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Properties

Label 16.0.358...625.4
Degree $16$
Signature $[0, 8]$
Discriminant $3.585\times 10^{22}$
Root discriminant \(25.68\)
Ramified primes $5,59$
Class number $6$ (GRH)
Class group [6] (GRH)
Galois group $C_2^2 : C_4$ (as 16T10)

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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 + 4*x^14 - 7*x^13 - 3*x^12 - 98*x^11 + 129*x^10 - 156*x^9 + 259*x^8 + 1081*x^7 + 932*x^6 + 735*x^5 - 4274*x^4 - 6735*x^3 + 225*x^2 + 3375*x + 50625)
 
gp: K = bnfinit(y^16 - 2*y^15 + 4*y^14 - 7*y^13 - 3*y^12 - 98*y^11 + 129*y^10 - 156*y^9 + 259*y^8 + 1081*y^7 + 932*y^6 + 735*y^5 - 4274*y^4 - 6735*y^3 + 225*y^2 + 3375*y + 50625, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 4*x^14 - 7*x^13 - 3*x^12 - 98*x^11 + 129*x^10 - 156*x^9 + 259*x^8 + 1081*x^7 + 932*x^6 + 735*x^5 - 4274*x^4 - 6735*x^3 + 225*x^2 + 3375*x + 50625);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 + 4*x^14 - 7*x^13 - 3*x^12 - 98*x^11 + 129*x^10 - 156*x^9 + 259*x^8 + 1081*x^7 + 932*x^6 + 735*x^5 - 4274*x^4 - 6735*x^3 + 225*x^2 + 3375*x + 50625)
 

\( x^{16} - 2 x^{15} + 4 x^{14} - 7 x^{13} - 3 x^{12} - 98 x^{11} + 129 x^{10} - 156 x^{9} + 259 x^{8} + \cdots + 50625 \) 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:   \(35847274805742431640625\) \(\medspace = 5^{12}\cdot 59^{8}\) 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:  \(25.68\)
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:  $5^{3/4}59^{1/2}\approx 25.683458749990002$
Ramified primes:   \(5\), \(59\) 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 over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5}a^{9}+\frac{1}{5}a^{8}-\frac{2}{5}a^{7}+\frac{1}{5}a^{5}+\frac{2}{5}a^{3}+\frac{2}{5}a^{2}+\frac{1}{5}a$, $\frac{1}{10}a^{10}+\frac{1}{5}a^{8}+\frac{1}{5}a^{7}-\frac{2}{5}a^{6}-\frac{1}{10}a^{5}+\frac{1}{5}a^{4}+\frac{2}{5}a^{2}+\frac{2}{5}a-\frac{1}{2}$, $\frac{1}{10}a^{11}-\frac{1}{10}a^{6}+\frac{3}{10}a$, $\frac{1}{70}a^{12}+\frac{1}{35}a^{11}+\frac{3}{70}a^{10}+\frac{1}{35}a^{9}+\frac{9}{35}a^{8}+\frac{3}{10}a^{7}+\frac{3}{35}a^{6}-\frac{31}{70}a^{5}+\frac{8}{35}a^{4}-\frac{13}{35}a^{3}-\frac{1}{70}a^{2}+\frac{2}{7}a-\frac{1}{14}$, $\frac{1}{2886346050}a^{13}+\frac{2120567}{577269210}a^{12}+\frac{10845949}{2886346050}a^{11}-\frac{137145739}{2886346050}a^{10}+\frac{8997769}{481057675}a^{9}+\frac{2975359}{6076518}a^{8}-\frac{359266857}{962115350}a^{7}+\frac{392901759}{962115350}a^{6}-\frac{151598741}{412335150}a^{5}+\frac{717508691}{1443173025}a^{4}-\frac{1364265289}{2886346050}a^{3}+\frac{19894309}{962115350}a^{2}+\frac{29695909}{115453842}a-\frac{6338349}{38484614}$, $\frac{1}{8659038150}a^{14}+\frac{1}{8659038150}a^{13}-\frac{43688081}{8659038150}a^{12}-\frac{2086963}{346361526}a^{11}-\frac{8067113}{288634605}a^{10}+\frac{724357909}{8659038150}a^{9}+\frac{810339893}{2886346050}a^{8}-\frac{70767133}{2886346050}a^{7}+\frac{284922077}{1731807630}a^{6}-\frac{54636242}{865903815}a^{5}+\frac{161547227}{455738850}a^{4}-\frac{1340832659}{2886346050}a^{3}+\frac{75086941}{1237005450}a^{2}+\frac{88043889}{192423070}a-\frac{8019218}{19242307}$, $\frac{1}{129885572250}a^{15}-\frac{1}{64942786125}a^{14}+\frac{2}{64942786125}a^{13}+\frac{344888884}{64942786125}a^{12}+\frac{522123187}{21647595375}a^{11}+\frac{1252199201}{64942786125}a^{10}-\frac{102833788}{3092513625}a^{9}+\frac{4566870874}{21647595375}a^{8}+\frac{8678222342}{64942786125}a^{7}-\frac{11338867147}{64942786125}a^{6}-\frac{137474084}{64942786125}a^{5}-\frac{16453909}{618502725}a^{4}-\frac{21159866362}{64942786125}a^{3}+\frac{224480363}{4329519075}a^{2}+\frac{10259920}{57726921}a+\frac{2581759}{5497802}$ 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:  $2$

