LMFDB - Number field 16.8.11654783511381160590876241.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 2*x^14 - 33*x^13 + 53*x^12 + 83*x^11 - 77*x^10 + 51*x^9 - 1774*x^8 + 1850*x^7 + 5383*x^6 - 6162*x^5 - 5048*x^4 + 6520*x^3 + 992*x^2 - 3360*x - 576)

gp: K = bnfinit(y^16 - 2*y^15 + 2*y^14 - 33*y^13 + 53*y^12 + 83*y^11 - 77*y^10 + 51*y^9 - 1774*y^8 + 1850*y^7 + 5383*y^6 - 6162*y^5 - 5048*y^4 + 6520*y^3 + 992*y^2 - 3360*y - 576, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 2*x^14 - 33*x^13 + 53*x^12 + 83*x^11 - 77*x^10 + 51*x^9 - 1774*x^8 + 1850*x^7 + 5383*x^6 - 6162*x^5 - 5048*x^4 + 6520*x^3 + 992*x^2 - 3360*x - 576);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 + 2*x^14 - 33*x^13 + 53*x^12 + 83*x^11 - 77*x^10 + 51*x^9 - 1774*x^8 + 1850*x^7 + 5383*x^6 - 6162*x^5 - 5048*x^4 + 6520*x^3 + 992*x^2 - 3360*x - 576)

$ x^{16} - 2 x^{15} + 2 x^{14} - 33 x^{13} + 53 x^{12} + 83 x^{11} - 77 x^{10} + 51 x^{9} - 1774 x^{8} + \cdots - 576 $ x^16 - 2*x^15 + 2*x^14 - 33*x^13 + 53*x^12 + 83*x^11 - 77*x^10 + 51*x^9 - 1774*x^8 + 1850*x^7 + 5383*x^6 - 6162*x^5 - 5048*x^4 + 6520*x^3 + 992*x^2 - 3360*x - 576

