Properties

Label 16.0.208...625.1
Degree $16$
Signature $[0, 8]$
Discriminant $2.082\times 10^{20}$
Root discriminant \(18.62\)
Ramified primes $5,31$
Class number $3$
Class group [3]
Galois group $C_2^2 : C_4$ (as 16T10)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 2*x^14 - x^13 - 8*x^12 - 46*x^11 + 93*x^10 - 78*x^9 + 80*x^8 + 271*x^7 + 434*x^6 - 866*x^5 - 47*x^4 - 776*x^3 - 960*x^2 - 512*x + 4096)
 
gp: K = bnfinit(y^16 - 2*y^15 + 2*y^14 - y^13 - 8*y^12 - 46*y^11 + 93*y^10 - 78*y^9 + 80*y^8 + 271*y^7 + 434*y^6 - 866*y^5 - 47*y^4 - 776*y^3 - 960*y^2 - 512*y + 4096, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 2*x^14 - x^13 - 8*x^12 - 46*x^11 + 93*x^10 - 78*x^9 + 80*x^8 + 271*x^7 + 434*x^6 - 866*x^5 - 47*x^4 - 776*x^3 - 960*x^2 - 512*x + 4096);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 + 2*x^14 - x^13 - 8*x^12 - 46*x^11 + 93*x^10 - 78*x^9 + 80*x^8 + 271*x^7 + 434*x^6 - 866*x^5 - 47*x^4 - 776*x^3 - 960*x^2 - 512*x + 4096)
 

\( x^{16} - 2 x^{15} + 2 x^{14} - x^{13} - 8 x^{12} - 46 x^{11} + 93 x^{10} - 78 x^{9} + 80 x^{8} + 271 x^{7} + 434 x^{6} - 866 x^{5} - 47 x^{4} - 776 x^{3} - 960 x^{2} - 512 x + 4096 \) Copy content Toggle raw display

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(208225350937744140625\) \(\medspace = 5^{12}\cdot 31^{8}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(18.62\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $5^{3/4}31^{1/2}\approx 18.616942190179014$
Ramified primes:   \(5\), \(31\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $16$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{41}a^{10}+\frac{2}{41}a^{5}-\frac{3}{41}$, $\frac{1}{41}a^{11}+\frac{2}{41}a^{6}-\frac{3}{41}a$, $\frac{1}{1025}a^{12}+\frac{11}{1025}a^{11}+\frac{1}{1025}a^{10}-\frac{6}{25}a^{8}+\frac{207}{1025}a^{7}+\frac{22}{1025}a^{6}-\frac{162}{1025}a^{5}+\frac{1}{25}a^{4}+\frac{4}{25}a^{3}-\frac{44}{1025}a^{2}-\frac{64}{205}a+\frac{161}{1025}$, $\frac{1}{155800}a^{13}-\frac{29}{77900}a^{12}+\frac{421}{77900}a^{11}-\frac{1569}{155800}a^{10}-\frac{7}{475}a^{9}-\frac{21647}{77900}a^{8}-\frac{53211}{155800}a^{7}-\frac{4563}{15580}a^{6}-\frac{6532}{19475}a^{5}-\frac{173}{760}a^{4}+\frac{4809}{15580}a^{3}-\frac{38617}{77900}a^{2}-\frac{7159}{155800}a-\frac{7873}{19475}$, $\frac{1}{1246400}a^{14}-\frac{1}{623200}a^{13}-\frac{63}{623200}a^{12}-\frac{1537}{1246400}a^{11}+\frac{83}{31160}a^{10}+\frac{13973}{124640}a^{9}+\frac{46649}{249280}a^{8}-\frac{1597}{32800}a^{7}+\frac{25889}{77900}a^{6}+\frac{434639}{1246400}a^{5}+\frac{49253}{124640}a^{4}+\frac{13077}{32800}a^{3}-\frac{537583}{1246400}a^{2}+\frac{72639}{155800}a+\frac{7209}{19475}$, $\frac{1}{49856000}a^{15}-\frac{1}{4985600}a^{14}+\frac{1}{608000}a^{13}+\frac{22383}{49856000}a^{12}-\frac{2339}{389500}a^{11}-\frac{9323}{4985600}a^{10}-\frac{2571827}{49856000}a^{9}+\frac{213789}{608000}a^{8}+\frac{17478}{97375}a^{7}+\frac{1116943}{49856000}a^{6}-\frac{719431}{4985600}a^{5}+\frac{8035687}{24928000}a^{4}+\frac{38521}{1216000}a^{3}-\frac{92129}{3116000}a^{2}+\frac{367003}{779000}a+\frac{23341}{97375}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