Class group and class number

$C_{6}$, which has order $6$ (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:   \( -\frac{679}{41233515} a^{15} + \frac{1358}{41233515} a^{14} - \frac{4753}{82467030} a^{13} - \frac{679}{27489010} a^{12} + \frac{14669}{41233515} a^{11} + \frac{29197}{27489010} a^{10} - \frac{17654}{13744505} a^{9} + \frac{175861}{82467030} a^{8} + \frac{733999}{82467030} a^{7} - \frac{245447}{8246703} a^{6} + \frac{33271}{5497802} a^{5} - \frac{1451023}{41233515} a^{4} - \frac{304871}{5497802} a^{3} + \frac{10185}{5497802} a^{2} - \frac{14247076}{41233515} a + \frac{2291625}{5497802} \)  (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:   $\frac{6463}{1237005450}a^{14}-\frac{109871}{1237005450}a^{13}+\frac{204037}{1237005450}a^{12}-\frac{433021}{1237005450}a^{11}+\frac{109871}{206167575}a^{10}-\frac{342539}{1237005450}a^{9}+\frac{3444779}{412335150}a^{8}-\frac{5830201}{412335150}a^{7}+\frac{16797337}{1237005450}a^{6}-\frac{9061126}{618502725}a^{5}-\frac{98774029}{1237005450}a^{4}-\frac{5706829}{82467030}a^{3}+\frac{101892223}{1237005450}a^{2}+\frac{1648065}{5497802}a-\frac{1294726}{2748901}$, $\frac{32902}{1443173025}a^{15}-\frac{1643}{68722525}a^{14}+\frac{29194}{1443173025}a^{13}-\frac{4843}{82467030}a^{12}-\frac{905717}{2886346050}a^{11}-\frac{2076409}{962115350}a^{10}+\frac{188141}{206167575}a^{9}+\frac{903472}{1443173025}a^{8}+\frac{549863}{2886346050}a^{7}+\frac{28100823}{962115350}a^{6}+\frac{14358169}{412335150}a^{5}+\frac{7797803}{1443173025}a^{4}+\frac{2296778}{96211535}a^{3}-\frac{30620279}{151912950}a^{2}-\frac{197505737}{577269210}a-\frac{1091103}{38484614}$, $\frac{9137}{1443173025}a^{15}+\frac{1861}{206167575}a^{14}-\frac{35989}{962115350}a^{13}-\frac{688}{8246703}a^{12}+\frac{40371}{962115350}a^{11}-\frac{527257}{481057675}a^{10}-\frac{76669}{206167575}a^{9}+\frac{7962079}{2886346050}a^{8}+\frac{13119914}{1443173025}a^{7}-\frac{1603981}{2886346050}a^{6}+\frac{443513}{10850925}a^{5}-\frac{14329334}{481057675}a^{4}-\frac{6076929}{192423070}a^{3}-\frac{288219088}{1443173025}a^{2}+\frac{180304409}{577269210}a+\frac{7475136}{19242307}$, $\frac{1809214}{64942786125}a^{15}-\frac{118469}{871715250}a^{14}+\frac{21850516}{64942786125}a^{13}-\frac{82161071}{129885572250}a^{12}+\frac{24428066}{21647595375}a^{11}-\frac{263088587}{64942786125}a^{10}+\frac{74294477}{6185027250}a^{9}-\frac{512724703}{21647595375}a^{8}+\frac{3876039557}{129885572250}a^{7}-\frac{1066300346}{64942786125}a^{6}+\frac{222178829}{9277540875}a^{5}-\frac{4074181}{1237005450}a^{4}-\frac{14216481326}{64942786125}a^{3}+\frac{147657431}{455738850}a^{2}-\frac{156731552}{288634605}a+\frac{31939294}{19242307}$, $\frac{2369723}{64942786125}a^{15}+\frac{2799029}{64942786125}a^{14}+\frac{4676792}{64942786125}a^{13}-\frac{1382036}{64942786125}a^{12}-\frac{1115197}{2405288375}a^{11}-\frac{586730933}{129885572250}a^{10}-\frac{44029087}{7215865125}a^{9}-\frac{110540246}{21647595375}a^{8}-\frac{6381343}{64942786125}a^{7}+\frac{1844732663}{64942786125}a^{6}+\frac{17884069247}{129885572250}a^{5}+\frac{89547148}{481057675}a^{4}+\frac{15714200948}{64942786125}a^{3}+\frac{899911738}{4329519075}a^{2}+\frac{153277837}{288634605}a+\frac{1556131}{5497802}$, $\frac{956093}{129885572250}a^{15}-\frac{128689}{6836082750}a^{14}-\frac{2859133}{129885572250}a^{13}-\frac{6474823}{64942786125}a^{12}-\frac{1323731}{14431730250}a^{11}+\frac{28702211}{129885572250}a^{10}+\frac{40530803}{14431730250}a^{9}+\frac{3422221}{2278694250}a^{8}+\frac{125114116}{64942786125}a^{7}+\frac{212019719}{18555081750}a^{6}-\frac{10416250349}{129885572250}a^{5}-\frac{35896573}{2886346050}a^{4}+\frac{8697857153}{129885572250}a^{3}-\frac{76322758}{4329519075}a^{2}-\frac{18374915}{115453842}a-\frac{4538727}{19242307}$, $\frac{1573849}{129885572250}a^{15}-\frac{4699643}{129885572250}a^{14}+\frac{9310321}{129885572250}a^{13}-\frac{13094024}{64942786125}a^{12}+\frac{1250579}{4810576750}a^{11}-\frac{6208114}{3418041375}a^{10}+\frac{5802067}{2061675750}a^{9}-\frac{45200233}{43295190750}a^{8}+\frac{231366113}{64942786125}a^{7}+\frac{1403912509}{129885572250}a^{6}+\frac{3490579189}{64942786125}a^{5}+\frac{3141007}{962115350}a^{4}+\frac{749341009}{129885572250}a^{3}+\frac{851962492}{4329519075}a^{2}+\frac{11837783}{82467030}a-\frac{1555937}{38484614}$ 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:  \( 52746.8654076 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 52746.8654076 \cdot 6}{10\cdot\sqrt{35847274805742431640625}}\cr\approx \mathstrut & 0.406030614643 \end{aligned}\] (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 + 4*x^14 - 7*x^13 - 3*x^12 - 98*x^11 + 129*x^10 - 156*x^9 + 259*x^8 + 1081*x^7 + 932*x^6 + 735*x^5 - 4274*x^4 - 6735*x^3 + 225*x^2 + 3375*x + 50625)
 
