Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 14*x^14 - 28*x^13 + 58*x^12 - 88*x^11 + 146*x^10 - 176*x^9 + 212*x^8 - 180*x^7 + 150*x^6 - 92*x^5 + 58*x^4 - 28*x^3 + 10*x^2 - 4*x + 1)
 
Copy content gp:K = bnfinit(y^16 - 4*y^15 + 14*y^14 - 28*y^13 + 58*y^12 - 88*y^11 + 146*y^10 - 176*y^9 + 212*y^8 - 180*y^7 + 150*y^6 - 92*y^5 + 58*y^4 - 28*y^3 + 10*y^2 - 4*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 14*x^14 - 28*x^13 + 58*x^12 - 88*x^11 + 146*x^10 - 176*x^9 + 212*x^8 - 180*x^7 + 150*x^6 - 92*x^5 + 58*x^4 - 28*x^3 + 10*x^2 - 4*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 4*x^15 + 14*x^14 - 28*x^13 + 58*x^12 - 88*x^11 + 146*x^10 - 176*x^9 + 212*x^8 - 180*x^7 + 150*x^6 - 92*x^5 + 58*x^4 - 28*x^3 + 10*x^2 - 4*x + 1)
 

\( x^{16} - 4 x^{15} + 14 x^{14} - 28 x^{13} + 58 x^{12} - 88 x^{11} + 146 x^{10} - 176 x^{9} + 212 x^{8} + \cdots + 1 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $16$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $(0, 8)$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(739537035580145664\) \(\medspace = 2^{34}\cdot 3^{16}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(13.09\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $2^{17/8}3^{4/3}\approx 18.87341170624049$
Ramified primes:   \(2\), \(3\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$:   $C_4$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
Maximal CM subfield:  \(\Q(\sqrt{-1}) \)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{5}a^{13}-\frac{2}{5}a^{12}+\frac{2}{5}a^{11}+\frac{1}{5}a^{10}+\frac{1}{5}a^{9}-\frac{1}{5}a^{8}+\frac{1}{5}a^{6}+\frac{2}{5}a^{4}-\frac{2}{5}a^{3}-\frac{2}{5}a^{2}-\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{25}a^{14}-\frac{1}{25}a^{13}-\frac{2}{5}a^{12}-\frac{12}{25}a^{11}+\frac{7}{25}a^{10}+\frac{2}{5}a^{9}+\frac{4}{25}a^{8}+\frac{1}{25}a^{7}+\frac{6}{25}a^{6}-\frac{8}{25}a^{5}+\frac{1}{25}a^{3}+\frac{11}{25}a^{2}+\frac{9}{25}a+\frac{6}{25}$, $\frac{1}{29125}a^{15}-\frac{71}{5825}a^{14}+\frac{2294}{29125}a^{13}-\frac{13022}{29125}a^{12}-\frac{369}{5825}a^{11}-\frac{46}{125}a^{10}-\frac{6611}{29125}a^{9}-\frac{1943}{5825}a^{8}+\frac{8377}{29125}a^{7}-\frac{4707}{29125}a^{6}-\frac{13643}{29125}a^{5}+\frac{451}{29125}a^{4}-\frac{12618}{29125}a^{3}-\frac{787}{5825}a^{2}+\frac{1299}{5825}a-\frac{2174}{29125}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order $1$
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $7$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -\frac{22158}{29125} a^{15} + \frac{14681}{5825} a^{14} - \frac{253142}{29125} a^{13} + \frac{425326}{29125} a^{12} - \frac{184204}{5825} a^{11} + \frac{5153}{125} a^{10} - \frac{2155812}{29125} a^{9} + \frac{418421}{5825} a^{8} - \frac{2670626}{29125} a^{7} + \frac{1592471}{29125} a^{6} - \frac{1516426}{29125} a^{5} + \frac{549992}{29125} a^{4} - \frac{492666}{29125} a^{3} + \frac{14839}{5825} a^{2} - \frac{86}{233} a + \frac{17382}{29125} \)  (order $4$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{11492}{29125}a^{15}-\frac{7422}{5825}a^{14}+\frac{115198}{29125}a^{13}-\frac{156024}{29125}a^{12}+\frac{55972}{5825}a^{11}-\frac{982}{125}a^{10}+\frac{380488}{29125}a^{9}+\frac{7589}{5825}a^{8}-\frac{315841}{29125}a^{7}+\frac{1198056}{29125}a^{6}-\frac{1298631}{29125}a^{5}+\frac{1315092}{29125}a^{4}-\frac{703331}{29125}a^{3}+\frac{81241}{5825}a^{2}-\frac{36317}{5825}a+\frac{46417}{29125}$, $\frac{46417}{29125}a^{15}-\frac{39432}{5825}a^{14}+\frac{686948}{29125}a^{13}-\frac{1414874}{29125}a^{12}+\frac{569642}{5825}a^{11}-\frac{18732}{125}a^{10}+\frac{7005688}{29125}a^{9}-\frac{1709976}{5825}a^{8}+\frac{9802459}{29125}a^{7}-\frac{8039219}{29125}a^{6}+\frac{5764494}{29125}a^{5}-\frac{2971733}{29125}a^{4}+\frac{1377094}{29125}a^{3}-\frac{119269}{5825}a^{2}+\frac{11593}{5825}a-\frac{4083}{29125}$, $\frac{391}{1165}a^{15}-\frac{9471}{5825}a^{14}+\frac{31441}{5825}a^{13}-\frac{2720}{233}a^{12}+\frac{125452}{5825}a^{11}-\frac{864}{25}a^{10}+\frac{58717}{1165}a^{9}-\frac{381484}{5825}a^{8}+\frac{367139}{5825}a^{7}-\frac{306711}{5825}a^{6}+\frac{122003}{5825}a^{5}-\frac{10059}{1165}a^{4}-\frac{37061}{5825}a^{3}+\frac{17744}{5825}a^{2}-\frac{25939}{5825}a+\frac{14429}{5825}$, $\frac{25258}{29125}a^{15}-\frac{20887}{5825}a^{14}+\frac{371047}{29125}a^{13}-\frac{758301}{29125}a^{12}+\frac{313416}{5825}a^{11}-\frac{10173}{125}a^{10}+\frac{3907387}{29125}a^{9}-\frac{187093}{1165}a^{8}+\frac{5622421}{29125}a^{7}-\frac{4507376}{29125}a^{6}+\frac{3558366}{29125}a^{5}-\frac{1679942}{29125}a^{4}+\frac{943761}{29125}a^{3}-\frac{10089}{1165}a^{2}+\frac{17256}{5825}a-\frac{2112}{29125}$, $\frac{2106}{29125}a^{15}+\frac{759}{5825}a^{14}-\frac{3586}{29125}a^{13}+\frac{52193}{29125}a^{12}-\frac{2389}{5825}a^{11}+\frac{549}{125}a^{10}+\frac{51409}{29125}a^{9}+\frac{46122}{5825}a^{8}+\frac{277637}{29125}a^{7}-\frac{22092}{29125}a^{6}+\frac{788942}{29125}a^{5}-\frac{343344}{29125}a^{4}+\frac{605942}{29125}a^{3}-\frac{19432}{5825}a^{2}+\frac{36389}{5825}a-\frac{46594}{29125}$, $\frac{46999}{29125}a^{15}-\frac{6924}{1165}a^{14}+\frac{585611}{29125}a^{13}-\frac{1054203}{29125}a^{12}+\frac{425716}{5825}a^{11}-\frac{12499}{125}a^{10}+\frac{4841511}{29125}a^{9}-\frac{1005461}{5825}a^{8}+\frac{5646793}{29125}a^{7}-\frac{3290073}{29125}a^{6}+\frac{2153408}{29125}a^{5}-\frac{373426}{29125}a^{4}+\frac{228888}{29125}a^{3}+\frac{10556}{5825}a^{2}-\frac{29948}{5825}a+\frac{44769}{29125}$, $\frac{71812}{29125}a^{15}-\frac{57464}{5825}a^{14}+\frac{988988}{29125}a^{13}-\frac{1935914}{29125}a^{12}+\frac{780546}{5825}a^{11}-\frac{24692}{125}a^{10}+\frac{9354968}{29125}a^{9}-\frac{2168129}{5825}a^{8}+\frac{12383189}{29125}a^{7}-\frac{9289869}{29125}a^{6}+\frac{6630114}{29125}a^{5}-\frac{2958888}{29125}a^{4}+\frac{1528849}{29125}a^{3}-\frac{98356}{5825}a^{2}-\frac{13}{233}a+\frac{44302}{29125}$ Copy content Toggle raw display
