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Properties

Label 16.0.26968313608...625.32
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 101^{10}$
Root discriminant $59.83$
Ramified primes $5, 101$
Class number $16$ (GRH)
Class group $[2, 2, 2, 2]$ (GRH)
Galois group $(C_2\times C_4).D_4$ (as 16T121)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![35940025, 46880900, 48334335, 44048875, 23495891, 6631093, 4075490, -329460, 621706, -124795, 56746, -9305, 3006, -280, 80, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 80*x^14 - 280*x^13 + 3006*x^12 - 9305*x^11 + 56746*x^10 - 124795*x^9 + 621706*x^8 - 329460*x^7 + 4075490*x^6 + 6631093*x^5 + 23495891*x^4 + 44048875*x^3 + 48334335*x^2 + 46880900*x + 35940025)
 
gp: K = bnfinit(x^16 - 3*x^15 + 80*x^14 - 280*x^13 + 3006*x^12 - 9305*x^11 + 56746*x^10 - 124795*x^9 + 621706*x^8 - 329460*x^7 + 4075490*x^6 + 6631093*x^5 + 23495891*x^4 + 44048875*x^3 + 48334335*x^2 + 46880900*x + 35940025, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 80 x^{14} - 280 x^{13} + 3006 x^{12} - 9305 x^{11} + 56746 x^{10} - 124795 x^{9} + 621706 x^{8} - 329460 x^{7} + 4075490 x^{6} + 6631093 x^{5} + 23495891 x^{4} + 44048875 x^{3} + 48334335 x^{2} + 46880900 x + 35940025 \)

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:  \(26968313608671985107666015625=5^{12}\cdot 101^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.83$
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:  $5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{71} a^{12} + \frac{14}{71} a^{11} - \frac{28}{71} a^{10} + \frac{29}{71} a^{9} - \frac{28}{71} a^{8} + \frac{3}{71} a^{7} - \frac{32}{71} a^{6} + \frac{8}{71} a^{5} + \frac{7}{71} a^{4} + \frac{13}{71} a^{3} - \frac{16}{71} a^{2} - \frac{7}{71} a - \frac{35}{71}$, $\frac{1}{71} a^{13} - \frac{11}{71} a^{11} - \frac{5}{71} a^{10} - \frac{8}{71} a^{9} - \frac{31}{71} a^{8} - \frac{3}{71} a^{7} + \frac{30}{71} a^{6} - \frac{34}{71} a^{5} - \frac{14}{71} a^{4} + \frac{15}{71} a^{3} + \frac{4}{71} a^{2} - \frac{8}{71} a - \frac{7}{71}$, $\frac{1}{500195} a^{14} + \frac{1412}{500195} a^{13} - \frac{222}{100039} a^{12} + \frac{31985}{100039} a^{11} - \frac{225719}{500195} a^{10} - \frac{787}{100039} a^{9} + \frac{77096}{500195} a^{8} + \frac{28007}{100039} a^{7} + \frac{220701}{500195} a^{6} - \frac{21530}{100039} a^{5} - \frac{7321}{100039} a^{4} - \frac{142842}{500195} a^{3} + \frac{89396}{500195} a^{2} + \frac{7088}{100039} a + \frac{2436}{100039}$, $\frac{1}{207053776623122502078799950096981352484630368000913405} a^{15} + \frac{12367480147613811215061376359270545301833994179}{207053776623122502078799950096981352484630368000913405} a^{14} - \frac{165696220860582311595478213194822671175617042035781}{207053776623122502078799950096981352484630368000913405} a^{13} + \frac{214401022487425805544598308362717745758448701206747}{41410755324624500415759990019396270496926073600182681} a^{12} - \frac{85236601349670666470239597880594133114648630228185839}{207053776623122502078799950096981352484630368000913405} a^{11} + \frac{62389461779653312657137212845125871984056461452080097}{207053776623122502078799950096981352484630368000913405} a^{10} - \frac{23977848865488255112020787943397626907186572368570379}{207053776623122502078799950096981352484630368000913405} a^{9} - \frac{809518070721982300297201945129911913377805822514868}{18823070602102045643527268190634668407693669818264855} a^{8} + \frac{25667342397338409773708915670713910879837725622938491}{207053776623122502078799950096981352484630368000913405} a^{7} + \frac{90225380203657776202580156127422303072731519657840177}{207053776623122502078799950096981352484630368000913405} a^{6} + \frac{9793586271247334390786863958229046133281689046888385}{41410755324624500415759990019396270496926073600182681} a^{5} - \frac{92097669935783281287573148352148700353351074479456797}{207053776623122502078799950096981352484630368000913405} a^{4} - \frac{42214438451382871222670538953845304204759031368952273}{207053776623122502078799950096981352484630368000913405} a^{3} - \frac{25786212096396005131023705155187886382199209265253688}{207053776623122502078799950096981352484630368000913405} a^{2} - \frac{20463472204632224714076782471798540756274551318803570}{41410755324624500415759990019396270496926073600182681} a - \frac{12894081791110711218418222829843976344745763505292}{34537744224040450722068382001164529188428751960119}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}$, which has order $16$ (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:  \( -\frac{740065321072581846625820383286902507369353}{146950870562897446471823953227098191969219565650045} a^{15} + \frac{10080828513456506290166751063688802892585218}{146950870562897446471823953227098191969219565650045} a^{14} - \frac{92690635271348315281836087588075879429380922}{146950870562897446471823953227098191969219565650045} a^{13} + \frac{174871723455625368541467860303618869789999031}{29390174112579489294364790645419638393843913130009} a^{12} - \frac{5265499865062320693536952210336583489996090318}{146950870562897446471823953227098191969219565650045} a^{11} + \frac{33981134991209879875916922978694542948362227564}{146950870562897446471823953227098191969219565650045} a^{10} - \frac{148495286768605760152890303086159407304441099773}{146950870562897446471823953227098191969219565650045} a^{9} + \frac{59705741410007946993404095554583407418026802969}{13359170051172495133802177566099835633565415059095} a^{8} - \frac{2104326496496485408589495536670032550386965006593}{146950870562897446471823953227098191969219565650045} a^{7} + \frac{6804860686424569387269233415017017909939815824469}{146950870562897446471823953227098191969219565650045} a^{6} - \frac{2529710950110394004197561498862788029273189748499}{29390174112579489294364790645419638393843913130009} a^{5} + \frac{33471364030572271971528906974281895228390209785786}{146950870562897446471823953227098191969219565650045} a^{4} - \frac{3683195190650265943898903471429792839333224678516}{146950870562897446471823953227098191969219565650045} a^{3} + \frac{89596960061195986729548007123709972187048830462824}{146950870562897446471823953227098191969219565650045} a^{2} + \frac{25710321350653479834567245782982923700140089364921}{29390174112579489294364790645419638393843913130009} a + \frac{4279722116258928542293798751472507539656829790}{24512238626004578227159958836880432355165899191} \) (order $10$)
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:  \( 45061837.8832 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4).D_4$ (as 16T121):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $(C_2\times C_4).D_4$
Character table for $(C_2\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, \(\Q(\zeta_{5})\), 4.0.12625.1, 8.4.164220320328125.2, 8.4.6568812813125.5, 8.0.159390625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
101Data not computed