Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + x^30 - 14*x^28 - 103*x^26 + 62*x^24 + 736*x^22 + 1892*x^20 + 4382*x^18 + 7111*x^16 + 4382*x^14 + 1892*x^12 + 736*x^10 + 62*x^8 - 103*x^6 - 14*x^4 + x^2 + 1)
gp: K = bnfinit(y^32 + y^30 - 14*y^28 - 103*y^26 + 62*y^24 + 736*y^22 + 1892*y^20 + 4382*y^18 + 7111*y^16 + 4382*y^14 + 1892*y^12 + 736*y^10 + 62*y^8 - 103*y^6 - 14*y^4 + y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 + x^30 - 14*x^28 - 103*x^26 + 62*x^24 + 736*x^22 + 1892*x^20 + 4382*x^18 + 7111*x^16 + 4382*x^14 + 1892*x^12 + 736*x^10 + 62*x^8 - 103*x^6 - 14*x^4 + x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 + x^30 - 14*x^28 - 103*x^26 + 62*x^24 + 736*x^22 + 1892*x^20 + 4382*x^18 + 7111*x^16 + 4382*x^14 + 1892*x^12 + 736*x^10 + 62*x^8 - 103*x^6 - 14*x^4 + x^2 + 1)
\( x^{32} + x^{30} - 14 x^{28} - 103 x^{26} + 62 x^{24} + 736 x^{22} + 1892 x^{20} + 4382 x^{18} + 7111 x^{16} + \cdots + 1 \)
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: |
\(5512695749115635425536000000000000000000000000\)
\(\medspace = 2^{32}\cdot 3^{16}\cdot 5^{24}\cdot 29^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(26.88\) | 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: | $2\cdot 3^{1/2}5^{3/4}29^{1/2}\approx 62.37594312593305$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(29\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{32768}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3}a^{16}-\frac{1}{3}a^{14}+\frac{1}{3}a^{10}-\frac{1}{3}a^{8}+\frac{1}{3}a^{6}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{17}-\frac{1}{3}a^{15}+\frac{1}{3}a^{11}-\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{3}a^{18}-\frac{1}{3}a^{14}+\frac{1}{3}a^{12}+\frac{1}{3}a^{6}-\frac{1}{3}a^{4}+\frac{1}{3}$, $\frac{1}{3}a^{19}-\frac{1}{3}a^{15}+\frac{1}{3}a^{13}+\frac{1}{3}a^{7}-\frac{1}{3}a^{5}+\frac{1}{3}a$, $\frac{1}{12}a^{20}-\frac{5}{12}a^{10}+\frac{1}{12}$, $\frac{1}{12}a^{21}-\frac{5}{12}a^{11}+\frac{1}{12}a$, $\frac{1}{12}a^{22}-\frac{5}{12}a^{12}+\frac{1}{12}a^{2}$, $\frac{1}{12}a^{23}-\frac{5}{12}a^{13}+\frac{1}{12}a^{3}$, $\frac{1}{1440}a^{24}+\frac{1}{96}a^{22}+\frac{1}{32}a^{20}-\frac{41}{360}a^{18}+\frac{1}{24}a^{16}-\frac{11}{96}a^{14}-\frac{419}{1440}a^{12}-\frac{35}{96}a^{10}-\frac{7}{24}a^{8}-\frac{161}{360}a^{6}+\frac{11}{96}a^{4}+\frac{11}{32}a^{2}-\frac{599}{1440}$, $\frac{1}{1440}a^{25}+\frac{1}{96}a^{23}+\frac{1}{32}a^{21}-\frac{41}{360}a^{19}+\frac{1}{24}a^{17}-\frac{11}{96}a^{15}-\frac{419}{1440}a^{13}-\frac{35}{96}a^{11}-\frac{7}{24}a^{9}-\frac{161}{360}a^{7}+\frac{11}{96}a^{5}+\frac{11}{32}a^{3}-\frac{599}{1440}a$, $\frac{1}{41760}a^{26}-\frac{1}{6960}a^{24}+\frac{35}{1392}a^{22}-\frac{389}{41760}a^{20}+\frac{21}{290}a^{18}-\frac{53}{928}a^{16}-\frac{3517}{20880}a^{14}-\frac{761}{6960}a^{12}-\frac{653}{2784}a^{10}+\frac{391}{2610}a^{8}+\frac{881}{4640}a^{6}-\frac{35}{1392}a^{4}-\frac{5797}{20880}a^{2}+\frac{2513}{13920}$, $\frac{1}{41760}a^{27}-\frac{1}{6960}a^{25}+\frac{35}{1392}a^{23}-\frac{389}{41760}a^{21}+\frac{21}{290}a^{19}-\frac{53}{928}a^{17}-\frac{3517}{20880}a^{15}-\frac{761}{6960}a^