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Properties

Label 16.0.64803441755...0625.3
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 61^{12}$
Root discriminant $72.98$
Ramified primes $5, 61$
Class number $144$ (GRH)
Class group $[12, 12]$ (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![16777216, -44564480, 65519616, -64197120, 46030480, -24600040, 10091364, -3163050, 771979, -149175, 27324, -5065, 1105, -195, 36, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 36*x^14 - 195*x^13 + 1105*x^12 - 5065*x^11 + 27324*x^10 - 149175*x^9 + 771979*x^8 - 3163050*x^7 + 10091364*x^6 - 24600040*x^5 + 46030480*x^4 - 64197120*x^3 + 65519616*x^2 - 44564480*x + 16777216)
 
gp: K = bnfinit(x^16 - 5*x^15 + 36*x^14 - 195*x^13 + 1105*x^12 - 5065*x^11 + 27324*x^10 - 149175*x^9 + 771979*x^8 - 3163050*x^7 + 10091364*x^6 - 24600040*x^5 + 46030480*x^4 - 64197120*x^3 + 65519616*x^2 - 44564480*x + 16777216, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 36 x^{14} - 195 x^{13} + 1105 x^{12} - 5065 x^{11} + 27324 x^{10} - 149175 x^{9} + 771979 x^{8} - 3163050 x^{7} + 10091364 x^{6} - 24600040 x^{5} + 46030480 x^{4} - 64197120 x^{3} + 65519616 x^{2} - 44564480 x + 16777216 \)

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:  \(648034417553121620683837890625=5^{12}\cdot 61^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.98$
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:  $5, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{3}{8} a^{8} - \frac{3}{8} a^{7} + \frac{3}{8} a^{6} + \frac{1}{8} a^{4} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{11} - \frac{3}{16} a^{9} + \frac{5}{16} a^{8} - \frac{5}{16} a^{7} - \frac{1}{2} a^{6} - \frac{7}{16} a^{5} - \frac{1}{16} a^{4} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{512} a^{13} - \frac{5}{512} a^{12} - \frac{7}{128} a^{11} - \frac{3}{512} a^{10} - \frac{47}{512} a^{9} + \frac{247}{512} a^{8} - \frac{1}{128} a^{7} - \frac{247}{512} a^{6} - \frac{245}{512} a^{5} + \frac{11}{256} a^{4} + \frac{9}{128} a^{3} - \frac{29}{64} a^{2} + \frac{9}{32} a$, $\frac{1}{26008575584026624} a^{14} - \frac{21322013041253}{26008575584026624} a^{13} + \frac{99749918616833}{6502143896006656} a^{12} - \frac{225319573323331}{26008575584026624} a^{11} + \frac{768709465723761}{26008575584026624} a^{10} - \frac{117808409977427}{1368872399159296} a^{9} + \frac{2185663802151943}{6502143896006656} a^{8} - \frac{7255097603247415}{26008575584026624} a^{7} - \frac{3310236731049429}{26008575584026624} a^{6} - \frac{458085208359653}{13004287792013312} a^{5} - \frac{3230170000784951}{6502143896006656} a^{4} + \frac{1070442588317843}{3251071948003328} a^{3} + \frac{710406344888409}{1625535974001664} a^{2} + \frac{281413249217}{668394726152} a + \frac{86439925466}{1587437474611}$, $\frac{1}{57904044518461362368412338618368} a^{15} - \frac{607775075731109}{57904044518461362368412338618368} a^{14} - \frac{3351985268076771113664436143}{14476011129615340592103084654592} a^{13} + \frac{1697215244675046268565902266557}{57904044518461362368412338618368} a^{12} + \frac{2962165388987533674693564731441}{57904044518461362368412338618368} a^{11} - \frac{4337006400331998157529440595049}{57904044518461362368412338618368} a^{10} - \frac{322008910487854531287612874857}{14476011129615340592103084654592} a^{9} + \frac{21611798067035106407616611130313}{57904044518461362368412338618368} a^{8} - \frac{8679738645740480014867921124373}{57904044518461362368412338618368} a^{7} - \frac{12760560847859463611427902971461}{28952022259230681184206169309184} a^{6} - \frac{1551583277204313184947126300567}{14476011129615340592103084654592} a^{5} + \frac{2875985121850729314135922636147}{7238005564807670296051542327296} a^{4} + \frac{699177159561725971785596147705}{3619002782403835148025771163648} a^{3} - \frac{5484208847663798990837812227}{56546918475059924187902674432} a^{2} + \frac{435020174335743795415132183}{883545601172811315435979288} a - \frac{48392705403046624027484128}{110443200146601414429497411}$

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

Class group and class number

$C_{12}\times C_{12}$, which has order $144$ (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:  \( \frac{15290731575489343349732979}{57904044518461362368412338618368} a^{15} - \frac{87153799645506472310168415}{57904044518461362368412338618368} a^{14} + \frac{142965293464809670101649395}{14476011129615340592103084654592} a^{13} - \frac{3198219463801529326182992409}{57904044518461362368412338618368} a^{12} + \frac{17978476061899386378137976771}{57904044518461362368412338618368} a^{11} - \frac{83187231147717388124996013675}{57904044518461362368412338618368} a^{10} + \frac{109963981239187596649466953125}{14476011129615340592103084654592} a^{9} - \frac{2424941846160671863200332761525}{57904044518461362368412338618368} a^{8} + \frac{12615259497529541800350206486865}{57904044518461362368412338618368} a^{7} - \frac{26074591441495183078010976990687}{28952022259230681184206169309184} a^{6} + \frac{41376604132656663609605780083275}{14476011129615340592103084654592} a^{5} - \frac{49272660077372967171441167211255}{7238005564807670296051542327296} a^{4} + \frac{44145742562434671376996703389947}{3619002782403835148025771163648} a^{3} - \frac{444048731104135470659561143635}{28273459237529962093951337216} a^{2} + \frac{24663752231540120075785824423}{1767091202345622630871958576} a - \frac{746700516269377327873238646}{110443200146601414429497411} \) (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:  \( 4866018.67571 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{305}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{5}, \sqrt{61})\), 4.4.5674525.1 x2, 4.4.1134905.1 x2, 4.0.465125.1, \(\Q(\zeta_{5})\), 8.8.32200233975625.1, 8.0.216341265625.2, 8.0.805005849390625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$61$61.4.3.1$x^{4} - 61$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.1$x^{4} - 61$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.1$x^{4} - 61$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.1$x^{4} - 61$$4$$1$$3$$C_4$$[\ ]_{4}$