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$16$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[8, 4]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$11654783511381160590876241$ 11654783511381160590876241 $\medspace = 13^{12}\cdot 29^{8}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$36.87$	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:	$13^{3/4}29^{1/2}\approx 36.868588674506164$
Ramified primes:	$13$, $29$ 13, 29	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Aut(K/\Q) }$:	$8$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{6}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{6}a$, $\frac{1}{12}a^{10}-\frac{1}{12}a^{9}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}+\frac{5}{12}a^{2}-\frac{1}{6}a$, $\frac{1}{12}a^{11}-\frac{1}{12}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}+\frac{1}{6}a^{3}+\frac{1}{4}a^{2}-\frac{1}{6}a$, $\frac{1}{72}a^{12}+\frac{1}{36}a^{11}-\frac{1}{36}a^{10}-\frac{1}{72}a^{9}-\frac{1}{24}a^{8}-\frac{1}{8}a^{7}+\frac{5}{24}a^{6}+\frac{1}{24}a^{5}+\frac{1}{36}a^{4}+\frac{5}{36}a^{3}+\frac{11}{72}a^{2}-\frac{13}{36}a-\frac{1}{6}$, $\frac{1}{144}a^{13}-\frac{1}{24}a^{11}+\frac{1}{48}a^{10}+\frac{11}{144}a^{9}+\frac{11}{48}a^{8}-\frac{1}{48}a^{7}-\frac{3}{16}a^{6}+\frac{17}{36}a^{5}+\frac{7}{24}a^{4}+\frac{3}{16}a^{3}-\frac{1}{12}a^{2}-\frac{1}{18}a+\frac{1}{6}$, $\frac{1}{145728}a^{14}-\frac{85}{36432}a^{13}-\frac{181}{72864}a^{12}+\frac{659}{145728}a^{11}-\frac{707}{48576}a^{10}+\frac{1747}{48576}a^{9}+\frac{6551}{48576}a^{8}+\frac{3079}{48576}a^{7}-\frac{8491}{36432}a^{6}+\frac{11255}{72864}a^{5}+\frac{1}{13248}a^{4}+\frac{295}{2277}a^{3}-\frac{333}{4048}a^{2}-\frac{25}{66}a-\frac{265}{1012}$, $\frac{1}{31\!\cdots\!68}a^{15}+\frac{4220154683}{15\!\cdots\!84}a^{14}-\frac{275904944855}{526916805505728}a^{13}-\frac{15227346880369}{31\!\cdots\!68}a^{12}+\frac{14191788313255}{10\!\cdots\!56}a^{11}+\frac{127705446040471}{31\!\cdots\!68}a^{10}+\frac{216942698160391}{31\!\cdots\!68}a^{9}-\frac{2827875056255}{15272950884224}a^{8}-\frac{7180571555}{634584671424}a^{7}+\frac{780734529811543}{15\!\cdots\!84}a^{6}+\frac{134117729733023}{351277870337152}a^{5}-\frac{447339078375385}{15\!\cdots\!84}a^{4}-\frac{33129623514013}{263458402752864}a^{3}+\frac{165953502262909}{395187604129296}a^{2}+\frac{76874340142937}{197593802064648}a-\frac{8294874976735}{32932300344108}$ 1, a, a^2, a^3, a^4, a^5, a^6, 1/2*a^7 - 1/2*a^5 - 1/2*a^4 - 1/2*a^3 - 1/2*a, 1/2*a^8 - 1/2*a^6 - 1/2*a^5 - 1/2*a^4 - 1/2*a^2, 1/6*a^9 - 1/2*a^6 - 1/2*a^4 - 1/6*a, 1/12*a^10 - 1/12*a^9 - 1/4*a^6 - 1/2*a^4 - 1/4*a^3 + 5/12*a^2 - 1/6*a, 1/12*a^11 - 1/12*a^9 - 1/4*a^7 - 1/4*a^6 - 1/2*a^5 + 1/4*a^4 + 1/6*a^3 + 1/4*a^2 - 1/6*a, 1/72*a^12 + 1/36*a^11 - 1/36*a^10 - 1/72*a^9 - 1/24*a^8 - 1/8*a^7 + 5/24*a^6 + 1/24*a^5 + 1/36*a^4 + 5/36*a^3 + 11/72*a^2 - 13/36*a - 1/6, 1/144*a^13 - 1/24*a^11 + 1/48*a^10 + 11/144*a^9 + 11/48*a^8 - 1/48*a^7 - 3/16*a^6 + 17/36*a^5 + 7/24*a^4 + 3/16*a^3 - 1/12*a^2 - 1/18*a + 1/6, 1/145728*a^14 - 85/36432*a^13 - 181/72864*a^12 + 659/145728*a^11 - 707/48576*a^10 + 1747/48576*a^9 + 6551/48576*a^8 + 3079/48576*a^7 - 8491/36432*a^6 + 11255/72864*a^5 + 1/13248*a^4 + 295/2277*a^3 - 333/4048*a^2 - 25/66*a - 265/1012, 1/3161500833034368*a^15 + 4220154683/1580750416517184*a^14 - 275904944855/526916805505728*a^13 - 15227346880369/3161500833034368*a^12 + 14191788313255/1053833611011456*a^11 + 127705446040471/3161500833034368*a^10 + 216942698160391/3161500833034368*a^9 - 2827875056255/15272950884224*a^8 - 7180571555/634584671424*a^7 + 780734529811543/1580750416517184*a^6 + 134117729733023/351277870337152*a^5 - 447339078375385/1580750416517184*a^4 - 33129623514013/263458402752864*a^3 + 165953502262909/395187604129296*a^2 + 76874340142937/197593802064648*a - 8294874976735/32932300344108

