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Properties

Label 16.0.236...000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2.361\times 10^{22}$
Root discriminant \(25.02\)
Ramified primes $2,5,7$
Class number $5$ (GRH)
Class group [5] (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 - 2*x^15 - 5*x^14 + 20*x^13 + 19*x^12 + 88*x^11 - 497*x^10 + 10*x^9 + 3711*x^8 - 3712*x^7 + 2758*x^6 - 1816*x^5 + 1084*x^4 - 464*x^3 + 184*x^2 - 64*x + 16)
 
gp: K = bnfinit(y^16 - 2*y^15 - 5*y^14 + 20*y^13 + 19*y^12 + 88*y^11 - 497*y^10 + 10*y^9 + 3711*y^8 - 3712*y^7 + 2758*y^6 - 1816*y^5 + 1084*y^4 - 464*y^3 + 184*y^2 - 64*y + 16, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 5*x^14 + 20*x^13 + 19*x^12 + 88*x^11 - 497*x^10 + 10*x^9 + 3711*x^8 - 3712*x^7 + 2758*x^6 - 1816*x^5 + 1084*x^4 - 464*x^3 + 184*x^2 - 64*x + 16);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 - 5*x^14 + 20*x^13 + 19*x^12 + 88*x^11 - 497*x^10 + 10*x^9 + 3711*x^8 - 3712*x^7 + 2758*x^6 - 1816*x^5 + 1084*x^4 - 464*x^3 + 184*x^2 - 64*x + 16)
 

\( x^{16} - 2 x^{15} - 5 x^{14} + 20 x^{13} + 19 x^{12} + 88 x^{11} - 497 x^{10} + 10 x^{9} + 3711 x^{8} + \cdots + 16 \) 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:   \(23612624896000000000000\) \(\medspace = 2^{24}\cdot 5^{12}\cdot 7^{8}\) 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:  \(25.02\)
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:  $2^{3/2}5^{3/4}7^{1/2}\approx 25.021971019484877$
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:  \(280=2^{3}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{280}(1,·)$, $\chi_{280}(267,·)$, $\chi_{280}(139,·)$, $\chi_{280}(209,·)$, $\chi_{280}(83,·)$, $\chi_{280}(153,·)$, $\chi_{280}(27,·)$, $\chi_{280}(97,·)$, $\chi_{280}(99,·)$, $\chi_{280}(41,·)$, $\chi_{280}(43,·)$, $\chi_{280}(113,·)$, $\chi_{280}(211,·)$, $\chi_{280}(169,·)$, $\chi_{280}(57,·)$, $\chi_{280}(251,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{11}a^{9}-\frac{4}{11}a^{8}+\frac{5}{11}a^{7}+\frac{2}{11}a^{6}+\frac{3}{11}a^{5}-\frac{1}{11}a^{4}+\frac{4}{11}a^{3}-\frac{5}{11}a^{2}-\frac{2}{11}a-\frac{3}{11}$, $\frac{1}{22}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}+\frac{5}{11}$, $\frac{1}{22}a^{11}-\frac{1}{22}a^{9}+\frac{2}{11}a^{8}-\frac{5}{22}a^{7}-\frac{1}{11}a^{6}-\frac{3}{22}a^{5}-\frac{5}{11}a^{4}+\frac{7}{22}a^{3}-\frac{3}{11}a^{2}-\frac{5}{11}a-\frac{4}{11}$, $\frac{1}{44}a^{12}-\frac{1}{44}a^{10}-\frac{1}{22}a^{9}+\frac{19}{44}a^{8}-\frac{5}{22}a^{7}+\frac{7}{44}a^{6}+\frac{4}{11}a^{5}-\frac{9}{44}a^{4}+\frac{7}{22}a^{3}-\frac{1}{22}a^{2}+\frac{1}{11}a-\frac{1}{11}$, $\frac{1}{1254836}a^{13}+\frac{10955}{1254836}a^{12}-\frac{3445}{1254836}a^{11}+\frac{107}{6004}a^{10}+\frac{12713}{1254836}a^{9}-\frac{364297}{1254836}a^{8}-\frac{418763}{1254836}a^{7}-\frac{276315}{1254836}a^{6}-\frac{318569}{1254836}a^{5}-\frac{326345}{1254836}a^{4}+\frac{4336}{16511}a^{3}+\frac{251633}{627418}a^{2}+\frac{109034}{313709}a+\frac{7419}{313709}$, $\frac{1}{27606392}a^{14}-\frac{1}{6901598}a^{13}-\frac{13209}{1452968}a^{12}+\frac{236633}{13803196}a^{11}+\frac{57401}{27606392}a^{10}+\frac{627417}{13803196}a^{9}+\frac{11543749}{27606392}a^{8}+\frac{188411}{6901598}a^{7}+\frac{12791}{349448}a^{6}+\frac{3381067}{13803196}a^{5}-\frac{1003923}{6901598}a^{4}+\frac{1945019}{6901598}a^{3}+\frac{564667}{6901598}a^{2}+\frac{9569}{181621}a+\frac{828475}{3450799}$, $\frac{1}{27606392}a^{15}-\frac{1}{2509672}a^{13}+\frac{4391}{627418}a^{12}-\frac{53593}{2509672}a^{11}+\frac{68985}{3450799}a^{10}-\frac{12713}{2509672}a^{9}+\frac{339003}{1254836}a^{8}+\frac{418763}{2509672}a^{7}-\frac{166203}{627418}a^{6}-\frac{71648}{181621}a^{5}-\frac{307391}{1254836}a^{4}+\frac{12175}{33022}a^{3}-\frac{54519}{627418}a^{2}+\frac{31040}{313709}a+\frac{1338806}{3450799}$ 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:  $11$