{13}-\frac{653}{2784}a^{11}+\frac{391}{2610}a^{9}+\frac{881}{4640}a^{7}-\frac{35}{1392}a^{5}-\frac{5797}{20880}a^{3}+\frac{2513}{13920}a$, $\frac{1}{187251840}a^{28}+\frac{61}{6241728}a^{26}+\frac{21589}{62417280}a^{24}-\frac{2103697}{93625920}a^{22}+\frac{29465}{4161152}a^{20}+\frac{4520329}{62417280}a^{18}+\frac{13057283}{93625920}a^{16}+\frac{1966453}{4161152}a^{14}+\frac{1974939}{10402880}a^{12}+\frac{79457851}{187251840}a^{10}+\frac{1500217}{4161152}a^{8}-\frac{10249163}{31208640}a^{6}+\frac{70804351}{187251840}a^{4}-\frac{233107}{6241728}a^{2}+\frac{6014539}{62417280}$, $\frac{1}{187251840}a^{29}+\frac{61}{6241728}a^{27}+\frac{21589}{62417280}a^{25}-\frac{2103697}{93625920}a^{23}+\frac{29465}{4161152}a^{21}+\frac{4520329}{62417280}a^{19}+\frac{13057283}{93625920}a^{17}+\frac{1966453}{4161152}a^{15}+\frac{1974939}{10402880}a^{13}+\frac{79457851}{187251840}a^{11}+\frac{1500217}{4161152}a^{9}-\frac{10249163}{31208640}a^{7}+\frac{70804351}{187251840}a^{5}-\frac{233107}{6241728}a^{3}+\frac{6014539}{62417280}a$, $\frac{1}{29024035200}a^{30}+\frac{1}{468129600}a^{28}+\frac{10907}{3224892800}a^{26}+\frac{289063}{2902403520}a^{24}+\frac{834797477}{29024035200}a^{22}+\frac{3838369}{333609600}a^{20}+\frac{78744139}{2902403520}a^{18}-\frac{3148591843}{29024035200}a^{16}-\frac{131615707}{4837339200}a^{14}+\frac{2496327887}{5804807040}a^{12}-\frac{4970558263}{29024035200}a^{10}-\frac{9887309}{1612446400}a^{8}-\frac{996093481}{5804807040}a^{6}+\frac{4077201151}{14512017600}a^{4}+\frac{31126021}{312086400}a^{2}-\frac{1088448923}{2418669600}$, $\frac{1}{29024035200}a^{31}+\frac{1}{468129600}a^{29}+\frac{10907}{3224892800}a^{27}+\frac{289063}{2902403520}a^{25}+\frac{834797477}{29024035200}a^{23}+\frac{3838369}{333609600}a^{21}+\frac{78744139}{2902403520}a^{19}-\frac{3148591843}{29024035200}a^{17}-\frac{131615707}{4837339200}a^{15}+\frac{2496327887}{5804807040}a^{13}-\frac{4970558263}{29024035200}a^{11}-\frac{9887309}{1612446400}a^{9}-\frac{996093481}{5804807040}a^{7}+\frac{4077201151}{14512017600}a^{5}+\frac{31126021}{312086400}a^{3}-\frac{1088448923}{2418669600}a$
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
$C_{2}\times C_{4}$, which has order $8$ (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: |
\( \frac{175336487}{403111600} a^{31} + \frac{5607073}{12319200} a^{29} - \frac{14720496029}{2418669600} a^{27} - \frac{7270586087}{161244640} a^{25} + \frac{91183167731}{3628004400} a^{23} + \frac{20557972577}{63649200} a^{21} + \frac{134761567491}{161244640} a^{19} + \frac{14022415998067}{7256008800} a^{17} + \frac{7634559615361}{2418669600} a^{15} + \frac{160623897541}{80622320} a^{13} + \frac{3010924729811}{3628004400} a^{11} + \frac{833451461371}{2418669600} a^{9} + \frac{1194068025}{32248928} a^{7} - \frac{347715096863}{7256008800} a^{5} - \frac{263695027}{39010800} a^{3} + \frac{81472599}{201555800} a \)
(order $60$)
| 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: |
