↑

Properties

Label 16.8.15760966028...1761.1
Degree $16$
Signature $[8, 4]$
Discriminant $37^{14}\cdot 53^{10}$
Root discriminant $281.74$
Ramified primes $37, 53$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2\times OD_{16}).C_2$ (as 16T123)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-605912492, -423649600, 1977927713, -494535179, -1689125860, 1679761614, -652229481, 93173012, 10029551, -4574878, 323635, 17814, -963, 536, -120, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 120*x^14 + 536*x^13 - 963*x^12 + 17814*x^11 + 323635*x^10 - 4574878*x^9 + 10029551*x^8 + 93173012*x^7 - 652229481*x^6 + 1679761614*x^5 - 1689125860*x^4 - 494535179*x^3 + 1977927713*x^2 - 423649600*x - 605912492)
 
gp: K = bnfinit(x^16 - 3*x^15 - 120*x^14 + 536*x^13 - 963*x^12 + 17814*x^11 + 323635*x^10 - 4574878*x^9 + 10029551*x^8 + 93173012*x^7 - 652229481*x^6 + 1679761614*x^5 - 1689125860*x^4 - 494535179*x^3 + 1977927713*x^2 - 423649600*x - 605912492, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 120 x^{14} + 536 x^{13} - 963 x^{12} + 17814 x^{11} + 323635 x^{10} - 4574878 x^{9} + 10029551 x^{8} + 93173012 x^{7} - 652229481 x^{6} + 1679761614 x^{5} - 1689125860 x^{4} - 494535179 x^{3} + 1977927713 x^{2} - 423649600 x - 605912492 \)

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:  \(1576096602835514136235637969229632141761=37^{14}\cdot 53^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $281.74$
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:  $37, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{56} a^{13} + \frac{3}{28} a^{12} + \frac{1}{56} a^{11} - \frac{3}{56} a^{10} + \frac{13}{56} a^{9} + \frac{1}{7} a^{8} - \frac{1}{14} a^{7} - \frac{5}{14} a^{6} + \frac{23}{56} a^{5} + \frac{23}{56} a^{4} + \frac{27}{56} a^{3} - \frac{3}{14} a^{2} - \frac{5}{56} a + \frac{13}{28}$, $\frac{1}{1176} a^{14} + \frac{3}{392} a^{13} + \frac{145}{1176} a^{12} + \frac{1}{84} a^{11} + \frac{107}{588} a^{10} + \frac{89}{1176} a^{9} + \frac{31}{588} a^{8} - \frac{46}{147} a^{7} - \frac{527}{1176} a^{6} - \frac{1}{196} a^{5} - \frac{43}{588} a^{4} - \frac{131}{392} a^{3} - \frac{149}{392} a^{2} - \frac{17}{1176} a - \frac{17}{588}$, $\frac{1}{16065701601006502784038827218317901105940638303900459534640} a^{15} + \frac{1385457672419799947697030194376075111570807885514588373}{16065701601006502784038827218317901105940638303900459534640} a^{14} + \frac{658533935676289582427723599645502009561002951576198283}{286887528589401835429264771755676805463225683998222491690} a^{13} + \frac{744653944488044350672071014056379368389183589580246448827}{8032850800503251392019413609158950552970319151950229767320} a^{12} + \frac{123253791719094890858938369545705602329428386252716905521}{16065701601006502784038827218317901105940638303900459534640} a^{11} + \frac{161449362223974802745169447848145561413857498172089963109}{803285080050325139201941360915895055297031915195022976732} a^{10} + \frac{71162935777778984911143061701081231108734330279106535343}{1071046773400433518935921814554526740396042553593363968976} a^{9} - \frac{390694252882239457695466198206459293559435916112091365501}{2008212700125812848004853402289737638242579787987557441830} a^{8} + \frac{925031162446350946996420286238668719004200104813237823741}{5355233867002167594679609072772633701980212767966819844880} a^{7} - \frac{383305969208685172509334824284102146013714757122037965615}{803285080050325139201941360915895055297031915195022976732} a^{6} + \frac{7819183586357991148493481202682780488652111300253608883919}{16065701601006502784038827218317901105940638303900459534640} a^{5} - \frac{389982636843613104193217814869985695394471743555713856362}{1004106350062906424002426701144868819121289893993778720915} a^{4} - \frac{33045443424200971033680026713692768352955705592181137421}{382516704785869113905686362340902407284300911997629988920} a^{3} + \frac{4945518411866546302723637921826896998330549905618428855479}{16065701601006502784038827218317901105940638303900459534640} a^{2} - \frac{5876755887593747245586157629646844257147706212739078933823}{16065701601006502784038827218317901105940638303900459534640} a - \frac{2442645760758704213997034246278641420642522503163786576539}{8032850800503251392019413609158950552970319151950229767320}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 276563092521000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).C_2$ (as 16T123):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $(C_2\times OD_{16}).C_2$
Character table for $(C_2\times OD_{16}).C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{37}) \), \(\Q(\sqrt{1961}) \), \(\Q(\sqrt{53}) \), 4.4.2684609.1 x2, 4.4.142284277.1 x2, \(\Q(\sqrt{37}, \sqrt{53})\), 8.8.20244815481412729.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$37$37.8.7.2$x^{8} - 148$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
37.8.7.2$x^{8} - 148$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$53$53.8.6.2$x^{8} + 477 x^{4} + 70225$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
53.8.4.1$x^{8} + 101124 x^{4} - 148877 x^{2} + 2556515844$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$