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$2$, $3$

Class group and class number

Trivial group, which has order $1$ (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:	$11$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -1 $ -1 (order $2$)	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:	$\frac{1923825376543}{31\!\cdots\!68}a^{15}-\frac{3317678706115}{790375208258592}a^{14}+\frac{3859456791983}{526916805505728}a^{13}-\frac{98810200001779}{31\!\cdots\!68}a^{12}+\frac{48404253572881}{351277870337152}a^{11}-\frac{44517073143137}{351277870337152}a^{10}-\frac{385686802021637}{31\!\cdots\!68}a^{9}+\frac{91438900592201}{10\!\cdots\!56}a^{8}-\frac{468376694791}{317292335712}a^{7}+\frac{10\!\cdots\!57}{15\!\cdots\!84}a^{6}-\frac{11\!\cdots\!23}{351277870337152}a^{5}-\frac{22\!\cdots\!31}{197593802064648}a^{4}+\frac{927166667412225}{87819467584288}a^{3}+\frac{56419442192967}{21954866896072}a^{2}-\frac{12\!\cdots\!13}{197593802064648}a+\frac{6251995957519}{16466150172054}$, $\frac{6106701078541}{790375208258592}a^{15}-\frac{35994086418703}{15\!\cdots\!84}a^{14}+\frac{4951504334881}{131729201376432}a^{13}-\frac{38711448624601}{131729201376432}a^{12}+\frac{10\!\cdots\!17}{15\!\cdots\!84}a^{11}-\frac{54327333288203}{15\!\cdots\!84}a^{10}-\frac{81556715162649}{175638935168576}a^{9}+\frac{466702991912449}{526916805505728}a^{8}-\frac{9355945869365}{634584671424}a^{7}+\frac{11\!\cdots\!33}{395187604129296}a^{6}+\frac{162324697676491}{11975381943312}a^{5}-\frac{98\!\cdots\!65}{175638935168576}a^{4}+\frac{76\!\cdots\!05}{395187604129296}a^{3}+\frac{85\!\cdots\!95}{395187604129296}a^{2}-\frac{117305032929279}{5488716724018}a-\frac{771991539113}{997948495276}$, $\frac{127883021}{14991444864}a^{15}-\frac{15664729}{624643536}a^{14}+\frac{100121317}{2498574144}a^{13}-\frac{531542809}{1665716096}a^{12}+\frac{11284129759}{14991444864}a^{11}+\frac{86611523}{4997148288}a^{10}-\frac{3328244177}{4997148288}a^{9}+\frac{1612192789}{1665716096}a^{8}-\frac{80199271}{4983858}a^{7}+\frac{25710447307}{832858048}a^{6}+\frac{29762791327}{1665716096}a^{5}-\frac{28478736771}{416429024}a^{4}+\frac{63201078923}{3747861216}a^{3}+\frac{982570463}{26026814}a^{2}-\frac{2336105835}{104107256}a-\frac{67513489}{13013407}$, $\frac{5520899751529}{790375208258592}a^{15}-\frac{5885948189845}{395187604129296}a^{14}+\frac{3297334969519}{131729201376432}a^{13}-\frac{197309958400853}{790375208258592}a^{12}+\frac{342898742039533}{790375208258592}a^{11}+\frac{17963580300331}{87819467584288}a^{10}-\frac{89088084996457}{790375208258592}a^{9}+\frac{177040193642005}{263458402752864}a^{8}-\frac{178532732041}{14422378896}a^{7}+\frac{61\!\cdots\!19}{395187604129296}a^{6}+\frac{52\!\cdots\!89}{263458402752864}a^{5}-\frac{12\!\cdots\!73}{395187604129296}a^{4}-\frac{54566386376305}{8981536457484}a^{3}+\frac{12\!\cdots\!95}{65864600688216}a^{2}-\frac{20278349914673}{8981536457484}a-\frac{80911373841055}{16466150172054}$, $\frac{3775430516819}{263458402752864}a^{15}-\frac{24845482447609}{526916805505728}a^{14}+\frac{1858502802195}{21954866896072}a^{13}-\frac{226608803867573}{395187604129296}a^{12}+\frac{23\!\cdots\!09}{15\!\cdots\!84}a^{11}-\frac{877750938450323}{15\!\cdots\!84}a^{10}-\frac{10\!\cdots\!65}{15\!\cdots\!84}a^{9}+\frac{704749334141969}{526916805505728}a^{8}-\frac{5758767350239}{211528223808}a^{7}+\frac{13\!\cdots\!79}{21954866896072}a^{6}+\frac{887943540197947}{131729201376432}a^{5}-\frac{72\!\cdots\!75}{68728278979008}a^{4}+\frac{93\!\cdots\!33}{197593802064648}a^{3}+\frac{17\!\cdots\!57}{395187604129296}a^{2}-\frac{34\!\cdots\!81}{98796901032324}a-\frac{225405325850965}{32932300344108}$, $\frac{103166102163}{15967175924416}a^{15}-\frac{36692161933}{1995896990552}a^{14}+\frac{2104339031875}{71852291659872}a^{13}-\frac{11403305801237}{47901527773248}a^{12}+\frac{8710795742201}{15967175924416}a^{11}+\frac{2435498049161}{47901527773248}a^{10}-\frac{78013164262001}{143704583319744}a^{9}+\frac{40201001975341}{47901527773248}a^{8}-\frac{58286680901}{4807459632}a^{7}+\frac{178656531108467}{7983587962208}a^{6}+\frac{21\!\cdots\!77}{143704583319744}a^{5}-\frac{211363604945607}{3991793981104}a^{4}+\frac{21924053443465}{1496922742914}a^{3}+\frac{179892378786617}{5987690971656}a^{2}-\frac{46736658202606}{2245384114371}a-\frac{2995398758423}{748461371457}$, $\frac{17238384020339}{31\!\cdots\!68}a^{15}-\frac{22869064767661}{15\!\cdots\!84}a^{14}+\frac{33643060908521}{15\!\cdots\!84}a^{13}-\frac{56832188771953}{287409166639488}a^{12}+\frac{149424688794723}{351277870337152}a^{11}+\frac{141022548893819}{10\!\cdots\!56}a^{10}-\frac{12\!\cdots\!75}{31\!\cdots\!68}a^{9}+\frac{153411743161241}{351277870337152}a^{8}-\frac{6349198248259}{634584671424}a^{7}+\frac{26\!\cdots\!01}{15\!\cdots\!84}a^{6}+\frac{52\!\cdots\!21}{31\!\cdots\!68}a^{5}-\frac{61\!\cdots\!93}{15\!\cdots\!84}a^{4}-\frac{895487528652047}{263458402752864}a^{3}+\frac{348931038254419}{11975381943312}a^{2}-\frac{13\!\cdots\!79}{197593802064648}a-\frac{362760581022529}{32932300344108}$, $\frac{21932419478083}{15\!\cdots\!84}a^{15}-\frac{69373126320409}{15\!\cdots\!84}a^{14}+\frac{58346726294929}{790375208258592}a^{13}-\frac{845891526752653}{15\!\cdots\!84}a^{12}+\frac{32116220525563}{23950763886624}a^{11}-\frac{31124579103197}{131729201376432}a^{10}-\frac{818870411502725}{790375208258592}a^{9}+\frac{13887204749579}{7983587962208}a^{8}-\frac{16898235624595}{634584671424}a^{7}+\frac{19\!\cdots\!35}{34364139489504}a^{6}+\frac{28\!\cdots\!43}{15\!\cdots\!84}a^{5}-\frac{18\!\cdots\!83}{15\!\cdots\!84}a^{4}+\frac{32\!\cdots\!97}{65864600688216}a^{3}+\frac{66\!\cdots\!33}{131729201376432}a^{2}-\frac{22\!\cdots\!61}{49398450516162}a-\frac{158035123033141}{32932300344108}$, $\frac{3273005675987}{526916805505728}a^{15}-\frac{1442354763137}{71852291659872}a^{14}+\frac{2188674987835}{71852291659872}a^{13}-\frac{371265693320969}{15\!\cdots\!84}a^{12}+\frac{951818775961553}{15\!\cdots\!84}a^{11}-\frac{1318460256121}{175638935168576}a^{10}-\frac{10\!\cdots\!33}{15\!\cdots\!84}a^{9}+\frac{126923649653175}{175638935168576}a^{8}-\frac{424512775815}{35254703968}a^{7}+\frac{18\!\cdots\!43}{71852291659872}a^{6}+\frac{20\!\cdots\!07}{15\!\cdots\!84}a^{5}-\frac{46\!\cdots\!15}{790375208258592}a^{4}+\frac{34\!\cdots\!77}{197593802064648}a^{3}+\frac{10\!\cdots\!09}{32932300344108}a^{2}-\frac{12\!\cdots\!45}{49398450516162}a-\frac{86613925935529}{16466150172054}$, $\frac{5226169128907}{526916805505728}a^{15}-\frac{47439026479565}{15\!\cdots\!84}a^{14}+\frac{4402736076229}{87819467584288}a^{13}-\frac{597607872925603}{15\!\cdots\!84}a^{12}+\frac{360483632230987}{395187604129296}a^{11}-\frac{75804385069387}{790375208258592}a^{10}-\frac{11741505909337}{17182069744752}a^{9}+\frac{25880474316443}{21954866896072}a^{8}-\frac{3980205481465}{211528223808}a^{7}+\frac{29\!\cdots\!09}{790375208258592}a^{6}+\frac{82\!\cdots\!91}{526916805505728}a^{5}-\frac{12\!\cdots\!71}{15\!\cdots\!84}a^{4}+\frac{11\!\cdots\!87}{395187604129296}a^{3}+\frac{14\!\cdots\!29}{395187604129296}a^{2}-\frac{27\!\cdots\!45}{98796901032324}a-\frac{179876661336653}{32932300344108}$, $\frac{6049519350865}{15\!\cdots\!84}a^{15}-\frac{9456238849933}{790375208258592}a^{14}+\frac{1775309749101}{87819467584288}a^{13}-\frac{232825263622297}{15\!