$\frac{1349327}{7637904}a^{30}+\frac{192031}{1231920}a^{28}-\frac{18919439}{7637904}a^{26}-\frac{1366472867}{76379040}a^{24}+\frac{16641793}{1294560}a^{22}+\frac{1940310323}{15275808}a^{20}+\frac{12229289849}{38189520}a^{18}+\frac{28526856001}{38189520}a^{16}+\frac{18266570483}{15275808}a^{14}+\frac{53282180833}{76379040}a^{12}+\frac{461294323}{1294560}a^{10}+\frac{1181155003}{7637904}a^{8}+\frac{1030608719}{38189520}a^{6}-\frac{761364583}{76379040}a^{4}-\frac{252679}{492768}a^{2}+\frac{7529273}{76379040}$, $\frac{753437549}{4837339200}a^{30}+\frac{199723943}{936259200}a^{28}-\frac{200937167}{95473800}a^{26}-\frac{97681228169}{5804807040}a^{24}+\frac{663596959}{190947600}a^{22}+\frac{3374833595767}{29024035200}a^{20}+\frac{1961386685359}{5804807040}a^{18}+\frac{5840262715247}{7256008800}a^{16}+\frac{40539573422437}{29024035200}a^{14}+\frac{342178380383}{290240352}a^{12}+\frac{20109847056233}{29024035200}a^{10}+\frac{9624918202957}{29024035200}a^{8}+\frac{81067030991}{725600880}a^{6}+\frac{331988665043}{29024035200}a^{4}-\frac{145547671}{117032400}a^{2}-\frac{35394589159}{29024035200}$, $\frac{172379353}{4837339200}a^{30}+\frac{13896151}{936259200}a^{28}-\frac{467216543}{907001100}a^{26}-\frac{3917610745}{1160961408}a^{24}+\frac{31110261769}{7256008800}a^{22}+\frac{710042366819}{29024035200}a^{20}+\frac{303994918351}{5804807040}a^{18}+\frac{872091803479}{7256008800}a^{16}+\frac{4967698026809}{29024035200}a^{14}+\frac{10460556253}{362800440}a^{12}+\frac{15388165999}{1527580800}a^{10}+\frac{266539075949}{29024035200}a^{8}-\frac{2147311417}{725600880}a^{6}-\frac{260234611}{491932800}a^{4}+\frac{53512049}{12319200}a^{2}+\frac{30254625917}{29024035200}$, $\frac{19860983}{151166850}a^{30}+\frac{399189}{3250900}a^{28}-\frac{3336271687}{1814002200}a^{26}-\frac{6483985279}{483733920}a^{24}+\frac{21545508811}{2418669600}a^{22}+\frac{691263750887}{7256008800}a^{20}+\frac{29367445093}{120933480}a^{18}+\frac{14255702931}{25194475}a^{16}+\frac{6624813686357}{7256008800}a^{14}+\frac{89154382223}{161244640}a^{12}+\frac{34709227489}{127298400}a^{10}+\frac{215127285263}{1814002200}a^{8}+\frac{252201859}{12093348}a^{6}-\frac{18464609239}{2418669600}a^{4}-\frac{4859467}{12319200}a^{2}+\frac{179715337}{2418669600}$, $\frac{115}{18848}a^{31}+\frac{41338}{1281075}a^{30}+\frac{24059}{317376}a^{29}-\frac{17552}{812725}a^{28}+\frac{295}{43152}a^{27}-\frac{12965323}{25194475}a^{26}-\frac{15535129}{9838656}a^{25}-\frac{682929}{265205}a^{24}-\frac{230269}{32364}a^{23}+\frac{193386523}{25194475}a^{22}+\frac{21153565}{3279552}a^{21}+\frac{536302137}{25194475}a^{20}+\frac{629384111}{9838656}a^{19}+\frac{5444508}{265205}a^{18}+\frac{53946997}{307458}a^{17}+\frac{838982143}{25194475}a^{16}+\frac{430129897}{1093184}a^{15}-\frac{536052493}{25194475}a^{14}+\frac{1539425189}{2459664}a^{13}-\frac{73225773}{265205}a^{12}+\frac{4804511849}{9838656}a^{11}-\frac{5736807612}{25194475}a^{10}+\frac{828068995}{3279552}a^{9}-\frac{2569458748}{25194475}a^{8}+\frac{253854209}{2459664}a^{7}-\frac{14095761}{265205}a^{6}+\frac{258381239}{9838656}a^{5}-\frac{335169527}{25194475}a^{4}-\frac{6785}{6612}a^{3}+\frac{2503577}{812725}a^{2}-\frac{23774599}{9838656}a+\frac{16468333}{75583425}$, $\frac{226503101}{2902403520}a^{31}-\frac{1349327}{7637904}a^{30