Properties

Label 16.0.605...000.6
Degree $16$
Signature $[0, 8]$
Discriminant $6.052\times 10^{23}$
Root discriminant \(30.65\)
Ramified primes $2,3,5,7$
Class number $32$ (GRH)
Class group [4, 8] (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 11*x^14 + 96*x^12 - 781*x^10 + 6191*x^8 - 19525*x^6 + 60000*x^4 - 171875*x^2 + 390625)
 
gp: K = bnfinit(y^16 - 11*y^14 + 96*y^12 - 781*y^10 + 6191*y^8 - 19525*y^6 + 60000*y^4 - 171875*y^2 + 390625, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 11*x^14 + 96*x^12 - 781*x^10 + 6191*x^8 - 19525*x^6 + 60000*x^4 - 171875*x^2 + 390625);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 11*x^14 + 96*x^12 - 781*x^10 + 6191*x^8 - 19525*x^6 + 60000*x^4 - 171875*x^2 + 390625)
 

\( x^{16} - 11x^{14} + 96x^{12} - 781x^{10} + 6191x^{8} - 19525x^{6} + 60000x^{4} - 171875x^{2} + 390625 \) 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:   \(605165749776000000000000\) \(\medspace = 2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 7^{8}\) 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:  \(30.65\)
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:  $2\cdot 3^{1/2}5^{3/4}7^{1/2}\approx 30.645530678223075$
Ramified primes:   \(2\), \(3\), \(5\), \(7\) 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:  \(420=2^{2}\cdot 3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{420}(1,·)$, $\chi_{420}(209,·)$, $\chi_{420}(337,·)$, $\chi_{420}(83,·)$, $\chi_{420}(251,·)$, $\chi_{420}(419,·)$, $\chi_{420}(293,·)$, $\chi_{420}(167,·)$, $\chi_{420}(41,·)$, $\chi_{420}(43,·)$, $\chi_{420}(211,·)$, $\chi_{420}(169,·)$, $\chi_{420}(377,·)$, $\chi_{420}(379,·)$, $\chi_{420}(253,·)$, $\chi_{420}(127,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5}a^{9}-\frac{1}{5}a^{7}+\frac{1}{5}a^{5}-\frac{1}{5}a^{3}+\frac{1}{5}a$, $\frac{1}{154775}a^{10}+\frac{4}{25}a^{8}+\frac{6}{25}a^{6}+\frac{9}{25}a^{4}+\frac{1}{25}a^{2}-\frac{781}{6191}$, $\frac{1}{773875}a^{11}+\frac{4}{125}a^{9}+\frac{56}{125}a^{7}+\frac{34}{125}a^{5}-\frac{24}{125}a^{3}+\frac{1082}{6191}a$, $\frac{1}{3869375}a^{12}-\frac{11}{3869375}a^{10}-\frac{44}{625}a^{8}-\frac{241}{625}a^{6}+\frac{1}{625}a^{4}-\frac{781}{154775}a^{2}+\frac{96}{6191}$, $\frac{1}{19346875}a^{13}-\frac{11}{19346875}a^{11}-\frac{44}{3125}a^{9}+\frac{1009}{3125}a^{7}-\frac{624}{3125}a^{5}+\frac{153994}{773875}a^{3}-\frac{1219}{6191}a$, $\frac{1}{96734375}a^{14}-\frac{11}{96734375}a^{12}+\frac{96}{96734375}a^{10}-\frac{2741}{15625}a^{8}+\frac{1}{15625}a^{6}-\frac{781}{3869375}a^{4}+\frac{96}{154775}a^{2}-\frac{11}{6191}$, $\frac{1}{483671875}a^{15}-\frac{11}{483671875}a^{13}+\frac{96}{483671875}a^{11}-\frac{2741}{78125}a^{9}+\frac{15626}{78125}a^{7}-\frac{3870156}{19346875}a^{5}+\frac{154871}{773875}a^{3}-\frac{6202}{30955}a$ 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

$C_{4}\times C_{8}$, which has order $32$ (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:   \( -\frac{3311}{483671875} a^{15} + \frac{28896}{483671875} a^{13} - \frac{235081}{483671875} a^{11} + \frac{301}{78125} a^{9} - \frac{2286}{78125} a^{7} + \frac{28896}{773875} a^{5} - \frac{3311}{30955} a^{3} + \frac{1505}{6191} a \)  (order $20$) 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:   $\frac{1}{154775}a^{12}+\frac{48576}{154775}a^{2}-1$, $\frac{1}{30955}a^{11}+\frac{17621}{30955}a+1$, $\frac{2286}{483671875}a^{15}-\frac{25146}{483671875}a^{13}+\frac{1}{154775}a^{12}+\frac{219456}{483671875}a^{11}-\frac{301}{78125}a^{9}+\frac{2286}{78125}a^{7}-\frac{1785366}{19346875}a^{5}+\frac{219456}{773875}a^{3}+\frac{48576}{154775}a^{2}-\frac{25146}{30955}a$, $\frac{27}{19346875}a^{15}+\frac{228127}{19346875}a^{5}$, $\frac{6077}{483671875}a^{15}-\frac{376}{96734375}a^{14}-\frac{89497}{483671875}a^{13}-\frac{2339}{96734375}a^{12}+\frac{860667}{483671875}a^{11}+\frac{69504}{96734375}a^{10}-\frac{1082}{78125}a^{9}-\frac{109}{15625}a^{8}+\frac{8477}{78125}a^{7}+\frac{724}{15625}a^{6}-\frac{9581308}{19346875}a^{5}-\frac{2083688}{3869375}a^{4}+\frac{950473}{773875}a^{3}+\frac{351913}{154775}a^{2}-\frac{40384}{30955}a-\frac{26537}{6191}$, $\frac{29401}{483671875}a^{15}-\frac{9271}{96734375}a^{14}-\frac{251161}{483671875}a^{13}+\frac{89656}{96734375}a^{12}+\frac{2082121}{483671875}a^{11}-\frac{738816}{96734375}a^{10}-\frac{2766}{78125}a^{9}+\frac{986}{15625}a^{8}+\frac{21526}{78125}a^{7}-\frac{7696}{15625}a^{6}-\frac{7701366}{19346875}a^{5}+\frac{4188488}{3869375}a^{4}+\frac{1413573}{773875}a^{3}-\frac{504983}{154775}a^{2}-\frac{43833}{6191}a+\frac{84656}{6191}$, $\frac{1334}{483671875}a^{15}+\frac{257}{96734375}a^{14}-\frac{23424}{483671875}a^{13}+\frac{3748}{96734375}a^{12}+\frac{96814}{483671875}a^{11}+\frac{36722}{96734375}a^{10}-\frac{244}{78125}a^{9}+\frac{13}{15625}a^{8}+\frac{1334}{78125}a^{7}+\frac{57}{15625}a^{6}-\frac{1041854}{19346875}a^{5}-\frac{120234}{3869375}a^{4}+\frac{2684}{30955}a^{3}+\frac{35954}{154775}a^{2}-\frac{18961}{30955}a+\frac{3660}{6191}$ 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:  \( 107520.786508 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 107520.786508 \cdot 32}{20\cdot\sqrt{605165749776000000000000}}\cr\approx \mathstrut & 0.537173020129 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 11*x^14 + 96*x^12 - 781*x^10 + 6191*x^8 - 19525*x^6 + 60000*x^4 - 171875*x^2 + 390625)
 
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 - 11*x^14 + 96*x^12 - 781*x^10 + 6191*x^8 - 19525*x^6 + 60000*x^4 - 171875*x^2 + 390625, 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 - 11*x^14 + 96*x^12 - 781*x^10 + 6191*x^8 - 19525*x^6 + 60000*x^4 - 171875*x^2 + 390625);
 
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 - 11*x^14 + 96*x^12 - 781*x^10 + 6191*x^8 - 19525*x^6 + 60000*x^4 - 171875*x^2 + 390625);
 
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

$C_2^2\times C_4$ (as 16T2):

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 $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-105}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{21}) \), \(\Q(i, \sqrt{105})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{21})\), \(\Q(\sqrt{5}, \sqrt{-21})\), \(\Q(\sqrt{-5}, \sqrt{21})\), \(\Q(\sqrt{5}, \sqrt{21})\), \(\Q(\sqrt{-5}, \sqrt{-21})\), 4.4.882000.1, 4.0.55125.1, \(\Q(\zeta_{5})\), \(\Q(\zeta_{20})^+\), 8.0.31116960000.8, 8.0.777924000000.8, \(\Q(\zeta_{20})\), 8.0.777924000000.10, 8.0.777924000000.6, 8.8.777924000000.2, 8.0.3038765625.3

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 R R R ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.1.0.1}{1} }^{16}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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
\(2\) Copy content Toggle raw display 2.8.8.1$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
\(3\) Copy content Toggle raw display 3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(5\) Copy content Toggle raw display 5.4.3.2$x^{4} + 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} + 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} + 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} + 5$$4$$1$$3$$C_4$$[\ ]_{4}$
\(7\) Copy content Toggle raw display 7.8.4.1$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$