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Properties

Label 16.0.181...536.1
Degree $16$
Signature $[0, 8]$
Discriminant $1.812\times 10^{22}$
Root discriminant \(24.61\)
Ramified primes $2,3,7,13$
Class number $6$
Class group [6]
Galois group $C_2^2 \times D_4$ (as 16T25)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 32*x^14 + 374*x^12 + 1968*x^10 + 4595*x^8 + 4080*x^6 + 1298*x^4 + 80*x^2 + 1)
 
gp: K = bnfinit(y^16 + 32*y^14 + 374*y^12 + 1968*y^10 + 4595*y^8 + 4080*y^6 + 1298*y^4 + 80*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 32*x^14 + 374*x^12 + 1968*x^10 + 4595*x^8 + 4080*x^6 + 1298*x^4 + 80*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 32*x^14 + 374*x^12 + 1968*x^10 + 4595*x^8 + 4080*x^6 + 1298*x^4 + 80*x^2 + 1)
 

\( x^{16} + 32x^{14} + 374x^{12} + 1968x^{10} + 4595x^{8} + 4080x^{6} + 1298x^{4} + 80x^{2} + 1 \) 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:   \(18123733314413355073536\) \(\medspace = 2^{24}\cdot 3^{8}\cdot 7^{8}\cdot 13^{4}\) 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:  \(24.61\)
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:  $2^{3/2}3^{1/2}7^{1/2}13^{1/2}\approx 46.73328578219169$
Ramified primes:   \(2\), \(3\), \(7\), \(13\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $8$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{4}a^{9}-\frac{1}{2}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a$, $\frac{1}{4}a^{10}-\frac{1}{2}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{3}-\frac{1}{2}$, $\frac{1}{24}a^{12}+\frac{1}{24}a^{10}-\frac{1}{2}a^{7}-\frac{7}{24}a^{6}+\frac{1}{4}a^{4}+\frac{7}{24}a^{2}-\frac{11}{24}$, $\frac{1}{24}a^{13}+\frac{1}{24}a^{11}-\frac{7}{24}a^{7}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{7}{24}a^{3}-\frac{11}{24}a-\frac{1}{2}$, $\frac{1}{8117208}a^{14}+\frac{9055}{901912}a^{12}+\frac{160625}{2029302}a^{10}-\frac{115933}{8117208}a^{8}-\frac{1}{2}a^{7}-\frac{951589}{4058604}a^{6}-\frac{1}{2}a^{5}-\frac{1603619}{8117208}a^{4}-\frac{442117}{2705736}a^{2}-\frac{342908}{1014651}$, $\frac{1}{8117208}a^{15}+\frac{9055}{901912}a^{13}+\frac{160625}{2029302}a^{11}-\frac{115933}{8117208}a^{9}-\frac{951589}{4058604}a^{7}-\frac{1}{2}a^{6}-\frac{1603619}{8117208}a^{5}-\frac{1}{2}a^{4}-\frac{442117}{2705736}a^{3}-\frac{342908}{1014651}a-\frac{1}{2}$ 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:  $2$, $3$