}-\frac{483299}{37450368}a^{29}-\frac{192031}{1231920}a^{28}-\frac{864907169}{725600880}a^{27}+\frac{18919439}{7637904}a^{26}-\frac{39276589183}{5804807040}a^{25}+\frac{1366472867}{76379040}a^{24}+\frac{4158438913}{290240352}a^{23}-\frac{16641793}{1294560}a^{22}+\frac{305155159973}{5804807040}a^{21}-\frac{1940310323}{15275808}a^{20}+\frac{461560139357}{5804807040}a^{19}-\frac{12229289849}{38189520}a^{18}+\frac{47749375063}{290240352}a^{17}-\frac{28526856001}{38189520}a^{16}+\frac{843303867383}{5804807040}a^{15}-\frac{18266570483}{15275808}a^{14}-\frac{240482924861}{725600880}a^{13}-\frac{53282180833}{76379040}a^{12}-\frac{333335065985}{1160961408}a^{11}-\frac{461294323}{1294560}a^{10}-\frac{35917215223}{305516160}a^{9}-\frac{1181155003}{7637904}a^{8}-\frac{13127844779}{181400220}a^{7}-\frac{1030608719}{38189520}a^{6}-\frac{1093778705}{61103232}a^{5}+\frac{761364583}{76379040}a^{4}+\frac{448709447}{46812960}a^{3}+\frac{252679}{492768}a^{2}+\frac{1895468255}{1160961408}a+\frac{68849767}{76379040}$, $\frac{2051447}{120933480}a^{31}+\frac{117455431}{907001100}a^{30}-\frac{103325}{1560432}a^{29}+\frac{23682281}{234064800}a^{28}-\frac{32912035}{96746784}a^{27}-\frac{703969169}{381895200}a^{26}-\frac{9241099}{15275808}a^{25}-\frac{18784094437}{1451201760}a^{24}+\frac{956902765}{96746784}a^{23}+\frac{2219948}{202275}a^{22}+\frac{453563327}{48373392}a^{21}+\frac{170088447721}{1814002200}a^{20}-\frac{229683227}{7637904}a^{19}+\frac{325107983609}{1451201760}a^{18}-\frac{3149620803}{32248928}a^{17}+\frac{3717848292071}{7256008800}a^{16}-\frac{27342305449}{96746784}a^{15}+\frac{5754581791849}{7256008800}a^{14}-\frac{9275605285}{15275808}a^{13}+\frac{64799699617}{181400220}a^{12}-\frac{23306909891}{48373392}a^{11}+\frac{3264415051}{30745800}a^{10}-\frac{11926249969}{48373392}a^{9}+\frac{233609191939}{7256008800}a^{8}-\frac{1605838687}{15275808}a^{7}-\frac{28397359543}{1451201760}a^{6}-\frac{857963673}{32248928}a^{5}-\frac{160304028469}{7256008800}a^{4}+\frac{6950445}{1040288}a^{3}-\frac{2464259}{58516200}a^{2}+\frac{1773300293}{725600880}a+\frac{201328969}{907001100}$, $\frac{9016238977}{14512017600}a^{31}-\frac{39419117}{1612446400}a^{30}+\frac{166273511}{312086400}a^{29}-\frac{95507917}{936259200}a^{28}-\frac{31853375491}{3628004400}a^{27}+\frac{2035564483}{7256008800}a^{26}-\frac{12557543807}{200165760}a^{25}+\frac{21009777007}{5804807040}a^{24}+\frac{57505952417}{1209334800}a^{23}+\frac{45339636227}{7256008800}a^{22}+\frac{221823001823}{491932800}a^{21}-\frac{709035523733}{29024035200}a^{20}+\frac{1288198139689}{1160961408}a^{19}-\frac{118745634097}{1160961408}a^{18}+\frac{2064730290423}{806223200}a^{17}-\frac{54872906849}{226750275}a^{16}+\frac{117588433743127}{29024035200}a^{15}-\frac{14057810737763}{29024035200}a^{14}+\frac{3110029239809}{1451201760}a^{13}-\frac{429318742303}{725600880}a^{12}+\frac{8538777281641}{9674678400}a^{11}-\frac{425863541833}{1527580800}a^{10}+\frac{193135093733}{491932800}a^{9}-\frac{3090581343743}{29024035200}a^{8}+\frac{977427478}{45350055}a^{7}-\frac{61362759899}{1451201760}a^{6}-\frac{490379878789}{9674678400}a^{5}+