac{29528117806601}{7626113676}a^{7}-\frac{91595543917}{401374404}a^{5}+\frac{371787129503}{7626113676}a^{3}-\frac{55937429119}{11439170514}a$, $\frac{236957825}{22878341028}a^{31}-\frac{280878}{211836491}a^{30}-\frac{783783575}{7626113676}a^{29}+\frac{2787174}{211836491}a^{28}+\frac{1184789125}{1271018946}a^{27}-\frac{25279020}{211836491}a^{26}-\frac{31041475075}{3813056838}a^{25}+\frac{220770108}{211836491}a^{24}+\frac{532839161155}{7626113676}a^{23}-\frac{17053227022}{1906528419}a^{22}-\frac{57945301975}{423672982}a^{21}+\frac{3709015596}{211836491}a^{20}+\frac{99759244325}{200687202}a^{19}-\frac{709497828}{11149289}a^{18}-\frac{191808245575}{133791468}a^{17}+\frac{2046239442}{11149289}a^{16}+\frac{99759244325}{200687202}a^{15}-\frac{709497828}{11149289}a^{14}+\frac{8942200611911}{200687202}a^{13}-\frac{572380912291}{100343601}a^{12}+\frac{264207974875}{7626113676}a^{11}-\frac{939536910}{211836491}a^{10}-\frac{31041475075}{3813056838}a^{9}+\frac{220770108}{211836491}a^{8}+\frac{1184789125}{1271018946}a^{7}-\frac{25279020}{211836491}a^{6}-\frac{783783575}{7626113676}a^{5}+\frac{2787174}{211836491}a^{4}+\frac{97557978185}{3813056838}a^{3}-\frac{16353810460}{1906528419}a^{2}-\frac{18227525}{22878341028}a+\frac{21606}{211836491}$, $\frac{236957825}{22878341028}a^{31}+\frac{943364305}{5719585257}a^{30}-\frac{783783575}{7626113676}a^{29}-\frac{4063723160}{1906528419}a^{28}+\frac{1184789125}{1271018946}a^{27}+\frac{40250212301}{1906528419}a^{26}-\frac{31041475075}{3813056838}a^{25}-\frac{364791720095}{1906528419}a^{24}+\frac{532839161155}{7626113676}a^{23}+\frac{3184797893680}{1906528419}a^{22}-\frac{57945301975}{423672982}a^{21}-\frac{13426904153065}{1906528419}a^{20}+\frac{99759244325}{200687202}a^{19}+\frac{2785754792665}{100343601}a^{18}-\frac{191808245575}{133791468}a^{17}-\frac{10107513797680}{100343601}a^{16}+\frac{99759244325}{200687202}a^{15}+\frac{28986754999735}{100343601}a^{14}+\frac{8942200611911}{200687202}a^{13}-\frac{8035214317565}{100343601}a^{12}+\frac{264207974875}{7626113676}a^{11}+\frac{38889250109320}{1906528419}a^{10}-\frac{31041475075}{3813056838}a^{9}-\frac{9521666196305}{1906528419}a^{8}+\frac{1184789125}{1271018946}a^{7}+\frac{2270909491370}{1906528419}a^{6}-\frac{783783575}{7626113676}a^{5}-\frac{120750631040}{1906528419}a^{4}+\frac{97557978185}{3813056838}a^{3}+\frac{12263735965}{1906528419}a^{2}-\frac{18227525}{22878341028}a-\frac{2902659400}{5719585257}$, $\frac{1045391557}{22878341028}a^{30}-\frac{119212075}{200687202}a^{28}+\frac{44951836951}{7626113676}a^{26}-\frac{67950451205}{1271018946}a^{24}+\frac{1780301821571}{3813056838}a^{22}-\frac{15152950618715}{7626113676}a^{20}+\frac{1574161149545}{200687202}a^{18}-\frac{5721427991461}{200687202}a^{16}+\frac{11000655354311}{133791468}a^{14}-\frac{5721427991461}{200687202}a^{12}+\frac{3323299759703}{423672982}a^{10}-\frac{882