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Properties

Label 16.8.90939508218...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 3^{8}\cdot 5^{8}\cdot 59^{2}\cdot 157^{4}$
Root discriminant $64.55$
Ramified primes $2, 3, 5, 59, 157$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1765

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5959, -148970, -547964, 336786, 458022, -378620, 6130, 88226, -35133, -1564, 5206, -1370, -96, 138, -22, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 22*x^14 + 138*x^13 - 96*x^12 - 1370*x^11 + 5206*x^10 - 1564*x^9 - 35133*x^8 + 88226*x^7 + 6130*x^6 - 378620*x^5 + 458022*x^4 + 336786*x^3 - 547964*x^2 - 148970*x + 5959)
 
gp: K = bnfinit(x^16 - 4*x^15 - 22*x^14 + 138*x^13 - 96*x^12 - 1370*x^11 + 5206*x^10 - 1564*x^9 - 35133*x^8 + 88226*x^7 + 6130*x^6 - 378620*x^5 + 458022*x^4 + 336786*x^3 - 547964*x^2 - 148970*x + 5959, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 22 x^{14} + 138 x^{13} - 96 x^{12} - 1370 x^{11} + 5206 x^{10} - 1564 x^{9} - 35133 x^{8} + 88226 x^{7} + 6130 x^{6} - 378620 x^{5} + 458022 x^{4} + 336786 x^{3} - 547964 x^{2} - 148970 x + 5959 \)

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:  \(90939508218265868697600000000=2^{24}\cdot 3^{8}\cdot 5^{8}\cdot 59^{2}\cdot 157^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.55$
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:  $2, 3, 5, 59, 157$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{10} a^{12} + \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{1}{10} a^{9} - \frac{1}{10} a^{8} + \frac{1}{5} a^{7} - \frac{1}{10} a^{5} - \frac{1}{10} a^{4} - \frac{2}{5} a^{2} + \frac{3}{10} a - \frac{2}{5}$, $\frac{1}{10} a^{13} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{1}{10} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{3}{10} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{2} a + \frac{3}{10}$, $\frac{1}{10} a^{14} + \frac{1}{10} a^{11} + \frac{1}{10} a^{10} + \frac{1}{10} a^{9} - \frac{1}{10} a^{8} + \frac{3}{10} a^{7} + \frac{1}{10} a^{6} - \frac{1}{2} a^{5} - \frac{1}{10} a^{4} + \frac{1}{10} a^{3} + \frac{1}{5} a^{2} - \frac{1}{10} a - \frac{3}{10}$, $\frac{1}{134646053714326824497611075173085550} a^{15} + \frac{1921329555266832668518624753422113}{134646053714326824497611075173085550} a^{14} + \frac{583384168512552564276295173880429}{134646053714326824497611075173085550} a^{13} + \frac{1643682353648711451038226289845621}{134646053714326824497611075173085550} a^{12} - \frac{14710589067508107262393766505556062}{67323026857163412248805537586542775} a^{11} - \frac{5730029393627292143646798330268724}{67323026857163412248805537586542775} a^{10} + \frac{3330953687666786246601739341551123}{13464605371432682449761107517308555} a^{9} - \frac{20313064719871858638073418688610329}{134646053714326824497611075173085550} a^{8} - \frac{4436035331015776853757265769897293}{67323026857163412248805537586542775} a^{7} + \frac{4375212753380131062682856567184087}{67323026857163412248805537586542775} a^{6} - \frac{29388262464478737718520498013241386}{67323026857163412248805537586542775} a^{5} - \frac{10891483804484566842485037522580619}{134646053714326824497611075173085550} a^{4} - \frac{23255985617199641882664643037507141}{134646053714326824497611075173085550} a^{3} + \frac{22705996959765676262446169746020959}{134646053714326824497611075173085550} a^{2} - \frac{65743416495975026689446915538139671}{134646053714326824497611075173085550} a + \frac{18812144397361608387558742451614469}{67323026857163412248805537586542775}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 762229199.577 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1765:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 12288
The 74 conjugacy class representatives for t16n1765 are not computed
Character table for t16n1765 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 8.4.5111216640000.1

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 R R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ R

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
3Data not computed
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
59Data not computed
$157$157.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
157.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
157.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
157.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
157.8.4.1$x^{8} + 739470 x^{4} - 3869893 x^{2} + 136703970225$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$