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Properties

Label 16.0.649...000.9
Degree $16$
Signature $[0, 8]$
Discriminant $6.491\times 10^{33}$
Root discriminant \(129.80\)
Ramified primes $2,5,7$
Class number $1427456$ (GRH)
Class group [2, 4, 4, 8, 5576] (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 280*x^14 + 30380*x^12 + 1646400*x^10 + 47899950*x^8 + 739508000*x^6 + 5529503000*x^4 + 16470860000*x^2 + 14412002500)
 
gp: K = bnfinit(y^16 + 280*y^14 + 30380*y^12 + 1646400*y^10 + 47899950*y^8 + 739508000*y^6 + 5529503000*y^4 + 16470860000*y^2 + 14412002500, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 280*x^14 + 30380*x^12 + 1646400*x^10 + 47899950*x^8 + 739508000*x^6 + 5529503000*x^4 + 16470860000*x^2 + 14412002500);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 280*x^14 + 30380*x^12 + 1646400*x^10 + 47899950*x^8 + 739508000*x^6 + 5529503000*x^4 + 16470860000*x^2 + 14412002500)
 

\( x^{16} + 280 x^{14} + 30380 x^{12} + 1646400 x^{10} + 47899950 x^{8} + 739508000 x^{6} + \cdots + 14412002500 \) 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:   \(6490588908866265677824000000000000\) \(\medspace = 2^{62}\cdot 5^{12}\cdot 7^{8}\) 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:  \(129.80\)
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:  $2^{31/8}5^{3/4}7^{1/2}\approx 129.79792701359696$
Ramified primes:   \(2\), \(5\), \(7\) 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 and abelian over $\Q$.
Conductor:  \(1120=2^{5}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{1120}(1,·)$, $\chi_{1120}(643,·)$, $\chi_{1120}(449,·)$, $\chi_{1120}(841,·)$, $\chi_{1120}(587,·)$, $\chi_{1120}(83,·)$, $\chi_{1120}(281,·)$, $\chi_{1120}(729,·)$, $\chi_{1120}(923,·)$, $\chi_{1120}(867,·)$, $\chi_{1120}(561,·)$, $\chi_{1120}(169,·)$, $\chi_{1120}(363,·)$, $\chi_{1120}(27,·)$, $\chi_{1120}(1009,·)$, $\chi_{1120}(307,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $\frac{1}{7}a^{2}$, $\frac{1}{7}a^{3}$, $\frac{1}{245}a^{4}$, $\frac{1}{245}a^{5}$, $\frac{1}{1715}a^{6}$, $\frac{1}{1715}a^{7}$, $\frac{1}{120050}a^{8}$, $\frac{1}{120050}a^{9}$, $\frac{1}{840350}a^{10}$, $\frac{1}{840350}a^{11}$, $\frac{1}{29412250}a^{12}$, $\frac{1}{29412250}a^{13}$, $\frac{1}{44677207750}a^{14}-\frac{38}{3191229125}a^{12}+\frac{3}{26050850}a^{10}+\frac{43}{26050850}a^{8}-\frac{13}{372155}a^{6}+\frac{1}{53165}a^{4}+\frac{78}{1519}a^{2}-\frac{71}{217}$, $\frac{1}{44677207750}a^{15}-\frac{38}{3191229125}a^{13}+\frac{3}{26050850}a^{11}+\frac{43}{26050850}a^{9}-\frac{13}{372155}a^{7}+\frac{1}{53165}a^{5}+\frac{78}{1519}a^{3}-\frac{71}{217}a$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{8}\times C_{5576}$, which has order $1427456$ (assuming GRH)

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:   \( -1 \)  (order $2$) 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:   $\frac{26}{22338603875}a^{14}+\frac{169}{638245825}a^{12}+\frac{264}{13025425}a^{10}+\frac{2921}{5210170}a^{8}+\frac{192}{372155}a^{6}-\frac{1986}{10633}a^{4}-\frac{2888}{1519}a^{2}-\frac{871}{217}$, $\frac{4}{3191229125}a^{14}+\frac{291}{911779750}a^{12}+\frac{394}{13025425}a^{10}+\frac{2528}{1860775}a^{8}+\frac{1632}{53165}a^{6}+\frac{2488}{7595}a^{4}+\frac{36}{31}a^{2}+\frac{21}{31}$, $\frac{83}{22338603875}a^{14}+\frac{3023}{3191229125}a^{12}+\frac{2327}{26050850}a^{10}+\frac{51092}{13025425}a^{8}+\frac{31043}{372155}a^{6}+\frac{42264}{53165}a^{4}+\frac{4051}{1519}a^{2}+\frac{583}{217}$, $\frac{2}{22338603875}a^{14}+\frac{347}{6382458250}a^{12}+\frac{26}{2605085}a^{10}+\frac{20787}{26050850}a^{8}+\frac{11232}{372155}a^{6}+\frac{27346}{53165}a^{4}+\frac{4652}{1519}a^{2}+\frac{584}{217}$, $\frac{13}{22338603875}a^{14}+\frac{531}{3191229125}a^{12}+\frac{481}{26050850}a^{10}+\frac{26507}{26050850}a^{8}+\frac{10946}{372155}a^{6}+\frac{22594}{53165}a^{4}+\frac{3981}{1519}a^{2}+\frac{1409}{217}$, $\frac{39}{22338603875}a^{14}+\frac{1376}{3191229125}a^{12}+\frac{1009}{26050850}a^{10}+\frac{20556}{13025425}a^{8}+\frac{11138}{372155}a^{6}+\frac{12664}{53165}a^{4}+\frac{1093}{1519}a^{2}+\frac{321}{217}$, $\frac{57}{22338603875}a^{14}+\frac{2178}{3191229125}a^{12}+\frac{257}{3721550}a^{10}+\frac{87579}{26050850}a^{8}+\frac{30851}{372155}a^{6}+\frac{52194}{53165}a^{4}+\frac{6939}{1519}a^{2}+\frac{1237}{217}$ Copy content Toggle raw display (assuming GRH)
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:  \( 12198.951274811623 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 12198.951274811623 \cdot 1427456}{2\cdot\sqrt{6490588908866265677824000000000000}}\cr\approx \mathstrut & 0.262513792728306 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 + 280*x^14 + 30380*x^12 + 1646400*x^10 + 47899950*x^8 + 739508000*x^6 + 5529503000*x^4 + 16470860000*x^2 + 14412002500)
 
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 + 280*x^14 + 30380*x^12 + 1646400*x^10 + 47899950*x^8 + 739508000*x^6 + 5529503000*x^4 + 16470860000*x^2 + 14412002500, 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 + 280*x^14 + 30380*x^12 + 1646400*x^10 + 47899950*x^8 + 739508000*x^6 + 5529503000*x^4 + 16470860000*x^2 + 14412002500);
 
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 + 280*x^14 + 30380*x^12 + 1646400*x^10 + 47899950*x^8 + 739508000*x^6 + 5529503000*x^4 + 16470860000*x^2 + 14412002500);
 
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

$C_2\times C_8$ (as 16T5):

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)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 4.4.51200.1, \(\Q(\zeta_{16})^+\), 8.8.2621440000.1, 8.0.80564191232000000.78, 8.0.80564191232000000.92

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.8.0.1}{8} }^{2}$ R R ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.1.0.1}{1} }^{16}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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

# 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
\(2\) Copy content Toggle raw display Deg $16$$8$$2$$62$
\(5\) Copy content Toggle raw display 5.16.12.2$x^{16} - 40 x^{12} + 500 x^{8} + 2500$$4$$4$$12$$C_8\times C_2$$[\ ]_{4}^{4}$
\(7\) Copy content Toggle raw display 7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$