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Properties

Label 16.8.17299727231...1968.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{34}\cdot 17^{8}\cdot 4129^{3}\cdot 5897^{3}$
Root discriminant $436.39$
Ramified primes $2, 17, 4129, 5897$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1472

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10795710088, 42989593936, 45607026072, -11048871344, -24625464628, 9016044960, 1087127760, -818116608, 34172043, 28578434, -3206305, -418126, 73174, 1854, -573, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 573*x^14 + 1854*x^13 + 73174*x^12 - 418126*x^11 - 3206305*x^10 + 28578434*x^9 + 34172043*x^8 - 818116608*x^7 + 1087127760*x^6 + 9016044960*x^5 - 24625464628*x^4 - 11048871344*x^3 + 45607026072*x^2 + 42989593936*x + 10795710088)
 
gp: K = bnfinit(x^16 - 2*x^15 - 573*x^14 + 1854*x^13 + 73174*x^12 - 418126*x^11 - 3206305*x^10 + 28578434*x^9 + 34172043*x^8 - 818116608*x^7 + 1087127760*x^6 + 9016044960*x^5 - 24625464628*x^4 - 11048871344*x^3 + 45607026072*x^2 + 42989593936*x + 10795710088, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 573 x^{14} + 1854 x^{13} + 73174 x^{12} - 418126 x^{11} - 3206305 x^{10} + 28578434 x^{9} + 34172043 x^{8} - 818116608 x^{7} + 1087127760 x^{6} + 9016044960 x^{5} - 24625464628 x^{4} - 11048871344 x^{3} + 45607026072 x^{2} + 42989593936 x + 10795710088 \)

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:  \(1729972723111358956551996577891668583251968=2^{34}\cdot 17^{8}\cdot 4129^{3}\cdot 5897^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $436.39$
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:  $2, 17, 4129, 5897$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{236784435508950380295416503184312469254154186099734400685714660427350718328} a^{15} - \frac{8814967016823185937092836210140629060763438147582701389183663366638725313}{78928145169650126765138834394770823084718062033244800228571553475783572776} a^{14} + \frac{2189934151317613163979088417844675466000832019473179259160949957973133421}{19732036292412531691284708598692705771179515508311200057142888368945893194} a^{13} - \frac{2116807501177733912202623259619948279213573265250765947734973353532064847}{19732036292412531691284708598692705771179515508311200057142888368945893194} a^{12} - \frac{649296109473734051042488014213446252475750302042130682200475307827890937}{118392217754475190147708251592156234627077093049867200342857330213675359164} a^{11} + \frac{10500638944461781026647119941118254238245302407168634891897338455067112029}{59196108877237595073854125796078117313538546524933600171428665106837679582} a^{10} + \frac{56209637363378944535884709619528151345017091547354515972812844494734293519}{236784435508950380295416503184312469254154186099734400685714660427350718328} a^{9} + \frac{21151202260676734249738344810097726583955396862206049934785355796968527}{236784435508950380295416503184312469254154186099734400685714660427350718328} a^{8} + \frac{4058513713119126108061108545155983804110498372650888126392916691417872527}{118392217754475190147708251592156234627077093049867200342857330213675359164} a^{7} - \frac{14285048821356545543360477205338334324434876548435916248896682194297883168}{29598054438618797536927062898039058656769273262466800085714332553418839791} a^{6} - \frac{673283529356068490223567600671567446886839550229631152313590070629584333}{29598054438618797536927062898039058656769273262466800085714332553418839791} a^{5} + \frac{1095377251076975250639826723541559770836846856062386495866746830468934502}{29598054438618797536927062898039058656769273262466800085714332553418839791} a^{4} + \frac{24771926089369629120331172404356977076592709880056630180822671516565234671}{59196108877237595073854125796078117313538546524933600171428665106837679582} a^{3} - \frac{237987133322325316097728018401752120925571727405944947889605163624874231}{9866018146206265845642354299346352885589757754155600028571444184472946597} a^{2} - \frac{1100380267192196432939379969697288177067210685109145161456940490969351700}{9866018146206265845642354299346352885589757754155600028571444184472946597} a - \frac{5686467445617748044067864362953860924416632750042734190810575222771145833}{29598054438618797536927062898039058656769273262466800085714332553418839791}$

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:  \( 1439115713660000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1472:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 74 conjugacy class representatives for t16n1472 are not computed
Character table for t16n1472 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{34}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{2}, \sqrt{17})\), 8.8.33318975217221632.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.8.22.84$x^{8} + 4 x^{7} + 10 x^{4} + 4 x^{2} + 14$$8$$1$$22$$D_4$$[2, 3, 7/2]$
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
4129Data not computed
5897Data not computed