465173105}{401374404}a^{8}+\frac{1780301821571}{3813056838}a^{6}-\frac{67950451205}{1271018946}a^{4}+\frac{44951836951}{7626113676}a^{2}-\frac{13590090241}{22878341028}$, $\frac{115618529}{45756682056}a^{30}-\frac{382430519}{15252227352}a^{28}+\frac{578092645}{2542037892}a^{26}-\frac{15146027299}{7626113676}a^{24}+\frac{86662509601}{5084075784}a^{22}-\frac{28273177207}{847345964}a^{20}+\frac{48675400709}{401374404}a^{18}-\frac{93588752359}{267582936}a^{16}+\frac{48675400709}{401374404}a^{14}+\frac{1454384437861}{133791468}a^{12}+\frac{128914659835}{15252227352}a^{10}-\frac{15146027299}{7626113676}a^{8}+\frac{578092645}{2542037892}a^{6}-\frac{382430519}{15252227352}a^{4}-\frac{3273272073}{847345964}a^{2}-\frac{8893733}{45756682056}$, $\frac{4716713491}{45756682056}a^{31}-\frac{1135539837}{847345964}a^{29}+\frac{202818680113}{15252227352}a^{27}-\frac{306586376915}{2542037892}a^{25}+\frac{8032563075173}{7626113676}a^{23}-\frac{68368762052045}{15252227352}a^{21}+\frac{2367299423201}{133791468}a^{19}-\frac{25814572936243}{401374404}a^{17}+\frac{49633976065793}{267582936}a^{15}-\frac{25814572936243}{401374404}a^{13}+\frac{14994432187889}{847345964}a^{11}-\frac{34481905213927}{5084075784}a^{9}+\frac{8032563075173}{7626113676}a^{7}-\frac{306586376915}{2542037892}a^{5}+\frac{202818680113}{15252227352}a^{3}-\frac{61317275383}{45756682056}a$, $\frac{12952198723}{22878341028}a^{31}-\frac{12587240427}{1694691928}a^{29}+\frac{11158009213}{151012152}a^{27}-\frac{1278922062496}{1906528419}a^{25}+\frac{9932621981551}{1694691928}a^{23}-\frac{385628518773389}{15252227352}a^{21}+\frac{1113776088866}{11149289}a^{19}-\frac{292099408436593}{802748808}a^{17}+\frac{848613089398867}{802748808}a^{15}-\frac{5141936230901}{11149289}a^{13}+\frac{15\!\cdots\!07}{15252227352}a^{11}-\frac{48833267201701}{1694691928}a^{9}+\frac{11452304098222}{1906528419}a^{7}-\frac{10490788019099}{15252227352}a^{5}+\frac{62925321273}{1694691928}a^{3}-\frac{48688757725}{22878341028}a$, $\frac{52116865}{1204123212}a^{31}-\frac{335803151}{635509473}a^{29}+\frac{13076484395}{2542037892}a^{27}-\frac{78464360939}{1694691928}a^{25}+\frac{13478253874}{33447867}a^{23}-\frac{11738373684521}{7626113676}a^{21}+\frac{1606377013751}{267582936}a^{19}-\frac{715533881683}{33447867}a^{17}+\frac{7702480829149}{133791468}a^{15}+\frac{2744937370735}{89194312}a^{13}-\frac{91622968171}{33447867}a^{11}-\frac{314186129543}{2542037892}a^{9}+\frac{71973591565}{5084075784}a^{7}+\frac{67157295683}{635509473}a^{5}+\frac{4697830553}{133791468}a^{3}+\frac{64047253093}{45756682056}a$, $\frac{8137980287}{45756682056}a^{31}-\frac{17783510501}{7626113676}a^{29}+\frac{2068768549}{89194312}a^{27}-\frac{3211445358109}{15252227352}a^{25}+\frac{1558782008453}{847345964}a^{23}-\frac{120925639331419}{15252227352}a^{21}+\frac{25143869818379}{802748808}a^{19}-\frac{5086888768659}{44597156