Class group and class number

$C_{6}$, which has order $6$

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:   \( -\frac{29041}{737928} a^{15} + \frac{101}{3324} a^{14} - \frac{154979}{122988} a^{13} + \frac{272}{277} a^{12} - \frac{10879241}{737928} a^{11} + \frac{38767}{3324} a^{10} - \frac{57360149}{737928} a^{9} + \frac{52450}{831} a^{8} - \frac{134518087}{737928} a^{7} + \frac{517007}{3324} a^{6} - \frac{120956893}{737928} a^{5} + \frac{128108}{831} a^{4} - \frac{556308}{10249} a^{3} + \frac{29069}{554} a^{2} - \frac{3481925}{737928} a + \frac{1420}{831} \)  (order $12$) 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:   $\frac{29041}{737928}a^{15}+\frac{101}{3324}a^{14}+\frac{154979}{122988}a^{13}+\frac{272}{277}a^{12}+\frac{10879241}{737928}a^{11}+\frac{38767}{3324}a^{10}+\frac{57360149}{737928}a^{9}+\frac{52450}{831}a^{8}+\frac{134518087}{737928}a^{7}+\frac{517007}{3324}a^{6}+\frac{120956893}{737928}a^{5}+\frac{128108}{831}a^{4}+\frac{556308}{10249}a^{3}+\frac{29069}{554}a^{2}+\frac{3481925}{737928}a+\frac{589}{831}$, $\frac{101116}{1014651}a^{15}-\frac{3392}{338217}a^{14}+\frac{8575099}{2705736}a^{13}-\frac{35917}{112739}a^{12}+\frac{297498145}{8117208}a^{11}-\frac{1242520}{338217}a^{10}+\frac{383716279}{2029302}a^{9}-\frac{12717299}{676434}a^{8}+\frac{3433964921}{8117208}a^{7}-\frac{13835000}{338217}a^{6}+\frac{1362026317}{4058604}a^{5}-\frac{9538439}{338217}a^{4}+\frac{238650197}{2705736}a^{3}-\frac{218792}{112739}a^{2}+\frac{25597777}{8117208}a+\frac{125384}{338217}$, $\frac{3587}{27423}a^{14}+\frac{152671}{36564}a^{12}+\frac{5330395}{109692}a^{10}+\frac{6956143}{27423}a^{8}+\frac{63833267}{109692}a^{6}+\frac{27036217}{54846}a^{4}+\frac{5200967}{36564}a^{2}+\frac{583885}{109692}$, $\frac{281}{184482}a^{14}+\frac{5969}{122988}a^{12}+\frac{207461}{368964}a^{10}+\frac{268673}{92241}a^{8}+\frac{2469613}{368964}a^{6}+\frac{1179479}{184482}a^{4}+\frac{386215}{122988}a^{2}+\frac{339923}{368964}$, $\frac{89959}{2029302}a^{15}-\frac{38875}{2705736}a^{14}+\frac{642635}{450956}a^{13}-\frac{1279615}{2705736}a^{12}+\frac{68137489}{4058604}a^{11}-\frac{2607585}{450956}a^{10}+\frac{182003411}{2029302}a^{9}-\frac{88797221}{2705736}a^{8}+\frac{877995983}{4058604}a^{7}-\frac{19789977}{225478}a^{6}+\frac{421153771}{2029302}a^{5}-\frac{270042415}{2705736}a^{4}+\frac{96165235}{1352868}a^{3}-\frac{101653537}{2705736}a^{2}+\frac{15421249}{4058604}a-\frac{854609}{450956}$, $\frac{75019}{338217}a^{15}-\frac{16273}{1014651}a^{14}+\frac{9608989}{1352868}a^{13}-\frac{117023}{225478}a^{12}+\frac{37477919}{450956}a^{11}-\frac{25091357}{4058604}a^{10}+\frac{148211474}{338217}a^{9}-\frac{34156634}{1014651}a^{8}+\frac{463270433}{450956}a^{7}-\frac{341044903}{4058604}a^{6}+\frac{311023477}{338217}a^{5}-\frac{173739485}{2029302}a^{4}+\frac{394470433}{1352868}a^{3}-\frac{41529023}{1352868}a^{2}+\frac{7184751}{450956}a-\frac{3973903}{2029302}$, $\frac{679057}{2029302}a^{15}+\frac{161}{1628}a^{14}+\frac{7218043}{676434}a^{13}+\frac{1280}{407}a^{12}+\frac{251575637}{2029302}a^{11}+\frac{59209}{1628}a^{10}+\frac{2618067445}{4058604}a^{9}+\frac{76366}{407}a^{8}+\frac{2982106315}{2029302}a^{7}+\frac{683273}{1628}a^{6}+\frac{4950706613}{4058604}a^{5}+\frac{135284}{407}a^{4}+\frac{37479128}{112739}a^{3}+\frac{35259}{407}a^{2}+\frac{15866605}{4058604}a+\frac{971}{814}$ 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:  \( 53298.19863992762 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 53298.19863992762 \cdot 6}{12\cdot\sqrt{18123733314413355073536}}\cr\approx \mathstrut & 0.480836747456989 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 + 32*x^14 + 374*x^12 + 1968*x^10 + 4595*x^8 + 4080*x^6 + 1298*x^4 + 80*x^2 + 1)
 
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 + 32*x^14 + 374*x^12 + 1968*x^10 + 4595*x^8 + 4080*x^6 + 1298*x^4 + 80*x^2 + 1, 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 + 32*x^14 + 374*x^12 + 1968*x^10 + 4595*x^8 + 4080*x^6 + 1298*x^4 + 80*x^2 + 1);
 
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 + 32*x^14 + 374*x^12 + 1968*x^10 + 4595*x^8 + 4080*x^6 + 1298*x^4 + 80*x^2 + 1);
 
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

$C_2^2\times D_4$ (as 16T25):

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 32
The 20 conjugacy class representatives for $C_2^2 \times D_4$
Character table for $C_2^2 \times D_4$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{21}) \), 4.0.366912.2, 4.4.7488.1, 4.4.366912.1, 4.0.7488.1, \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{3}, \sqrt{-7})\), \(\Q(i, \sqrt{7})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{-3}, \sqrt{7})\), \(\Q(\sqrt{3}, \sqrt{7})\), \(\Q(i, \sqrt{21})\), 8.0.49787136.1, 8.0.134624415744.43, 8.0.134624415744.27, 8.0.134624415744.23, 8.0.56070144.2, 8.8.134624415744.1, 8.0.134624415744.42

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

Galois closure: deg 32
Degree 16 siblings: deg 16, deg 16, deg 16, deg 16, deg 16, deg 16, deg 16
Minimal sibling: This field is its own minimal sibling

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 ${\href{/padicField/5.4.0.1}{4} }^{4}$ R ${\href{/padicField/11.2.0.1}{2} }^{8}$ R ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{8}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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 2.8.12.14$x^{8} + 8 x^{7} + 28 x^{6} + 58 x^{5} + 95 x^{4} + 130 x^{3} + 58 x^{2} - 58 x + 13$$4$$2$$12$$D_4$$[2, 2]^{2}$
2.8.12.14$x^{8} + 8 x^{7} + 28 x^{6} + 58 x^{5} + 95 x^{4} + 130 x^{3} + 58 x^{2} - 58 x + 13$$4$$2$$12$$D_4$$[2, 2]^{2}$
\(3\) Copy content Toggle raw display 3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{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}$
\(13\) Copy content Toggle raw display 13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$