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 + 4*x^14 - 7*x^13 - 3*x^12 - 98*x^11 + 129*x^10 - 156*x^9 + 259*x^8 + 1081*x^7 + 932*x^6 + 735*x^5 - 4274*x^4 - 6735*x^3 + 225*x^2 + 3375*x + 50625, 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 + 4*x^14 - 7*x^13 - 3*x^12 - 98*x^11 + 129*x^10 - 156*x^9 + 259*x^8 + 1081*x^7 + 932*x^6 + 735*x^5 - 4274*x^4 - 6735*x^3 + 225*x^2 + 3375*x + 50625);
 
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 + 4*x^14 - 7*x^13 - 3*x^12 - 98*x^11 + 129*x^10 - 156*x^9 + 259*x^8 + 1081*x^7 + 932*x^6 + 735*x^5 - 4274*x^4 - 6735*x^3 + 225*x^2 + 3375*x + 50625);
 
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

$C_2^2:C_4$ (as 16T10):

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 16
The 10 conjugacy class representatives for $C_2^2 : C_4$
Character table for $C_2^2 : C_4$

Intermediate fields

\(\Q(\sqrt{-59}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-295}) \), \(\Q(\sqrt{5}, \sqrt{-59})\), 4.2.7375.1 x2, 4.0.435125.1 x2, 4.0.17405.1 x2, 4.2.1475.1 x2, 4.4.435125.1, \(\Q(\zeta_{5})\), 8.0.189333765625.2, 8.0.7573350625.1, 8.0.189333765625.1, 8.4.189333765625.1 x2, 8.0.54390625.1 x2

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

Degree 8 siblings: 8.0.54390625.1, 8.4.189333765625.1
Minimal sibling: 8.0.54390625.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{4}$ ${\href{/padicField/3.4.0.1}{4} }^{4}$ R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ R

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
\(5\) 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}$
\(59\) Copy content Toggle raw display 59.4.2.1$x^{4} + 116 x^{3} + 3486 x^{2} + 7076 x + 201725$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 116 x^{3} + 3486 x^{2} + 7076 x + 201725$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 116 x^{3} + 3486 x^{2} + 7076 x + 201725$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 116 x^{3} + 3486 x^{2} + 7076 x + 201725$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$