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Properties

Label 16.16.1357398247...7041.1
Degree $16$
Signature $[16, 0]$
Discriminant $13^{12}\cdot 17^{12}$
Root discriminant $57.32$
Ramified primes $13, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1223, 9255, -11509, -133892, 73203, 214708, -93555, -124859, 44659, 32075, -9962, -3878, 1089, 210, -55, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 55*x^14 + 210*x^13 + 1089*x^12 - 3878*x^11 - 9962*x^10 + 32075*x^9 + 44659*x^8 - 124859*x^7 - 93555*x^6 + 214708*x^5 + 73203*x^4 - 133892*x^3 - 11509*x^2 + 9255*x + 1223)
 
gp: K = bnfinit(x^16 - 4*x^15 - 55*x^14 + 210*x^13 + 1089*x^12 - 3878*x^11 - 9962*x^10 + 32075*x^9 + 44659*x^8 - 124859*x^7 - 93555*x^6 + 214708*x^5 + 73203*x^4 - 133892*x^3 - 11509*x^2 + 9255*x + 1223, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 55 x^{14} + 210 x^{13} + 1089 x^{12} - 3878 x^{11} - 9962 x^{10} + 32075 x^{9} + 44659 x^{8} - 124859 x^{7} - 93555 x^{6} + 214708 x^{5} + 73203 x^{4} - 133892 x^{3} - 11509 x^{2} + 9255 x + 1223 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(13573982477229290545823357041=13^{12}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.32$
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:  $13, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{12720} a^{12} - \frac{1}{4240} a^{11} - \frac{187}{1272} a^{10} + \frac{547}{3180} a^{9} - \frac{73}{1060} a^{8} - \frac{4339}{12720} a^{7} - \frac{1181}{4240} a^{6} - \frac{1367}{3180} a^{5} + \frac{1737}{4240} a^{4} - \frac{329}{2544} a^{3} - \frac{497}{12720} a^{2} - \frac{2681}{12720} a + \frac{1379}{12720}$, $\frac{1}{12720} a^{13} - \frac{1879}{12720} a^{11} + \frac{1469}{6360} a^{10} - \frac{14}{265} a^{9} - \frac{607}{12720} a^{8} - \frac{16}{53} a^{7} + \frac{2983}{12720} a^{6} - \frac{1611}{4240} a^{5} + \frac{317}{3180} a^{4} - \frac{679}{1590} a^{3} + \frac{547}{3180} a^{2} + \frac{757}{1590} a - \frac{741}{4240}$, $\frac{1}{12720} a^{14} - \frac{2699}{12720} a^{11} + \frac{1339}{6360} a^{10} + \frac{139}{848} a^{9} - \frac{217}{1060} a^{8} + \frac{587}{2120} a^{7} - \frac{107}{424} a^{6} + \frac{97}{265} a^{5} + \frac{4357}{12720} a^{4} - \frac{1389}{4240} a^{3} + \frac{251}{4240} a^{2} + \frac{1829}{6360} a + \frac{2621}{12720}$, $\frac{1}{363690537475353386160} a^{15} - \frac{2286701725001031}{60615089579225564360} a^{14} - \frac{3930767249569699}{181845268737676693080} a^{13} - \frac{11664203627730059}{363690537475353386160} a^{12} + \frac{4402876634961157127}{181845268737676693080} a^{11} + \frac{26751949058893218253}{363690537475353386160} a^{10} - \frac{1859200059791691493}{60615089579225564360} a^{9} + \frac{13105321872047778803}{90922634368838346540} a^{8} - \frac{3660901249844159277}{60615089579225564360} a^{7} + \frac{13900998833281943861}{181845268737676693080} a^{6} - \frac{31354037892778338673}{72738107495070677232} a^{5} - \frac{60046644313935378313}{363690537475353386160} a^{4} + \frac{169175552793693006631}{363690537475353386160} a^{3} - \frac{10182731527951545844}{22730658592209586635} a^{2} + \frac{20120255269863389645}{72738107495070677232} a - \frac{2982322163081862804}{7576886197403195545}$

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:  $15$
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:  \( 639742572.352 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{17}) \), \(\Q(\sqrt{221}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{13}, \sqrt{17})\), 4.4.634933.1 x2, 4.4.37349.1 x2, 4.4.4913.1, 4.4.830297.1, 8.8.403139914489.1, 8.8.689393108209.1, 8.8.116507435287321.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$13$13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
$17$17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$