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Properties

Label 16.0.93675802669...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 13^{4}\cdot 29^{6}\cdot 109^{4}$
Root discriminant $48.50$
Ramified primes $5, 13, 29, 109$
Class number $256$ (GRH)
Class group $[2, 2, 2, 32]$ (GRH)
Galois group 16T1177

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3748096, 1192576, 3693888, 1021152, 1669808, 264632, 425144, 28096, 66017, -428, 8451, -789, 778, -101, 42, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 42*x^14 - 101*x^13 + 778*x^12 - 789*x^11 + 8451*x^10 - 428*x^9 + 66017*x^8 + 28096*x^7 + 425144*x^6 + 264632*x^5 + 1669808*x^4 + 1021152*x^3 + 3693888*x^2 + 1192576*x + 3748096)
 
gp: K = bnfinit(x^16 - 2*x^15 + 42*x^14 - 101*x^13 + 778*x^12 - 789*x^11 + 8451*x^10 - 428*x^9 + 66017*x^8 + 28096*x^7 + 425144*x^6 + 264632*x^5 + 1669808*x^4 + 1021152*x^3 + 3693888*x^2 + 1192576*x + 3748096, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 42 x^{14} - 101 x^{13} + 778 x^{12} - 789 x^{11} + 8451 x^{10} - 428 x^{9} + 66017 x^{8} + 28096 x^{7} + 425144 x^{6} + 264632 x^{5} + 1669808 x^{4} + 1021152 x^{3} + 3693888 x^{2} + 1192576 x + 3748096 \)

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:  \(936758026692429703922265625=5^{8}\cdot 13^{4}\cdot 29^{6}\cdot 109^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.50$
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:  $5, 13, 29, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} + \frac{1}{4} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{16} a^{12} - \frac{1}{8} a^{10} - \frac{1}{16} a^{9} + \frac{7}{16} a^{7} - \frac{7}{16} a^{6} + \frac{1}{8} a^{5} + \frac{1}{16} a^{4} + \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{20768} a^{13} - \frac{265}{10384} a^{12} + \frac{21}{10384} a^{11} - \frac{1157}{20768} a^{10} + \frac{125}{10384} a^{9} + \frac{6955}{20768} a^{8} + \frac{5107}{20768} a^{7} + \frac{1785}{5192} a^{6} - \frac{9311}{20768} a^{5} - \frac{235}{1298} a^{4} + \frac{871}{2596} a^{3} + \frac{1047}{2596} a^{2} + \frac{611}{1298} a - \frac{28}{59}$, $\frac{1}{26956864} a^{14} - \frac{3}{228448} a^{13} + \frac{46837}{13478432} a^{12} + \frac{1579411}{26956864} a^{11} - \frac{189955}{13478432} a^{10} - \frac{4922981}{26956864} a^{9} + \frac{12449099}{26956864} a^{8} + \frac{1106537}{6739216} a^{7} - \frac{9899895}{26956864} a^{6} - \frac{562911}{3369608} a^{5} + \frac{808579}{3369608} a^{4} + \frac{209243}{1684804} a^{3} + \frac{458783}{1684804} a^{2} + \frac{1005}{6962} a + \frac{1251}{3481}$, $\frac{1}{10009727085337798365458869866770048} a^{15} - \frac{24805763856481982283156589}{5004863542668899182729434933385024} a^{14} - \frac{63685662800990488799577825559}{5004863542668899182729434933385024} a^{13} - \frac{253252685717037010027252548810933}{10009727085337798365458869866770048} a^{12} - \frac{303789305410152683882586593402983}{5004863542668899182729434933385024} a^{11} + \frac{440400009961355990226485016213763}{10009727085337798365458869866770048} a^{10} + \frac{1406475449310696700235532199175387}{10009727085337798365458869866770048} a^{9} + \frac{8382103568712956513831250366541}{2502431771334449591364717466692512} a^{8} - \frac{1817131882864296790524131431271079}{10009727085337798365458869866770048} a^{7} - \frac{477808984355297817553361831544829}{1251215885667224795682358733346256} a^{6} + \frac{29685037836823527195397670767601}{625607942833612397841179366673128} a^{5} - \frac{485189383390849425136803281020175}{1251215885667224795682358733346256} a^{4} + \frac{224561870617894126176378027683257}{625607942833612397841179366673128} a^{3} - \frac{6122933114831746335258439962473}{28436724674255108992780880303324} a^{2} + \frac{26359235100461004901473144299}{1292578394284323136035494559242} a - \frac{26546870359030986475869207450}{58753563376560142547067934511}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{32}$, 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:  \( 30078.3569453 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1177:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 76 conjugacy class representatives for t16n1177 are not computed
Character table for t16n1177 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.0.30606503013125.1, 8.4.280793605625.1, 8.4.57293125.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
109Data not computed