\frac{203951585483}{29024035200}a^{4}+\frac{922408259}{117032400}a^{3}+\frac{3147683}{424800}a^{2}+\frac{18166263011}{29024035200}a-\frac{14566357579}{29024035200}$, $\frac{1117593917}{3628004400}a^{30}+\frac{32613749}{117032400}a^{28}-\frac{15659276899}{3628004400}a^{26}-\frac{4541494031}{145120176}a^{24}+\frac{78956888099}{3628004400}a^{22}+\frac{201613894289}{907001100}a^{20}+\frac{408561177689}{725600880}a^{18}+\frac{4762872542159}{3628004400}a^{16}+\frac{7650717332891}{3628004400}a^{14}+\frac{907375437917}{725600880}a^{12}+\frac{30026188469}{47736900}a^{10}+\frac{991303196951}{3628004400}a^{8}+\frac{34713677201}{725600880}a^{6}-\frac{63861731551}{3628004400}a^{4}-\frac{5588221}{6159600}a^{2}+\frac{627213473}{3628004400}$, $\frac{4711005337}{29024035200}a^{30}+\frac{62201657}{468129600}a^{28}-\frac{3509464411}{1527580800}a^{26}-\frac{47305797203}{2902403520}a^{24}+\frac{337513409}{25891200}a^{22}+\frac{3399844198271}{29024035200}a^{20}+\frac{828399670699}{2902403520}a^{18}+\frac{19129672088449}{29024035200}a^{16}+\frac{15030480988153}{14512017600}a^{14}+\frac{3053986232771}{5804807040}a^{12}+\frac{108329535251}{491932800}a^{10}+\frac{1436698047883}{14512017600}a^{8}+\frac{1687409575}{1160961408}a^{6}-\frac{240102655843}{14512017600}a^{4}+\frac{1647500291}{936259200}a^{2}+\frac{40227301}{83402400}$, $\frac{7529273}{76379040}a^{31}-\frac{1526655901}{9674678400}a^{30}-\frac{64129}{821280}a^{29}-\frac{54342797}{312086400}a^{28}-\frac{1222039}{795615}a^{27}+\frac{1107907673}{509193600}a^{26}-\frac{195440243}{25459680}a^{25}+\frac{95564600689}{5804807040}a^{24}+\frac{611095931}{25459680}a^{23}-\frac{1334618071}{169731200}a^{22}+\frac{1519893047}{25459680}a^{21}-\frac{279043897657}{2418669600}a^{20}+\frac{1514610967}{25459680}a^{19}-\frac{1808001257747}{5804807040}a^{18}+\frac{711224549}{6364920}a^{17}-\frac{7112638381127}{9674678400}a^{16}-\frac{1171017233}{25459680}a^{15}-\frac{11870475690283}{9674678400}a^{14}-\frac{19446526043}{25459680}a^{13}-\frac{5135782952099}{5804807040}a^{12}-\frac{4337421813}{8486560}a^{11}-\frac{1203665569583}{2418669600}a^{10}-\frac{7224940043}{25459680}a^{9}-\frac{2231842938463}{9674678400}a^{8}-\frac{118174324}{795615}a^{7}-\frac{66869864335}{1160961408}a^{6}-\frac{945577519}{25459680}a^{5}-\frac{50110966097}{9674678400}a^{4}+\frac{7053277}{821280}a^{3}+\frac{260113987}{312086400}a^{2}+\frac{23347259}{38189520}a+\frac{17865701213}{29024035200}$, $\frac{166856371}{1000828800}a^{31}+\frac{329520751}{14512017600}a^{30}+\frac{134562949}{468129600}a^{29}-\frac{841621}{936259200}a^{28}-\frac{64971939023}{29024035200}a^{27}-\frac{2496260291}{7256008800}a^{26}-\frac{54798792289}{2902403520}a^{25}-\frac{68136231}{33946240}a^{24}-\frac{50534975077}{29024035200}a^{23}+\frac{14062128313}{3628004400}a^{22}+\frac{3851195214937}{29024035200}a^{21}+\frac{448608691591}{29024035200}a^{20}+\frac{233381161375}{580480704}a^{19}+\frac{2554620619}{101838720}a^{18}+\frac{27314568499343}{29024035200}a^{17}+\frac{193811535433}{3628004400}a^{16}+\frac{24259255670291}{14512017600}a^{15}+\frac{1616794678201}{29024035200}a^{14}+\frac{8647238