\cdots\!84}a^{12}+\frac{64591110615765}{175638935168576}a^{11}-\frac{105053551403309}{15\!\cdots\!84}a^{10}-\frac{18348063746179}{68728278979008}a^{9}+\frac{69726915558387}{175638935168576}a^{8}-\frac{2277802039529}{317292335712}a^{7}+\frac{12\!\cdots\!57}{790375208258592}a^{6}+\frac{26\!\cdots\!13}{526916805505728}a^{5}-\frac{24\!\cdots\!81}{790375208258592}a^{4}+\frac{27335026585704}{2744358362009}a^{3}+\frac{17\!\cdots\!65}{98796901032324}a^{2}-\frac{230989542370189}{24699225258081}a-\frac{66044768919607}{16466150172054}$ 1923825376543/3161500833034368a^15 - 3317678706115/790375208258592a^14 + 3859456791983/526916805505728a^13 - 98810200001779/3161500833034368a^12 + 48404253572881/351277870337152a^11 - 44517073143137/351277870337152a^10 - 385686802021637/3161500833034368a^9 + 91438900592201/1053833611011456a^8 - 468376694791/317292335712a^7 + 10081262869408457/1580750416517184a^6 - 1135326445312723/351277870337152a^5 - 2248815534213731/197593802064648a^4 + 927166667412225/87819467584288a^3 + 56419442192967/21954866896072a^2 - 1271656108448413/197593802064648a + 6251995957519/16466150172054, 6106701078541/790375208258592a^15 - 35994086418703/1580750416517184a^14 + 4951504334881/131729201376432a^13 - 38711448624601/131729201376432a^12 + 1089638336717017/1580750416517184a^11 - 54327333288203/1580750416517184a^10 - 81556715162649/175638935168576a^9 + 466702991912449/526916805505728a^8 - 9355945869365/634584671424a^7 + 11082425934807133/395187604129296a^6 + 162324697676491/11975381943312a^5 - 9862888727486365/175638935168576a^4 + 7635967196801105/395187604129296a^3 + 8543740511197895/395187604129296a^2 - 117305032929279/5488716724018a - 771991539113/997948495276, 127883021/14991444864a^15 - 15664729/624643536a^14 + 100121317/2498574144a^13 - 531542809/1665716096a^12 + 11284129759/14991444864a^11 + 86611523/4997148288a^10 - 3328244177/4997148288a^9 + 1612192789/1665716096a^8 - 80199271/4983858a^7 + 25710447307/832858048a^6 + 29762791327/1665716096a^5 - 28478736771/416429024a^4 + 63201078923/3747861216a^3 + 982570463/26026814a^2 - 2336105835/104107256a - 67513489/13013407, 5520899751529/790375208258592a^15 - 5885948189845/395187604129296a^14 + 3297334969519/131729201376432a^13 - 197309958400853/790375208258592a^12 + 342898742039533/790375208258592a^11 + 17963580300331/87819467584288a^10 - 89088084996457/790375208258592a^9 + 177040193642005/263458402752864a^8 - 178532732041/14422378896a^7 + 6191549469854119/395187604129296a^6 + 5284953582666089/263458402752864a^5 - 12116932002866873/395187604129296a^4 - 54566386376305/8981536457484a^3 + 1262826579721595/65864600688216a^2 - 20278349914673/8981536457484a - 80911373841055/16466150172054, 3775430516819/263458402752864a^15 - 24845482447609/526916805505728a^14 + 1858502802195/21954866896072a^13 - 226608803867573/395187604129296a^12 + 2347012627135109/1580750416517184a^11 - 877750938450323/1580750416517184a^10 - 1000279786784665/1580750416517184a^9 + 704749334141969/526916805505728a^8 - 5758767350239/211528223808a^7 + 1342983013587679/21954866896072a^6 + 887943540197947/131729201376432a^5 - 7208776414281575/68728278979008a^4 + 9353951786923133/197593802064648a^3 + 17693661284759657/395187604129296a^2 - 3499843431161081/98796901032324a - 225405325850965/32932300344108, 103166102163/15967175924416a^15 - 36692161933/1995896990552a^14 + 2104339031875/71852291659872a^13 - 11403305801237/47901527773248a^12 + 8710795742201/15967175924416a^11 + 2435498049161/47901527773248a^10 - 78013164262001/143704583319744a^9 + 40201001975341/47901527773248a^8 - 58286680901/4807459632a^7 + 178656531108467/7983587962208a^6 + 2114286961250377/143704583319744a^5 - 211363604945607/3991793981104a^4 + 21924053443465/1496922742914a^3 + 179892378786617/5987690971656a^2 - 46736658202606/2245384114371a - 2995398758423/748461371457, 17238384020339/3161500833034368a^15 - 22869064767661/1580750416517184a^14 + 33643060908521/1580750416517184a^13 - 56832188771953/287409166639488a^12 + 149424688794723/351277870337152a^11 + 141022548893819/1053833611011456a^10 - 1204794596642375/3161500833034368a^9 + 153411743161241/351277870337152a^8 - 6349198248259/634584671424a^7 + 26402594077865701/1580750416517184a^6 + 52448075714393021/3161500833034368a^5 - 61731554119141493/1580750416517184a^4 - 895487528652047/263458402752864a^3 + 348931038254419/11975381943312a^2 - 1385300446617979/197593802064648a - 362760581022529/32932300344108, 21932419478083/1580750416517184a^15 - 69373126320409/1580750416517184a^14 + 58346726294929/790375208258592a^13 - 845891526752653/1580750416517184a^12 + 32116220525563/23950763886624a^11 - 31124579103197/131729201376432a^10 - 818870411502725/790375208258592a^9 + 13887204749579/7983587962208a^8 - 16898235624595/634584671424a^7 + 1921047024954235/34364139489504a^6 + 28681973151420643/1580750416517184a^5 - 181526911847466983/1580750416517184a^4 + 3269385783456197/65864600688216a^3 + 6635806278793733/131729201376432a^2 - 2254419468231361/49398450516162a - 158035123033141/32932300344108, 3273005675987/526916805505728a^15 - 1442354763137/71852291659872a^14 + 2188674987835/71852291659872a^13 - 371265693320969/1580750416517184a^12 + 951818775961553/1580750416517184a^11 - 1318460256121/175638935168576a^10 - 1040177211726833/1580750416517184a^9 + 126923649653175/175638935168576a^8 - 424512775815/35254703968a^7 + 1813242745477043/71852291659872a^6 + 20845080056186207/1580750416517184a^5 - 46645350025577615/790375208258592a^4 + 3441282399710777/197593802064648a^3 + 1054769358628009/32932300344108a^2 - 1225315776426245/49398450516162a - 86613925935529/16466150172054, 5226169128907/526916805505728a^15 - 47439026479565/1580750416517184a^14 + 4402736076229/87819467584288a^13 - 597607872925603/1580750416517184a^12 + 360483632230987/395187604129296a^11 - 75804385069387/790375208258592a^10 - 11741505909337/17182069744752a^9 + 25880474316443/21954866896072a^8 - 3980205481465/211528223808a^7 + 29692545807757909/790375208258592a^6 + 8278330577508991/526916805505728a^5 - 122833401972114371/1580750416517184a^4 + 11094207656845487/395187604129296a^3 + 14737075225413029/395187604129296a^2 - 2789349034527745/98796901032324a - 179876661336653/32932300344108, 6049519350865/1580750416517184a^15 - 9456238849933/790375208258592a^14 + 1775309749101/87819467584288a^13 - 232825263622297/1580750416517184a^12 + 64591110615765/175638935168576a^11 - 105053551403309/1580750416517184a^10 - 18348063746179/68728278979008a^9 + 69726915558387/175638935168576a^8 - 2277802039529/317292335712a^7 + 12026720594442757/790375208258592a^6 + 2691705732753413/526916805505728a^5 - 24442150643022781/790375208258592a^4 + 27335026585704/2744358362009a^3 + 1762951157288165/98796901032324a^2 - 230989542370189/24699225258081*a - 66044768919607/16466150172054 (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:	$ 19014964.3925 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 19014964.3925 \cdot 1}{2\cdot\sqrt{11654783511381160590876241}}\cr\approx \mathstrut & 1.11115041839 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 2*x^14 - 33*x^13 + 53*x^12 + 83*x^11 - 77*x^10 + 51*x^9 - 1774*x^8 + 1850*x^7 + 5383*x^6 - 6162*x^5 - 5048*x^4 + 6520*x^3 + 992*x^2 - 3360*x - 576)