Class group and class number

$C_{5}$, which has order $5$ (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:   \( \frac{13159}{13803196} a^{15} + \frac{65795}{27606392} a^{14} - \frac{65795}{6901598} a^{13} - \frac{13159}{1452968} a^{12} + \frac{1145401}{13803196} a^{11} + \frac{6540023}{27606392} a^{10} - \frac{65795}{13803196} a^{9} - \frac{48833049}{27606392} a^{8} + \frac{6105776}{3450799} a^{7} + \frac{4179365}{349448} a^{6} + \frac{2987093}{3450799} a^{5} - \frac{3566089}{6901598} a^{4} + \frac{763222}{3450799} a^{3} - \frac{302657}{3450799} a^{2} + \frac{407678}{181621} a - \frac{26318}{3450799} \)  (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:   $\frac{13159}{13803196}a^{15}+\frac{65795}{27606392}a^{14}-\frac{65795}{6901598}a^{13}-\frac{13159}{1452968}a^{12}+\frac{1145401}{13803196}a^{11}+\frac{6540023}{27606392}a^{10}-\frac{65795}{13803196}a^{9}-\frac{48833049}{27606392}a^{8}+\frac{6105776}{3450799}a^{7}+\frac{4179365}{349448}a^{6}+\frac{2987093}{3450799}a^{5}-\frac{3566089}{6901598}a^{4}+\frac{763222}{3450799}a^{3}-\frac{302657}{3450799}a^{2}+\frac{407678}{181621}a-\frac{3477117}{3450799}$, $\frac{60209}{3450799}a^{15}-\frac{575455}{27606392}a^{14}-\frac{1371907}{13803196}a^{13}+\frac{7183985}{27606392}a^{12}+\frac{3517321}{6901598}a^{11}+\frac{56526705}{27606392}a^{10}-\frac{47505639}{6901598}a^{9}-\frac{140250811}{27606392}a^{8}+\frac{803737203}{13803196}a^{7}-\frac{518744035}{27606392}a^{6}+\frac{184642714}{3450799}a^{5}-\frac{18145427}{6901598}a^{4}+\frac{3976616}{3450799}a^{3}+\frac{29167981}{6901598}a^{2}-\frac{7032358}{3450799}a+\frac{5146896}{3450799}$, $\frac{171854}{3450799}a^{15}-\frac{1538939}{27606392}a^{14}-\frac{4238057}{13803196}a^{13}+\frac{20014987}{27606392}a^{12}+\frac{11461491}{6901598}a^{11}+\frac{159289291}{27606392}a^{10}-\frac{69645415}{3450799}a^{9}-\frac{511395713}{27606392}a^{8}+\frac{2363250601}{13803196}a^{7}-\frac{704172401}{27606392}a^{6}+\frac{290981359}{3450799}a^{5}-\frac{714753955}{13803196}a^{4}+\frac{109952978}{3450799}a^{3}-\frac{61184543}{6901598}a^{2}+\frac{19940034}{3450799}a-\frac{6289166}{3450799}$, $\frac{89008}{3450799}a^{15}-\frac{1407699}{27606392}a^{14}-\frac{494430}{3450799}a^{13}+\frac{14519019}{27606392}a^{12}+\frac{8160695}{13803196}a^{11}+\frac{5247065}{2509672}a^{10}-\frac{183126167}{13803196}a^{9}-\frac{41448529}{27606392}a^{8}+\frac{697310395}{6901598}a^{7}-\frac{2410963989}{27606392}a^{6}+\frac{297348737}{13803196}a^{5}-\frac{630310195}{13803196}a^{4}+\frac{100823468}{3450799}a^{3}-\frac{43132873}{3450799}a^{2}+\frac{24730278}{3450799}a-\frac{9781472}{3450799}$, $\frac{73089}{726484}a^{15}-\frac{4748567}{27606392}a^{14}-\frac{3851695}{6901598}a^{13}+\frac{51332429}{27606392}a^{12}+\frac{34158169}{13803196}a^{11}+\frac{23791019}{2509672}a^{10}-\frac{653443495}{13803196}a^{9}-\frac{366491007}{27606392}a^{8}+\frac{2566230593}{6901598}a^{7}-\frac{7329480107}{27606392}a^{6}+\frac{632292983}{3450799}a^{5}-\frac{389167244}{3450799}a^{4}+\frac{346260529}{6901598}a^{3}-\frac{66672076}{3450799}a^{2}+\frac{31129926}{3450799}a-\frac{2765413}{3450799}$, $\frac{1974805}{27606392}a^{15}-\frac{2825183}{13803196}a^{14}-\frac{368359}{1452968}a^{13}+\frac{6184362}{3450799}a^{12}+\frac{5307343}{27606392}a^{11}+\frac{32172801}{6901598}a^{10}-\frac{1132529373}{27606392}a^{9}+\frac{413778711}{13803196}a^{8}+\frac{7608142755}{27606392}a^{7}-\frac{3464459761}{6901598}a^{6}+\frac{4897612011}{13803196}a^{5}-\frac{587167608}{3450799}a^{4}+\frac{281256226}{3450799}a^{3}-\frac{7361122}{181621}a^{2}+\frac{44682881}{3450799}a-\frac{5868349}{3450799}$, $\frac{351}{43681}a^{15}-\frac{390}{43681}a^{14}-\frac{2301}{43681}a^{13}+\frac{10609}{87362}a^{12}+\frac{12519}{43681}a^{11}+\frac{38376}{43681}a^{10}-\frac{145509}{43681}a^{9}-\frac{144690}{43681}a^{8}+\frac{1266766}{43681}a^{7}-\frac{144300}{43681}a^{6}+\frac{99372}{43681}a^{5}-\frac{66456}{43681}a^{4}+\frac{29016}{43681}a^{3}+\frac{5}{87362}a^{2}+\frac{4368}{43681}a-\frac{1248}{43681}$ 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:  \( 85253.3675979 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 85253.3675979 \cdot 5}{10\cdot\sqrt{23612624896000000000000}}\cr\approx \mathstrut & 0.673826924346 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 5*x^14 + 20*x^13 + 19*x^12 + 88*x^11 - 497*x^10 + 10*x^9 + 3711*x^8 - 3712*x^7 + 2758*x^6 - 1816*x^5 + 1084*x^4 - 464*x^3 + 184*x^2 - 64*x + 16)
 