295601}{5804807040}a^{13}-\frac{627794031}{8486560}a^{12}+\frac{20060320963463}{29024035200}a^{11}-\frac{1913738300551}{29024035200}a^{10}+\frac{4068791261351}{14512017600}a^{9}-\frac{21544123891}{1000828800}a^{8}+\frac{394501041949}{5804807040}a^{7}-\frac{88105481}{5091936}a^{6}-\frac{265065585401}{14512017600}a^{5}-\frac{123486939121}{29024035200}a^{4}-\frac{9425082623}{936259200}a^{3}+\frac{99427967}{117032400}a^{2}+\frac{1063139171}{806223200}a+\frac{11241886633}{29024035200}$, $\frac{115}{18848}a^{31}-\frac{8543373751}{29024035200}a^{30}+\frac{24059}{317376}a^{29}-\frac{159466841}{468129600}a^{28}+\frac{295}{43152}a^{27}+\frac{39508955909}{9674678400}a^{26}-\frac{15535129}{9838656}a^{25}+\frac{29959976399}{967467840}a^{24}-\frac{230269}{32364}a^{23}-\frac{395135903707}{29024035200}a^{22}+\frac{21153565}{3279552}a^{21}-\frac{710547490457}{3224892800}a^{20}+\frac{629384111}{9838656}a^{19}-\frac{190145199511}{322489280}a^{18}+\frac{53946997}{307458}a^{17}-\frac{39808889954287}{29024035200}a^{16}+\frac{430129897}{1093184}a^{15}-\frac{3683737474551}{1612446400}a^{14}+\frac{1539425189}{2459664}a^{13}-\frac{3082250764699}{1934935680}a^{12}+\frac{4804511849}{9838656}a^{11}-\frac{21107715795967}{29024035200}a^{10}+\frac{828068995}{3279552}a^{9}-\frac{1515322636033}{4837339200}a^{8}+\frac{253854209}{2459664}a^{7}-\frac{110993591263}{1934935680}a^{6}+\frac{258381239}{9838656}a^{5}+\frac{292556963509}{14512017600}a^{4}-\frac{6785}{6612}a^{3}+\frac{1692898709}{312086400}a^{2}-\frac{33613255}{9838656}a-\frac{3413912081}{7256008800}$, $\frac{1022086609}{7256008800}a^{31}-\frac{9737657951}{29024035200}a^{30}-\frac{2978593}{58516200}a^{29}-\frac{9536923}{32284800}a^{28}-\frac{15608957183}{7256008800}a^{27}+\frac{136879059457}{29024035200}a^{26}-\frac{5719440703}{483733920}a^{25}+\frac{197199742157}{5804807040}a^{24}+\frac{102666451789}{3628004400}a^{23}-\frac{715072092617}{29024035200}a^{22}+\frac{328894421801}{3628004400}a^{21}-\frac{439417476413}{1814002200}a^{20}+\frac{15403778327}{120933480}a^{19}-\frac{703929131183}{1160961408}a^{18}+\frac{1906480578923}{7256008800}a^{17}-\frac{40928104945397}{29024035200}a^{16}+\frac{44183739293}{250207200}a^{15}-\frac{65347893161213}{29024035200}a^{14}-\frac{11486543251}{16124464}a^{13}-\frac{7435797702199}{5804807040}a^{12}-\frac{101900011739}{190947600}a^{11}-\frac{58951253063}{95473800}a^{10}-\frac{255619647151}{907001100}a^{9}-\frac{7990383110693}{29024035200}a^{8}-\frac{72262662629}{483733920}a^{7}-\frac{3347892689}{98386560}a^{6}-\frac{157273627}{4240800}a^{5}+\frac{511648249033}{29024035200}a^{4}+\frac{69323459}{6159600}a^{3}-\frac{57008297}{49276800}a^{2}+\frac{311763919}{250207200}a+\frac{3252979147}{9674678400}$, $\frac{6790193}{40994400}a^{31}-\frac{551143}{226750275}a^{30}+\frac{2916179}{29258100}a^{29}-\frac{345121}{19505400}a^{28}-\frac{937045699}{403111600}a^{27}+\frac{11496413}{806223200}a^{26}-\frac{11673628297}{725600880}a^{25}+\frac{34954589}{76379040}a^{24}+\frac{117627530707}{7256008800}a^{23}+\frac{1196101197}{806223200}a^{22}+\frac{135162815293}{1209334800}a^{21}-\frac{1357149529}{604667