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Properties

Label 16.0.74015667246...1393.6
Degree $16$
Signature $[0, 8]$
Discriminant $43^{8}\cdot 97^{15}$
Root discriminant $477.90$
Ramified primes $43, 97$
Class number $256$ (GRH)
Class group $[8, 32]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T260)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2866358222303, -5241373973535, 3007775646743, -447528584932, 170125614243, -1279043207, 3912204275, 534487532, 28754614, 15125453, 24577, 28405, -3665, -1156, 14, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 14*x^14 - 1156*x^13 - 3665*x^12 + 28405*x^11 + 24577*x^10 + 15125453*x^9 + 28754614*x^8 + 534487532*x^7 + 3912204275*x^6 - 1279043207*x^5 + 170125614243*x^4 - 447528584932*x^3 + 3007775646743*x^2 - 5241373973535*x + 2866358222303)
 
gp: K = bnfinit(x^16 - 2*x^15 + 14*x^14 - 1156*x^13 - 3665*x^12 + 28405*x^11 + 24577*x^10 + 15125453*x^9 + 28754614*x^8 + 534487532*x^7 + 3912204275*x^6 - 1279043207*x^5 + 170125614243*x^4 - 447528584932*x^3 + 3007775646743*x^2 - 5241373973535*x + 2866358222303, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 14 x^{14} - 1156 x^{13} - 3665 x^{12} + 28405 x^{11} + 24577 x^{10} + 15125453 x^{9} + 28754614 x^{8} + 534487532 x^{7} + 3912204275 x^{6} - 1279043207 x^{5} + 170125614243 x^{4} - 447528584932 x^{3} + 3007775646743 x^{2} - 5241373973535 x + 2866358222303 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7401566724659785057512888367473350870971393=43^{8}\cdot 97^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $477.90$
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:  $43, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{106} a^{14} + \frac{9}{53} a^{13} + \frac{9}{106} a^{12} + \frac{33}{106} a^{11} - \frac{18}{53} a^{10} - \frac{24}{53} a^{9} + \frac{14}{53} a^{8} - \frac{51}{106} a^{7} + \frac{43}{106} a^{6} - \frac{7}{106} a^{5} + \frac{6}{53} a^{4} - \frac{51}{106} a^{3} + \frac{26}{53} a^{2} - \frac{24}{53} a + \frac{31}{106}$, $\frac{1}{16145008487309927797445579473836968719437761751856623576479824757864653592182137124645449898} a^{15} - \frac{11610496627106335137141406721853518031864402485976697864003909655359779088201275279711625}{16145008487309927797445579473836968719437761751856623576479824757864653592182137124645449898} a^{14} + \frac{1756552590285583943572562324290958205897543594834576053036364947227828052075304646675039995}{16145008487309927797445579473836968719437761751856623576479824757864653592182137124645449898} a^{13} + \frac{607305054062099656840891308766482094097442687718950145889129075487676254011686452874423284}{8072504243654963898722789736918484359718880875928311788239912378932326796091068562322724949} a^{12} + \frac{6497479479375036148863537204406533700542207946092895212473071370167623755755073105176702543}{16145008487309927797445579473836968719437761751856623576479824757864653592182137124645449898} a^{11} + \frac{1193189927280147458928975435854870847357936899836250285130174545404392644510043789116784114}{8072504243654963898722789736918484359718880875928311788239912378932326796091068562322724949} a^{10} - \frac{4030830616134977063762477484814464626681287367140896649499768474102855391473505787794643117}{8072504243654963898722789736918484359718880875928311788239912378932326796091068562322724949} a^{9} + \frac{2056598193098961674817724068678594630932907356850162424440320024466422358993534478069674297}{16145008487309927797445579473836968719437761751856623576479824757864653592182137124645449898} a^{8} + \frac{395147100966933267029816523709643025572851432306297663720437725787532008593731985806902777}{8072504243654963898722789736918484359718880875928311788239912378932326796091068562322724949} a^{7} + \frac{3307075818566459429136976285338141408329242240673827973166491745857996168891348834967730926}{8072504243654963898722789736918484359718880875928311788239912378932326796091068562322724949} a^{6} + \frac{30182144191119036886488849563174517258705893796036019361332377928570633801265459247599427}{16145008487309927797445579473836968719437761751856623576479824757864653592182137124645449898} a^{5} - \frac{1218474033646665581833996884254666652686974137163358442993492065637971919471216821067763855}{16145008487309927797445579473836968719437761751856623576479824757864653592182137124645449898} a^{4} - \frac{5205872041325880794896726114276296748771415353407252352718135890911799207059528159046007455}{16145008487309927797445579473836968719437761751856623576479824757864653592182137124645449898} a^{3} + \frac{817096586970872457588823446859515581988638745229503658932104809899061986667006278365950545}{8072504243654963898722789736918484359718880875928311788239912378932326796091068562322724949} a^{2} + \frac{2793172542205639341182632680269975430444630094397769648437575207411193120744959281924264225}{16145008487309927797445579473836968719437761751856623576479824757864653592182137124645449898} a + \frac{5459685909788241090337341558144457656468538024085642864274093935379144931106090754574630883}{16145008487309927797445579473836968719437761751856623576479824757864653592182137124645449898}$

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

Class group and class number

$C_{8}\times C_{32}$, which has order $256$ (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:  $7$
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:  \( 2348245511120 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T260):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-4171}) \), 4.0.1687532377.1, 8.0.276233255772057202513.4

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

Sibling fields

Degree 16 sibling: data not computed
Degree 32 siblings: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ $16$ R ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ $16$

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
$43$43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
97Data not computed