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Properties

Label 16.8.27537614904...2129.3
Degree $16$
Signature $[8, 4]$
Discriminant $41^{12}\cdot 73^{14}$
Root discriminant $691.82$
Ramified primes $41, 73$
Class number $24$ (GRH)
Class group $[2, 2, 6]$ (GRH)
Galois group $OD_{16}.C_2$ (as 16T40)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![940107776688, 8083094211792, 5867704308348, 6649957871712, -48125809891, -1881824439314, 250720847712, -26650450836, 4637668682, 166021436, 8567282, 1209554, -118952, -844, -390, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 390*x^14 - 844*x^13 - 118952*x^12 + 1209554*x^11 + 8567282*x^10 + 166021436*x^9 + 4637668682*x^8 - 26650450836*x^7 + 250720847712*x^6 - 1881824439314*x^5 - 48125809891*x^4 + 6649957871712*x^3 + 5867704308348*x^2 + 8083094211792*x + 940107776688)
 
gp: K = bnfinit(x^16 - 4*x^15 - 390*x^14 - 844*x^13 - 118952*x^12 + 1209554*x^11 + 8567282*x^10 + 166021436*x^9 + 4637668682*x^8 - 26650450836*x^7 + 250720847712*x^6 - 1881824439314*x^5 - 48125809891*x^4 + 6649957871712*x^3 + 5867704308348*x^2 + 8083094211792*x + 940107776688, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 390 x^{14} - 844 x^{13} - 118952 x^{12} + 1209554 x^{11} + 8567282 x^{10} + 166021436 x^{9} + 4637668682 x^{8} - 26650450836 x^{7} + 250720847712 x^{6} - 1881824439314 x^{5} - 48125809891 x^{4} + 6649957871712 x^{3} + 5867704308348 x^{2} + 8083094211792 x + 940107776688 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2753761490493877153393821007665572209609402129=41^{12}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $691.82$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $41, 73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{3}{8} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{6} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{5} - \frac{1}{2} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{9} + \frac{3}{16} a^{6} + \frac{5}{16} a^{3} - \frac{1}{2}$, $\frac{1}{96} a^{13} + \frac{1}{48} a^{12} - \frac{1}{16} a^{11} + \frac{5}{96} a^{10} - \frac{1}{48} a^{9} - \frac{1}{24} a^{8} + \frac{23}{96} a^{7} + \frac{1}{48} a^{6} - \frac{5}{48} a^{5} + \frac{1}{32} a^{4} + \frac{5}{16} a^{3} - \frac{11}{24} a^{2} + \frac{1}{3} a$, $\frac{1}{576} a^{14} - \frac{1}{576} a^{13} - \frac{1}{96} a^{12} + \frac{11}{576} a^{11} - \frac{29}{576} a^{10} + \frac{5}{144} a^{9} - \frac{37}{576} a^{8} + \frac{53}{576} a^{7} - \frac{59}{288} a^{6} - \frac{1}{192} a^{5} + \frac{1}{64} a^{4} + \frac{11}{72} a^{3} + \frac{23}{144} a^{2} + \frac{1}{6} a + \frac{1}{4}$, $\frac{1}{43060353219330515889849279918079980743066048259786524284836481649294013172097520914325794521342347456} a^{15} - \frac{33103169180155788696290955946614726491875047784201657931174263054225800364676447672278401079404073}{43060353219330515889849279918079980743066048259786524284836481649294013172097520914325794521342347456} a^{14} + \frac{365657350651050760492303911459412451075816868400290124996996258904575275033010584788039225862097}{598060461379590498470128887751110843653695114719257281733840022906861294056910012698969368351977048} a^{13} - \frac{1143908060371361113415592884917347979989388954573591574356040187652247623030422275555143040788428901}{43060353219330515889849279918079980743066048259786524284836481649294013172097520914325794521342347456} a^{12} + \frac{2109463330698049678358863480516435815090076140571831256736993642833964234297935359532828957571042295}{43060353219330515889849279918079980743066048259786524284836481649294013172097520914325794521342347456} a^{11} - \frac{213886829016651541024923112889418706842892786651712801318091389773585875565573719574068659155150643}{21530176609665257944924639959039990371533024129893262142418240824647006586048760457162897260671173728} a^{10} + \frac{2173350328083870757566042701463560135407528436787318710459880853888681692435989823046124154258338459}{43060353219330515889849279918079980743066048259786524284836481649294013172097520914325794521342347456} a^{9} + \frac{2436272088222881356956249220841265473011702075887096090889992229681613436640351694498948126774313989}{43060353219330515889849279918079980743066048259786524284836481649294013172097520914325794521342347456} a^{8} - \frac{2298896148002196290686327576545536031982055905354476286620047594940747658180620302231256969105798157}{10765088304832628972462319979519995185766512064946631071209120412323503293024380228581448630335586864} a^{7} - \frac{1739308755341140834505618175271691995947472349982954133675250707828833020441983299963313386505697369}{14353451073110171963283093306026660247688682753262174761612160549764671057365840304775264840447449152} a^{6} - \frac{74926195490037058831255687007641590248234208044939614127724479573113928094354529034653464641373139}{4784483691036723987761031102008886749229560917754058253870720183254890352455280101591754946815816384} a^{5} + \frac{907253106371903323338499201828254037896119775044270576710141072379989515026527405387250870101730993}{21530176609665257944924639959039990371533024129893262142418240824647006586048760457162897260671173728} a^{4} - \frac{60910468397605868181707171541134945949133978207720350170256333346423622046549416893493394634982167}{290948332563044026282765404851891761777473299052611650573219470603337926838496762934633746765826672} a^{3} - \frac{694184682281854650935696120649105233791457543896820525651760518124628014331437261606802293201226351}{1794181384138771495410386663253332530961085344157771845201520068720583882170730038096908105055931144} a^{2} - \frac{16702925140334413560921525201260150790642160700146536387362911175713396226158741972495875280721305}{299030230689795249235064443875555421826847557359628640866920011453430647028455006349484684175988524} a - \frac{18667682935031613053353103770302538643704479431755653534387864784598443239483417774677921185697945}{49838371781632541539177407312592570304474592893271440144486668575571774504742501058247447362664754}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{6}$, which has order $24$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 16536498877900000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}.C_2$ (as 16T40):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $OD_{16}.C_2$
Character table for $OD_{16}.C_2$

Intermediate fields

\(\Q(\sqrt{73}) \), \(\Q(\sqrt{2993}) \), \(\Q(\sqrt{41}) \), 4.4.389017.1, 4.4.653937577.1, \(\Q(\sqrt{41}, \sqrt{73})\), 8.8.427634354612630929.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
41Data not computed
$73$73.8.7.1$x^{8} - 73$$8$$1$$7$$C_8$$[\ ]_{8}$
73.8.7.1$x^{8} - 73$$8$$1$$7$$C_8$$[\ ]_{8}$