400}a^{20}+\frac{97850861467}{362800440}a^{19}-\frac{38026352}{2386845}a^{18}+\frac{2337789017231}{3628004400}a^{17}-\frac{104133265619}{2418669600}a^{16}+\frac{63317521967}{63649200}a^{15}-\frac{75748591167}{806223200}a^{14}+\frac{728492262637}{1451201760}a^{13}-\frac{2167209935}{15275808}a^{12}+\frac{1484865600221}{3628004400}a^{11}-\frac{22237951567}{201555800}a^{10}+\frac{58697712607}{302333700}a^{9}-\frac{18212781667}{302333700}a^{8}+\frac{34852816997}{725600880}a^{7}-\frac{1769611273}{76379040}a^{6}+\frac{48002540441}{3628004400}a^{5}-\frac{14275137359}{2418669600}a^{4}+\frac{387667231}{78021600}a^{3}+\frac{120935189}{78021600}a^{2}-\frac{24569755261}{7256008800}a+\frac{164244239}{302333700}$
| 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: | \( 22391784240.2237 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 22391784240.2237 \cdot 8}{60\cdot\sqrt{5512695749115635425536000000000000000000000000}}\cr\approx \mathstrut & 0.237259340616545 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^32 + x^30 - 14*x^28 - 103*x^26 + 62*x^24 + 736*x^22 + 1892*x^20 + 4382*x^18 + 7111*x^16 + 4382*x^14 + 1892*x^12 + 736*x^10 + 62*x^8 - 103*x^6 - 14*x^4 + 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 + x^30 - 14*x^28 - 103*x^26 + 62*x^24 + 736*x^22 + 1892*x^20 + 4382*x^18 + 7111*x^16 + 4382*x^14 + 1892*x^12 + 736*x^10 + 62*x^8 - 103*x^6 - 14*x^4 + 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 + x^30 - 14*x^28 - 103*x^26 + 62*x^24 + 736*x^22 + 1892*x^20 + 4382*x^18 + 7111*x^16 + 4382*x^14 + 1892*x^12 + 736*x^10 + 62*x^8 - 103*x^6 - 14*x^4 + 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 + x^30 - 14*x^28 - 103*x^26 + 62*x^24 + 736*x^22 + 1892*x^20 + 4382*x^18 + 7111*x^16 + 4382*x^14 + 1892*x^12 + 736*x^10 + 62*x^8 - 103*x^6 - 14*x^4 + 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
$C_2^4:C_4$ (as 32T262):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A solvable group of order 64 |
| The 40 conjugacy class representatives for $C_2^4:C_4$ |
| Character table for $C_2^4:C_4$ is not computed |
Intermediate fields
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]
Sibling fields
| Degree 32 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | R | ${\href{/padicField/7.4.0.1}{4} }^{8}$ | ${\href{/padicField/11.2.0.1}{2} }^{16}$ | ${\href{/padicField/13.4.0.1}{4} }^{8}$ | ${\href{/padicField/17.4.0.1}{4} }^{8}$ | ${\href{/padicField/19.2.0.1}{2} }^{16}$ | ${\href{/padicField/23.4.0.1}{4} }^{8}$ | R | ${\href{/padicField/31.2.0.1}{2} }^{16}$ | ${\href{/padicField/37.4.0.1}{4} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{16}$ | ${\href{/padicField/43.4.0.1}{4} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{8}$ | ${\href{/padicField/59.2.0.1}{2} }^{16}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 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 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
|
\(3\)
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(5\)
| 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
|
\(29\)
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |