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Properties

Label 16.0.605...000.4
Degree $16$
Signature $[0, 8]$
Discriminant $6.052\times 10^{23}$
Root discriminant \(30.65\)
Ramified primes $2,3,5,7$
Class number $8$ (GRH)
Class group [2, 2, 2] (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 + 4*x^14 + 50*x^12 - 114*x^10 + 1259*x^8 - 3294*x^6 + 11675*x^4 - 15881*x^2 + 22801)
 
gp: K = bnfinit(y^16 + 4*y^14 + 50*y^12 - 114*y^10 + 1259*y^8 - 3294*y^6 + 11675*y^4 - 15881*y^2 + 22801, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 4*x^14 + 50*x^12 - 114*x^10 + 1259*x^8 - 3294*x^6 + 11675*x^4 - 15881*x^2 + 22801);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 4*x^14 + 50*x^12 - 114*x^10 + 1259*x^8 - 3294*x^6 + 11675*x^4 - 15881*x^2 + 22801)
 

\( x^{16} + 4x^{14} + 50x^{12} - 114x^{10} + 1259x^{8} - 3294x^{6} + 11675x^{4} - 15881x^{2} + 22801 \) 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$$\chi_{420}(1,·)$, $\chi_{420}(323,·)$, $\chi_{420}(197,·)$, $\chi_{420}(391,·)$, $\chi_{420}(139,·)$, $\chi_{420}(83,·)$, $\chi_{420}(407,·)$, $\chi_{420}(349,·)$, $\chi_{420}(293,·)$, $\chi_{420}(167,·)$, $\chi_{420}(169,·)$, $\chi_{420}(113,·)$, $\chi_{420}(211,·)$, $\chi_{420}(181,·)$, $\chi_{420}(377,·)$, $\chi_{420}(379,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{10}+\frac{1}{3}a^{6}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{11}+\frac{1}{3}a^{7}-\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{24573148562793}a^{14}-\frac{2552008684120}{24573148562793}a^{12}+\frac{114174152058}{264227403901}a^{10}-\frac{8115531350084}{24573148562793}a^{8}+\frac{1684474300064}{8191049520931}a^{6}-\frac{343598590231}{24573148562793}a^{4}-\frac{5135159655305}{24573148562793}a^{2}+\frac{1516452978428}{8191049520931}$, $\frac{1}{37\!\cdots\!43}a^{15}+\frac{407000467362430}{37\!\cdots\!43}a^{13}+\frac{40240860445225}{119695013967153}a^{11}+\frac{14\!\cdots\!96}{37\!\cdots\!43}a^{9}-\frac{838624677755701}{37\!\cdots\!43}a^{7}+\frac{11\!\cdots\!54}{37\!\cdots\!43}a^{5}+\frac{568238306809865}{37\!\cdots\!43}a^{3}-\frac{716262998906644}{37\!\cdots\!43}a$ 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:  $3$

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:   \( -\frac{437437}{3838986401} a^{15} + \frac{1584030}{3838986401} a^{13} + \frac{34600}{123838271} a^{11} + \frac{261130710}{3838986401} a^{9} - \frac{455030440}{3838986401} a^{7} + \frac{4579621900}{3838986401} a^{5} - \frac{7029315280}{3838986401} a^{3} + \frac{18471719510}{3838986401} a \)  (order $4$) 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:   $\frac{50644}{56103207}a^{14}+\frac{239786}{56103207}a^{12}+\frac{817616}{18701069}a^{10}-\frac{5550197}{56103207}a^{8}+\frac{15116734}{18701069}a^{6}-\frac{126037294}{56103207}a^{4}+\frac{283867327}{56103207}a^{2}-\frac{99308186}{18701069}$, $\frac{82653508894}{37\!\cdots\!43}a^{15}-\frac{1213404164165}{37\!\cdots\!43}a^{13}-\frac{219056591975}{119695013967153}a^{11}-\frac{108301361946230}{37\!\cdots\!43}a^{9}+\frac{44971765426310}{37\!\cdots\!43}a^{7}-\frac{16\!\cdots\!58}{37\!\cdots\!43}a^{5}+\frac{14\!\cdots\!85}{37\!\cdots\!43}a^{3}-\frac{77\!\cdots\!58}{37\!\cdots\!43}a$, $\frac{81428440}{264227403901}a^{14}+\frac{1364672159}{792682211703}a^{12}+\frac{11982022327}{792682211703}a^{10}-\frac{6971930378}{264227403901}a^{8}+\frac{163846353965}{792682211703}a^{6}-\frac{92575539237}{264227403901}a^{4}+\frac{675807890200}{792682211703}a^{2}-\frac{26189237044}{792682211703}$, $\frac{94251889468}{12\!\cdots\!81}a^{15}+\frac{205114046}{346100683983}a^{14}+\frac{444150173405}{12\!\cdots\!81}a^{13}+\frac{263323819}{346100683983}a^{12}+\frac{136789064742}{39898337989051}a^{11}+\frac{57903361}{3721512731}a^{10}-\frac{9772317582108}{12\!\cdots\!81}a^{9}-\frac{65579049718}{346100683983}a^{8}+\frac{84762598464381}{12\!\cdots\!81}a^{7}+\frac{61359875541}{115366894661}a^{6}-\frac{65870425448707}{12\!\cdots\!81}a^{5}-\frac{1172400834941}{346100683983}a^{4}+\frac{261001229345088}{12\!\cdots\!81}a^{3}+\frac{1978195696313}{346100683983}a^{2}+\frac{15\!\cdots\!40}{12\!\cdots\!81}a-\frac{1472994827356}{115366894661}$, $\frac{762949831688}{37\!\cdots\!43}a^{15}-\frac{9290}{29962833}a^{14}-\frac{1848662052415}{37\!\cdots\!43}a^{13}-\frac{105265}{29962833}a^{12}+\frac{152171816375}{119695013967153}a^{11}-\frac{9073}{322181}a^{10}-\frac{396486757724830}{37\!\cdots\!43}a^{9}-\frac{2713175}{29962833}a^{8}+\frac{834641207711530}{37\!\cdots\!43}a^{7}-\frac{2761255}{9987611}a^{6}-\frac{71\!\cdots\!42}{37\!\cdots\!43}a^{5}-\frac{34185505}{29962833}a^{4}+\frac{12\!\cdots\!95}{37\!\cdots\!43}a^{3}+\frac{19653550}{29962833}a^{2}-\frac{31\!\cdots\!45}{37\!\cdots\!43}a-\frac{84471433}{9987611}$, $\frac{1356324175016}{37\!\cdots\!43}a^{15}+\frac{3117333082}{8191049520931}a^{14}+\frac{2056499574730}{12\!\cdots\!81}a^{13}+\frac{21359547241}{24573148562793}a^{12}+\frac{2171801565931}{119695013967153}a^{11}+\frac{11853297983}{792682211703}a^{10}-\frac{125933034021223}{37\!\cdots\!43}a^{9}-\frac{608460954389}{8191049520931}a^{8}+\frac{15\!\cdots\!95}{37\!\cdots\!43}a^{7}+\frac{13311140382523}{24573148562793}a^{6}-\frac{25\!\cdots\!98}{37\!\cdots\!43}a^{5}-\frac{11902822003806}{8191049520931}a^{4}+\frac{37\!\cdots\!42}{12\!\cdots\!81}a^{3}+\frac{130071339824174}{24573148562793}a^{2}-\frac{230229368324617}{37\!\cdots\!43}a-\frac{125085770734325}{24573148562793}$, $\frac{44340142241}{119695013967153}a^{15}+\frac{19969154780}{24573148562793}a^{14}+\frac{230307292316}{119695013967153}a^{13}+\frac{108223048063}{24573148562793}a^{12}+\frac{2277354347969}{119695013967153}a^{11}+\frac{12411993889}{264227403901}a^{10}-\frac{2643404935801}{119695013967153}a^{9}-\frac{498954388528}{24573148562793}a^{8}+\frac{44893433256067}{119695013967153}a^{7}+\frac{7915130099852}{8191049520931}a^{6}-\frac{50711982480311}{119695013967153}a^{5}-\frac{36118765604945}{24573148562793}a^{4}+\frac{95927733520009}{119695013967153}a^{3}+\frac{101742935776442}{24573148562793}a^{2}+\frac{201168123984196}{119695013967153}a+\frac{6700064623145}{8191049520931}$ 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:  \( 67203.8653181 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 67203.8653181 \cdot 8}{4\cdot\sqrt{605165749776000000000000}}\cr\approx \mathstrut & 0.419687491016 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 + 4*x^14 + 50*x^12 - 114*x^10 + 1259*x^8 - 3294*x^6 + 11675*x^4 - 15881*x^2 + 22801)
 
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 + 4*x^14 + 50*x^12 - 114*x^10 + 1259*x^8 - 3294*x^6 + 11675*x^4 - 15881*x^2 + 22801, 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 + 4*x^14 + 50*x^12 - 114*x^10 + 1259*x^8 - 3294*x^6 + 11675*x^4 - 15881*x^2 + 22801);
 
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 + 4*x^14 + 50*x^12 - 114*x^10 + 1259*x^8 - 3294*x^6 + 11675*x^4 - 15881*x^2 + 22801);
 
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{35}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-7}) \), \(\Q(i, \sqrt{35})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{7})\), \(\Q(\sqrt{5}, \sqrt{7})\), \(\Q(\sqrt{-5}, \sqrt{-7})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{-5}, \sqrt{7})\), 4.4.882000.1, 4.0.55125.1, \(\Q(\zeta_{15})^+\), 4.0.18000.1, 8.0.384160000.1, 8.0.777924000000.8, 8.0.324000000.1, 8.8.777924000000.3, 8.0.777924000000.3, 8.0.777924000000.4, 8.0.3038765625.2

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 ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.1.0.1}{1} }^{16}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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 3.4.2.2$x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
\(5\) 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}$
\(7\) 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}$