}a^{17}+\frac{265864322244989}{802748808}a^{15}-\frac{38057000645989}{267582936}a^{13}+\frac{25478906396513}{847345964}a^{11}-\frac{87339797302939}{15252227352}a^{9}+\frac{140142720043}{89194312}a^{7}-\frac{553811239121}{7626113676}a^{5}-\frac{136129967075}{5084075784}a^{3}+\frac{39802923667}{11439170514}a$, $\frac{6417190871}{15252227352}a^{31}+\frac{1498897913}{15252227352}a^{30}-\frac{20798794015}{3813056838}a^{29}-\frac{6393387791}{5084075784}a^{28}+\frac{412476633655}{7626113676}a^{27}+\frac{2631003498}{211836491}a^{26}-\frac{31694862395}{64628082}a^{25}-\frac{857447797727}{7626113676}a^{24}+\frac{5443068835477}{1271018946}a^{23}+\frac{14964195503201}{15252227352}a^{22}-\frac{46139529132065}{2542037892}a^{21}-\frac{7763922915851}{1906528419}a^{20}+\frac{14377883900261}{200687202}a^{19}+\frac{2144665481567}{133791468}a^{18}-\frac{52222631509321}{200687202}a^{17}-\frac{15521452454549}{267582936}a^{16}+\frac{5094426842447}{6802956}a^{15}+\frac{16541644684756}{100343601}a^{14}-\frac{16026010849697}{66895734}a^{13}-\frac{10886273094689}{401374404}a^{12}+\frac{31446342766243}{423672982}a^{11}+\frac{147721535138489}{15252227352}a^{10}-\frac{132645297565607}{7626113676}a^{9}-\frac{1505627208304}{635509473}a^{8}+\frac{14501429701811}{3813056838}a^{7}+\frac{1236528759661}{2542037892}a^{6}-\frac{23422283105}{64628082}a^{5}+\frac{264707245403}{15252227352}a^{4}+\frac{100176909437}{2542037892}a^{3}-\frac{1553561696}{1906528419}a^{2}-\frac{33266622209}{15252227352}a-\frac{1222977979}{5084075784}$, $\frac{8528428501}{22878341028}a^{31}-\frac{149303821}{15252227352}a^{30}-\frac{73806293443}{15252227352}a^{29}+\frac{1929405167}{15252227352}a^{28}+\frac{183014042059}{3813056838}a^{27}-\frac{324140465}{258512328}a^{26}-\frac{3319206129889}{7626113676}a^{25}+\frac{28896363839}{2542037892}a^{24}+\frac{57970903987375}{15252227352}a^{23}-\frac{168208890331}{1694691928}a^{22}-\frac{6837497578067}{423672982}a^{21}+\frac{2130276344053}{5084075784}a^{20}+\frac{25563155408527}{401374404}a^{19}-\frac{665020689095}{401374404}a^{18}-\frac{185748673349129}{802748808}a^{17}+\frac{81862163839}{13605912}a^{16}+\frac{133749816390949}{200687202}a^{15}-\frac{4628321118155}{267582936}a^{14}-\frac{874958418143}{3974004}a^{13}+\frac{236608417067}{44597156}a^{12}+\frac{919961409676663}{15252227352}a^{11}-\frac{14285443167073}{5084075784}a^{10}-\frac{65686783406365}{3813056838}a^{9}+\frac{10518222176879}{15252227352}a^{8}+\frac{28857192507221}{7626113676}a^{7}-\frac{48498917395}{129256164}a^{6}-\frac{6608502635699}{15252227352}a^{5}+\frac{348701903029}{5084075784}a^{4}+\frac{167417178763}{3813056838}a^{3}-\frac{16040772099}{1694691928}a^{2}-\frac{38918541107}{11439170514}a+\frac{16323302785}{15252227352}$, $\frac{25232979689}{45756682056}a^{31}+\frac{3261921773}{15252227352}a^{30}-\frac{54338139901}{7626113676}a^{29}-\frac{42656536645