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 - 5*x^14 + 20*x^13 + 19*x^12 + 88*x^11 - 497*x^10 + 10*x^9 + 3711*x^8 - 3712*x^7 + 2758*x^6 - 1816*x^5 + 1084*x^4 - 464*x^3 + 184*x^2 - 64*x + 16, 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 - 5*x^14 + 20*x^13 + 19*x^12 + 88*x^11 - 497*x^10 + 10*x^9 + 3711*x^8 - 3712*x^7 + 2758*x^6 - 1816*x^5 + 1084*x^4 - 464*x^3 + 184*x^2 - 64*x + 16);
 
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 - 5*x^14 + 20*x^13 + 19*x^12 + 88*x^11 - 497*x^10 + 10*x^9 + 3711*x^8 - 3712*x^7 + 2758*x^6 - 1816*x^5 + 1084*x^4 - 464*x^3 + 184*x^2 - 64*x + 16);
 
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

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{70}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-2}, \sqrt{-35})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{-10}, \sqrt{14})\), \(\Q(\sqrt{5}, \sqrt{14})\), \(\Q(\sqrt{-7}, \sqrt{-10})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{-7})\), 4.0.392000.2, 4.4.8000.1, \(\Q(\zeta_{5})\), 4.4.6125.1, 8.0.6146560000.1, 8.0.153664000000.4, 8.0.37515625.1, 8.0.153664000000.6, 8.8.153664000000.2, 8.0.153664000000.3, 8.0.64000000.1

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 ${\href{/padicField/3.4.0.1}{4} }^{4}$ R R ${\href{/padicField/11.1.0.1}{1} }^{16}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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 2.8.12.2$x^{8} + 16 x^{7} + 120 x^{6} + 546 x^{5} + 1646 x^{4} + 3352 x^{3} + 4457 x^{2} + 3470 x + 1203$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 16 x^{7} + 120 x^{6} + 546 x^{5} + 1646 x^{4} + 3352 x^{3} + 4457 x^{2} + 3470 x + 1203$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
\(5\) 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}$