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^16 - 2*x^15 + 2*x^14 - 33*x^13 + 53*x^12 + 83*x^11 - 77*x^10 + 51*x^9 - 1774*x^8 + 1850*x^7 + 5383*x^6 - 6162*x^5 - 5048*x^4 + 6520*x^3 + 992*x^2 - 3360*x - 576, 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^16 - 2*x^15 + 2*x^14 - 33*x^13 + 53*x^12 + 83*x^11 - 77*x^10 + 51*x^9 - 1774*x^8 + 1850*x^7 + 5383*x^6 - 6162*x^5 - 5048*x^4 + 6520*x^3 + 992*x^2 - 3360*x - 576);

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^16 - 2*x^15 + 2*x^14 - 33*x^13 + 53*x^12 + 83*x^11 - 77*x^10 + 51*x^9 - 1774*x^8 + 1850*x^7 + 5383*x^6 - 6162*x^5 - 5048*x^4 + 6520*x^3 + 992*x^2 - 3360*x - 576);

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

$C_2^3:C_4$ (as 16T33):

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 32

The 11 conjugacy class representatives for $C_2^3:C_4$

Character table for $C_2^3:C_4$

Intermediate fields

$\Q(\sqrt{377}) $, $\Q(\sqrt{13}) $, $\Q(\sqrt{29}) $, 4.4.1847677.1 x2, 4.4.63713.1 x2, $\Q(\sqrt{13}, \sqrt{29})$, 8.4.262608484333.1 x2, 8.4.3413910296329.1 x2, 8.8.3413910296329.1

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

Galois closure:	deg 32
Degree 8 siblings:	8.4.262608484333.1, 8.4.3413910296329.1, 8.0.20200652641.2, 8.0.4059346369.1
Degree 16 siblings:	16.0.11654783511381160590876241.5, 16.0.68963216043675506454889.2, 16.0.13858244365494840179401.1
Minimal sibling:	8.0.20200652641.2

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }^{4}$	${\href{/padicField/3.2.0.1}{2} }^{8}$	${\href{/padicField/5.4.0.1}{4} }^{4}$	${\href{/padicField/7.4.0.1}{4} }^{4}$	${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$	R	${\href{/padicField/17.4.0.1}{4} }^{4}$	${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$	${\href{/padicField/23.2.0.1}{2} }^{8}$	R	${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$	${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$	${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$	${\href{/padicField/43.4.0.1}{4} }^{4}$	${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$	${\href{/padicField/53.2.0.1}{2} }^{8}$	${\href{/padicField/59.4.0.1}{4} }^{4}$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:

p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:

p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:

p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:

p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$13$ 13	13.8.6.1	$x^{8} + 48 x^{7} + 872 x^{6} + 7200 x^{5} + 24242 x^{4} + 15024 x^{3} + 14408 x^{2} + 86496 x + 254881$	$4$	$2$	$6$	$C_4\times C_2$	$[\ ]_{4}^{2}$
$13$ 13	13.8.6.1	$x^{8} + 48 x^{7} + 872 x^{6} + 7200 x^{5} + 24242 x^{4} + 15024 x^{3} + 14408 x^{2} + 86496 x + 254881$	$4$	$2$	$6$	$C_4\times C_2$	$[\ ]_{4}^{2}$
$29$ 29	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	$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

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