}{15252227352}a^{28}+\frac{269067672163}{3813056838}a^{27}+\frac{141342082325}{5084075784}a^{26}-\frac{9753852304859}{15252227352}a^{25}-\frac{641414227225}{2542037892}a^{24}+\frac{42576813670261}{7626113676}a^{23}+\frac{33620532487273}{15252227352}a^{22}-\frac{89712665589785}{3813056838}a^{21}-\frac{144377845077947}{15252227352}a^{20}+\frac{74426901848101}{802748808}a^{19}+\frac{15015132392245}{401374404}a^{18}-\frac{135006645442471}{401374404}a^{17}-\frac{12146132134615}{89194312}a^{16}+\frac{193511864876689}{200687202}a^{15}+\frac{105633576461407}{267582936}a^{14}-\frac{210761497476937}{802748808}a^{13}-\frac{64948740128545}{401374404}a^{12}+\frac{453402426777193}{7626113676}a^{11}+\frac{659907804206089}{15252227352}a^{10}-\frac{68001750190937}{3813056838}a^{9}-\frac{166030669989979}{15252227352}a^{8}+\frac{58775028030775}{15252227352}a^{7}+\frac{6405326984525}{2542037892}a^{6}-\frac{1658534151937}{7626113676}a^{5}-\frac{1597151972495}{5084075784}a^{4}+\frac{117228676847}{3813056838}a^{3}+\frac{391448611015}{15252227352}a^{2}-\frac{102215317961}{11439170514}a-\frac{53047729837}{15252227352}$, $\frac{1384418795}{22878341028}a^{31}-\frac{1398080977}{22878341028}a^{30}-\frac{12384051961}{15252227352}a^{29}+\frac{1487338049}{1906528419}a^{28}+\frac{61990871605}{7626113676}a^{27}-\frac{58713612607}{7626113676}a^{26}-\frac{9898347529}{133791468}a^{25}+\frac{1062713073373}{15252227352}a^{24}+\frac{9870150670157}{15252227352}a^{23}-\frac{1158991604996}{1906528419}a^{22}-\frac{21984333608929}{7626113676}a^{21}+\frac{19166178397363}{7626113676}a^{20}+\frac{4586226262735}{401374404}a^{19}-\frac{7935762464075}{802748808}a^{18}-\frac{33586776994075}{802748808}a^{17}+\frac{3586525486865}{100343601}a^{16}+\frac{5514705558895}{44597156}a^{15}-\frac{40664693057101}{401374404}a^{14}-\frac{31949311128695}{401374404}a^{13}+\frac{10739441284727}{802748808}a^{12}+\frac{26609545516429}{1694691928}a^{11}-\frac{8185437264049}{1906528419}a^{10}-\frac{35489817473681}{7626113676}a^{9}+\frac{1576471149227}{7626113676}a^{8}+\frac{8786191294153}{7626113676}a^{7}-\frac{361143973097}{15252227352}a^{6}-\frac{54848153089}{267582936}a^{5}-\frac{52465242553}{1906528419}a^{4}+\frac{140625306421}{7626113676}a^{3}+\frac{10567832989}{7626113676}a^{2}-\frac{6073857176}{5719585257}a-\frac{20726610979}{45756682056}$, $\frac{1066513459}{5084075784}a^{31}-\frac{67637707}{1906528419}a^{30}-\frac{556514483}{211836491}a^{29}+\frac{195444277}{401374404}a^{28}+\frac{65527968415}{2542037892}a^{27}-\frac{3119824019}{635509473}a^{26}-\frac{3550000319927}{15252227352}a^{25}+\frac{341671167485}{7626113676}a^{24}+\frac{1289207889743}{635509473}a^{23}-\frac{997404760217}{2542037892}a^{22}-\frac{20640447318899}{2542037892}a^{21}+\frac{1146087814729}{635509473}a^{20}+\frac{2839158988793}{89194312}a^{19}-\frac{2885645558987}{401374404}a^{18}-\frac{3827390420057}{33447867}a^{17}+\frac{3541130442031}{133791468}a^{16}+\frac{127593027462911}{401374404}a^{15}-\frac{7983208617016}{100343601}a^{14}+\frac{10403863207849}{267582936}a^{13}+\frac{8993234160625}{133791468}a^{12}+\frac{1772661825025}{635509473}a^{11}-\frac{47250002999953}{2542037892}a^{10}-\frac{560549969369}{847345964}a^{9}+\frac{458047855051}{100343601}a^{8}+\frac{385235865545}{5084075784}a^{7}-\frac{2902836956999}{2542037892}a^{6}+\frac{189101710019}{1906528419}a^{5}+\frac{966586914719}{7626113676}a^{4}+\frac{35530031831}{2542037892}a^{3}-\frac{9190099193}{635509473}a^{2}-\frac{166019599}{2542037892}a+\frac{3546841897}{7626113676}$, $\frac{211814885}{211836491}a^{31}+\frac{5184804385}{45756682056}a^{30}-13a^{29}-\frac{2817224452}{1906528419}a^{28}+129a^{27}+\frac{3108554917}{211836491}a^{26}-1170a^{25}-\frac{28202374902}{211836491}a^{24}+10218a^{23}+\frac{2217005970764}{1906528419}a^{22}-\frac{82913985695047}{1906528419}a^{21}-\frac{3159205502539}{635509473}a^{20}+171666a^{19}+\frac{1969047364037}{100343601}a^{18}-623922a^{17}-\frac{795696697172}{11149289}a^{16}+1799433a^{15}+\frac{2300034770943}{11149289}a^{14}-623922a^{13}-\frac{7752067341817}{100343601}a^{12}+\frac{316377489102161}{1906528419}a^{11}+\frac{33139906594375}{1906528419}a^{10}-43485a^{9}-\frac{9197960070289}{1906528419}a^{8}+10218a^{7}+\frac{248203520734}{211836491}a^{6}-1170a^{5}-\frac{28420357836}{211836491}a^{4}+129a^{3}+\frac{11850694973}{1906528419}a^{2}-\frac{41136152005}{1906528419}a-\frac{255954322843}{45756682056}$ Copy content Toggle raw display (assuming GRH)
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:  \( 19962612841466.996 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 19962612841466.996 \cdot 320}{20\cdot\sqrt{86898859856540647588737053982976000000000000000000000000}}\cr\approx \mathstrut & 0.202166047903848 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 - 13*x^30 + 129*x^28 - 1170*x^26 + 10218*x^24 - 43485*x^22 + 171666*x^20 - 623922*x^18 + 1799433*x^16 - 623922*x^14 + 171666*x^12 - 43485*x^10 + 10218*x^8 - 1170*x^6 + 129*x^4 - 13*x^2 + 1)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^32 - 13*x^30 + 129*x^28 - 1170*x^26 + 10218*x^24 - 43485*x^22 + 171666*x^20 - 623922*x^18 + 1799433*x^16 - 623922*x^14 + 171666*x^12 - 43485*x^10 + 10218*x^8 - 1170*x^6 + 129*x^4 - 13*x^2 + 1, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^32 - 13*x^30 + 129*x^28 - 1170*x^26 + 10218*x^24 - 43485*x^22 + 171666*x^20 - 623922*x^18 + 1799433*x^16 - 623922*x^14 + 171666*x^12 - 43485*x^10 + 10218*x^8 - 1170*x^6 + 129*x^4 - 13*x^2 + 1);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 13*x^30 + 129*x^28 - 1170*x^26 + 10218*x^24 - 43485*x^22 + 171666*x^20 - 623922*x^18 + 1799433*x^16 - 623922*x^14 + 171666*x^12 - 43485*x^10 + 10218*x^8 - 1170*x^6 + 129*x^4 - 13*x^2 + 1);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_2\times C_4^2$ (as 32T36):

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 32
The 32 conjugacy class representatives for $C_2\times C_4^2$
Character table for $C_2\times C_4^2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{-85}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-17}) \), \(\Q(i, \sqrt{85})\), 4.4.9826000.2, 4.0.614125.2, \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{17})\), \(\Q(\sqrt{5}, \sqrt{17})\), \(\Q(\sqrt{-5}, \sqrt{-17})\), 4.4.9826000.1, 4.0.614125.1, \(\Q(\sqrt{5}, \sqrt{-17})\), \(\Q(\sqrt{-5}, \sqrt{17})\), 4.0.36125.1, 4.4.578000.1, \(\Q(\zeta_{5})\), \(\Q(\zeta_{20})^+\), 4.0.1965200.1, 4.4.122825.1, 4.0.78608.1, 4.4.4913.1, 8.0.96550276000000.11, 8.0.13363360000.1, 8.0.96550276000000.12, 8.8.96550276000000.1, 8.0.96550276000000.4, 8.0.377149515625.1, 8.0.96550276000000.5, 8.0.334084000000.2, \(\Q(\zeta_{20})\), 8.0.3862011040000.3, 8.0.6179217664.1, 8.0.1305015625.1, 8.8.334084000000.1, 8.0.3862011040000.4, 8.8.15085980625.1, 8.0.334084000000.1, 8.0.334084000000.3, 8.0.3862011040000.2, 8.0.3862011040000.1, 16.0.9321955795676176000000000000.1, 16.0.111612119056000000000000.1, 16.0.14915129273081881600000000.1, 16.0.9321955795676176000000000000.2, 16.16.9321955795676176000000000000.1, 16.0.142241757136172119140625.1, 16.0.9321955795676176000000000000.3

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 R ${\href{/padicField/3.4.0.1}{4} }^{8}$ R ${\href{/padicField/7.4.0.1}{4} }^{8}$ ${\href{/padicField/11.4.0.1}{4} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{8}$ R ${\href{/padicField/19.2.0.1}{2} }^{16}$ ${\href{/padicField/23.4.0.1}{4} }^{8}$ ${\href{/padicField/29.4.0.1}{4} }^{8}$ ${\href{/padicField/31.4.0.1}{4} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{8}$ ${\href{/padicField/41.4.0.1}{4} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{8}$ ${\href{/padicField/59.2.0.1}{2} }^{16}$

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
\(2\) Copy content Toggle raw display 2.8.8.1$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
\(5\) Copy content Toggle raw display 5.16.12.1$x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$$4$$4$$12$$C_4^2$$[\ ]_{4}^{4}$
5.16.12.1$x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$$4$$4$$12$$C_4^2$$[\ ]_{4}^{4}$
\(17\) Copy content Toggle raw display 17.16.12.1$x^{16} + 28 x^{14} + 40 x^{13} + 374 x^{12} + 840 x^{11} + 1748 x^{10} - 1240 x^{9} + 19691 x^{8} + 32280 x^{7} + 73844 x^{6} + 54200 x^{5} + 457786 x^{4} + 211320 x^{3} + 955096 x^{2} + 33040 x + 1895241$$4$$4$$12$$C_4^2$$[\ ]_{4}^{4}$
17.16.12.1$x^{16} + 28 x^{14} + 40 x^{13} + 374 x^{12} + 840 x^{11} + 1748 x^{10} - 1240 x^{9} + 19691 x^{8} + 32280 x^{7} + 73844 x^{6} + 54200 x^{5} + 457786 x^{4} + 211320 x^{3} + 955096 x^{2} + 33040 x + 1895241$$4$$4$$12$$C_4^2$$[\ ]_{4}^{4}$