$C_{3}$, which has order $3$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{7}{124640} a^{15} + \frac{21}{249280} a^{14} - \frac{7}{24928} a^{13} + \frac{43}{15580} a^{12} + \frac{7}{249280} a^{11} + \frac{189}{62320} a^{10} - \frac{7}{3280} a^{9} + \frac{3017}{249280} a^{8} - \frac{15179}{124640} a^{7} + \frac{7}{124640} a^{6} - \frac{12453}{249280} a^{5} - \frac{609}{24928} a^{4} - \frac{1099}{15580} a^{3} + \frac{192809}{249280} a^{2} + \frac{336}{3895} a + \frac{896}{3895} \)  (order $10$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{89}{1994240}a^{15}-\frac{153}{997120}a^{14}+\frac{953}{997120}a^{13}-\frac{5849}{1994240}a^{12}+\frac{39}{249280}a^{11}-\frac{1759}{997120}a^{10}+\frac{10581}{1994240}a^{9}-\frac{25359}{997120}a^{8}+\frac{15961}{124640}a^{7}-\frac{4213}{398848}a^{6}+\frac{19889}{997120}a^{5}-\frac{5609}{997120}a^{4}-\frac{505303}{1994240}a^{3}-\frac{189497}{249280}a^{2}-\frac{14}{3895}a-\frac{528}{3895}$, $\frac{9477}{24928000}a^{15}-\frac{1089}{2492800}a^{14}+\frac{2917}{12464000}a^{13}+\frac{22651}{24928000}a^{12}+\frac{3387}{3116000}a^{11}-\frac{489}{26240}a^{10}+\frac{485841}{24928000}a^{9}-\frac{9233}{656000}a^{8}-\frac{43883}{1558000}a^{7}-\frac{1555509}{24928000}a^{6}+\frac{15601}{60800}a^{5}-\frac{112119}{656000}a^{4}+\frac{2267797}{24928000}a^{3}+\frac{223899}{3116000}a^{2}+\frac{45204}{97375}a-\frac{125721}{97375}$, $\frac{4369}{49856000}a^{15}+\frac{599}{4985600}a^{14}-\frac{34871}{24928000}a^{13}-\frac{63233}{49856000}a^{12}-\frac{3793}{3116000}a^{11}-\frac{34811}{4985600}a^{10}-\frac{192323}{49856000}a^{9}+\frac{1573661}{24928000}a^{8}+\frac{68067}{1558000}a^{7}+\frac{2959487}{49856000}a^{6}+\frac{672097}{4985600}a^{5}-\frac{1453497}{24928000}a^{4}-\frac{21137711}{49856000}a^{3}-\frac{36057}{389500}a^{2}-\frac{145619}{389500}a-\frac{34426}{97375}$, $\frac{7153}{24928000}a^{15}-\frac{1823}{2492800}a^{14}+\frac{10693}{12464000}a^{13}-\frac{30361}{24928000}a^{12}+\frac{7091}{6232000}a^{11}-\frac{31019}{2492800}a^{10}+\frac{31591}{1312000}a^{9}-\frac{312313}{12464000}a^{8}+\frac{140061}{3116000}a^{7}-\frac{1696801}{24928000}a^{6}+\frac{227831}{2492800}a^{5}+\frac{523691}{12464000}a^{4}-\frac{6370327}{24928000}a^{3}+\frac{488377}{6232000}a^{2}+\frac{433463}{779000}a-\frac{35729}{97375}$, $\frac{2969}{49856000}a^{15}-\frac{241}{4985600}a^{14}+\frac{32529}{24928000}a^{13}-\frac{84233}{49856000}a^{12}-\frac{2393}{3116000}a^{11}-\frac{19411}{4985600}a^{10}+\frac{192677}{49856000}a^{9}-\frac{1350139}{24928000}a^{8}+\frac{91867}{1558000}a^{7}+\frac{1684087}{49856000}a^{6}+\frac{307817}{4985600}a^{5}-\frac{3277697}{24928000}a^{4}+\frac{15251289}{49856000}a^{3}-\frac{23457}{389500}a^{2}-\frac{67219}{389500}a+\frac{85349}{97375}$, $\frac{1277}{3116000}a^{15}-\frac{489}{623200}a^{14}+\frac{991}{779000}a^{13}-\frac{1179}{3116000}a^{12}-\frac{9629}{3116000}a^{11}-\frac{7193}{311600}a^{10}+\frac{110771}{3116000}a^{9}-\frac{187109}{3116000}a^{8}+\frac{59091}{1558000}a^{7}+\frac{385371}{3116000}a^{6}+\frac{184969}{623200}a^{5}-\frac{47643}{779000}a^{4}+\frac{488267}{3116000}a^{3}-\frac{515523}{3116000}a^{2}-\frac{737779}{779000}a-\frac{56398}{97375}$, $\frac{41517}{49856000}a^{15}-\frac{13773}{4985600}a^{14}+\frac{98037}{24928000}a^{13}-\frac{218109}{49856000}a^{12}-\frac{2541}{779000}a^{11}-\frac{26179}{997120}a^{10}+\frac{5599881}{49856000}a^{9}-\frac{3785887}{24928000}a^{8}+\frac{73829}{389500}a^{7}+\frac{4265891}{49856000}a^{6}-\frac{274907}{4985600}a^{5}-\frac{14618021}{24928000}a^{4}+\frac{12545677}{49856000}a^{3}-\frac{697553}{3116000}a^{2}-\frac{53831}{41000}a+\frac{257522}{97375}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 5085.12397787 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 5085.12397787 \cdot 3}{10\cdot\sqrt{208225350937744140625}}\cr\approx \mathstrut & 0.256799956563 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 2*x^14 - x^13 - 8*x^12 - 46*x^11 + 93*x^10 - 78*x^9 + 80*x^8 + 271*x^7 + 434*x^6 - 866*x^5 - 47*x^4 - 776*x^3 - 960*x^2 - 512*x + 4096)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^16 - 2*x^15 + 2*x^14 - x^13 - 8*x^12 - 46*x^11 + 93*x^10 - 78*x^9 + 80*x^8 + 271*x^7 + 434*x^6 - 866*x^5 - 47*x^4 - 776*x^3 - 960*x^2 - 512*x + 4096, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^16 - 2*x^15 + 2*x^14 - x^13 - 8*x^12 - 46*x^11 + 93*x^10 - 78*x^9 + 80*x^8 + 271*x^7 + 434*x^6 - 866*x^5 - 47*x^4 - 776*x^3 - 960*x^2 - 512*x + 4096);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 + 2*x^14 - x^13 - 8*x^12 - 46*x^11 + 93*x^10 - 78*x^9 + 80*x^8 + 271*x^7 + 434*x^6 - 866*x^5 - 47*x^4 - 776*x^3 - 960*x^2 - 512*x + 4096);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_2^2:C_4$ (as 16T10):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_2^2 : C_4$
Character table for $C_2^2 : C_4$

Intermediate fields

\(\Q(\sqrt{-155}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-31}) \), \(\Q(\sqrt{5}, \sqrt{-31})\), 4.2.775.1 x2, 4.0.4805.1 x2, 4.0.120125.1 x2, 4.2.3875.1 x2, \(\Q(\zeta_{5})\), 4.4.120125.1, 8.0.577200625.1, 8.0.14430015625.2, 8.0.14430015625.1, 8.0.15015625.1 x2, 8.4.14430015625.1 x2

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Degree 8 siblings: 8.4.14430015625.1, 8.0.15015625.1
Minimal sibling: 8.0.15015625.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{4}$ ${\href{/padicField/3.4.0.1}{4} }^{4}$ R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ R ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.1.0.1}{1} }^{16}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(5\) Copy content Toggle raw display 5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
\(31\) Copy content Toggle raw display 31.4.2.1$x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$