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Properties

Label 16.8.46705271798...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 5^{14}\cdot 61^{6}\cdot 97^{4}$
Root discriminant $169.56$
Ramified primes $2, 5, 61, 97$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T852

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![63700077901, 49830668338, 869375994, -3989723662, -1989121957, 78373888, 168371486, -9218806, -5379957, -91542, 248404, 32504, -10157, -324, 76, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 76*x^14 - 324*x^13 - 10157*x^12 + 32504*x^11 + 248404*x^10 - 91542*x^9 - 5379957*x^8 - 9218806*x^7 + 168371486*x^6 + 78373888*x^5 - 1989121957*x^4 - 3989723662*x^3 + 869375994*x^2 + 49830668338*x + 63700077901)
 
gp: K = bnfinit(x^16 - 6*x^15 + 76*x^14 - 324*x^13 - 10157*x^12 + 32504*x^11 + 248404*x^10 - 91542*x^9 - 5379957*x^8 - 9218806*x^7 + 168371486*x^6 + 78373888*x^5 - 1989121957*x^4 - 3989723662*x^3 + 869375994*x^2 + 49830668338*x + 63700077901, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 76 x^{14} - 324 x^{13} - 10157 x^{12} + 32504 x^{11} + 248404 x^{10} - 91542 x^{9} - 5379957 x^{8} - 9218806 x^{7} + 168371486 x^{6} + 78373888 x^{5} - 1989121957 x^{4} - 3989723662 x^{3} + 869375994 x^{2} + 49830668338 x + 63700077901 \)

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:  \(467052717980688838758400000000000000=2^{24}\cdot 5^{14}\cdot 61^{6}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $169.56$
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:  $2, 5, 61, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{12763225191744047212720469135571886243124553799920501954714752684159040223720787921} a^{15} + \frac{6318504421098891043523982365039934256609050391882680182823336765592480386096418697}{12763225191744047212720469135571886243124553799920501954714752684159040223720787921} a^{14} - \frac{4420333156356904591118121371732645011852197981698821537430789229163264163310501565}{12763225191744047212720469135571886243124553799920501954714752684159040223720787921} a^{13} + \frac{6250782488211657662026921243313718682992096682401666850354986113030048839452514682}{12763225191744047212720469135571886243124553799920501954714752684159040223720787921} a^{12} - \frac{2905733941774983517596828299061427696412612973849041222899371005283662736761321823}{12763225191744047212720469135571886243124553799920501954714752684159040223720787921} a^{11} + \frac{5028725958591000023679560244257506994992499782571342293775484108140096358398126550}{12763225191744047212720469135571886243124553799920501954714752684159040223720787921} a^{10} + \frac{4601026782279330415433671765821897472978461419340739309135929524148081582180624604}{12763225191744047212720469135571886243124553799920501954714752684159040223720787921} a^{9} + \frac{2973756715702768114767181236748344475224636962427974585322045004644971831378153152}{12763225191744047212720469135571886243124553799920501954714752684159040223720787921} a^{8} - \frac{330919216868132704184479532552860711124496607036282860624580046258031063077776628}{12763225191744047212720469135571886243124553799920501954714752684159040223720787921} a^{7} - \frac{5907180270928861566768555109144377141723394861987964018302587001600634870738619734}{12763225191744047212720469135571886243124553799920501954714752684159040223720787921} a^{6} + \frac{1316214844411422188974280020168148271121474006612328247802161262408381409997145427}{12763225191744047212720469135571886243124553799920501954714752684159040223720787921} a^{5} - \frac{4411585938287426691568119876734111946033001214897216913502207786585758868781125321}{12763225191744047212720469135571886243124553799920501954714752684159040223720787921} a^{4} - \frac{5060323617681711475930190331772717986652741166547442540546254852601141239233532663}{12763225191744047212720469135571886243124553799920501954714752684159040223720787921} a^{3} + \frac{888292738275450414307229634156886344162949295573430088842464430883715886530971320}{12763225191744047212720469135571886243124553799920501954714752684159040223720787921} a^{2} - \frac{3758756777545847740835435676790484129222971443926348021080397150338320137926395721}{12763225191744047212720469135571886243124553799920501954714752684159040223720787921} a - \frac{1856279830908604131701064929047664656671423210605781607740078198762248775505051780}{12763225191744047212720469135571886243124553799920501954714752684159040223720787921}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 207398901381 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T852:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 41 conjugacy class representatives for t16n852
Character table for t16n852 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.8.14884000000.1

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

Sibling fields

Degree 16 siblings: 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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
5Data not computed
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.4$x^{4} + 488$$4$$1$$3$$C_4$$[\ ]_{4}$
$97$97.8.4.2$x^{8} - 912673 x^{2} + 2036173463$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
97.8.0.1$x^{8} - x + 84$$1$$8$$0$$C_8$$[\ ]^{8}$