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 926.908388825 \)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 926.908388825 \cdot 1}{4\cdot\sqrt{739537035580145664}}\cr\approx \mathstrut & 0.65453941529 \end{aligned}\]

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 14*x^14 - 28*x^13 + 58*x^12 - 88*x^11 + 146*x^10 - 176*x^9 + 212*x^8 - 180*x^7 + 150*x^6 - 92*x^5 + 58*x^4 - 28*x^3 + 10*x^2 - 4*x + 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^16 - 4*x^15 + 14*x^14 - 28*x^13 + 58*x^12 - 88*x^11 + 146*x^10 - 176*x^9 + 212*x^8 - 180*x^7 + 150*x^6 - 92*x^5 + 58*x^4 - 28*x^3 + 10*x^2 - 4*x + 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 14*x^14 - 28*x^13 + 58*x^12 - 88*x^11 + 146*x^10 - 176*x^9 + 212*x^8 - 180*x^7 + 150*x^6 - 92*x^5 + 58*x^4 - 28*x^3 + 10*x^2 - 4*x + 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 4*x^15 + 14*x^14 - 28*x^13 + 58*x^12 - 88*x^11 + 146*x^10 - 176*x^9 + 212*x^8 - 180*x^7 + 150*x^6 - 92*x^5 + 58*x^4 - 28*x^3 + 10*x^2 - 4*x + 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

$\SL(2,3):C_2$ (as 16T60):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 48
The 14 conjugacy class representatives for $\SL(2,3):C_2$
Character table for $\SL(2,3):C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.0.5184.1, 8.0.107495424.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 24 sibling: data not computed
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 ${\href{/padicField/5.3.0.1}{3} }^{4}{,}\,{\href{/padicField/5.1.0.1}{1} }^{4}$ ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.4.0.1}{4} }$ ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.4.0.1}{4} }$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.4.0.1}{4} }$ ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.4.0.1}{4} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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 2.1.16.34d1.5$x^{16} + 2 x^{14} + 4 x^{8} + 4 x^{5} + 4 x^{3} + 2$$16$$1$$34$16T60$$[2, 2, 2, \frac{5}{2}]^{3}$$
\(3\) Copy content Toggle raw display 3.4.1.0a1.1$x^{4} + 2 x^{3} + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
3.4.3.16a2.1$x^{12} + 6 x^{11} + 12 x^{10} + 8 x^{9} + 12 x^{8} + 48 x^{7} + 48 x^{6} + 36 x^{4} + 72 x^{3} + 35$$3$$4$$16$$C_{12}$$$[2]^{4}$$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
*48 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
*48 1.4.2t1.a.a$1$ $ 2^{2}$ \(\Q(\sqrt{-1}) \) $C_2$ (as 2T1) $1$ $-1$
1.9.3t1.a.a$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
1.36.6t1.b.a$1$ $ 2^{2} \cdot 3^{2}$ 6.0.419904.1 $C_6$ (as 6T1) $0$ $-1$
1.9.3t1.a.b$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
1.36.6t1.b.b$1$ $ 2^{2} \cdot 3^{2}$ 6.0.419904.1 $C_6$ (as 6T1) $0$ $-1$
2.2592.24t21.a.a$2$ $ 2^{5} \cdot 3^{4}$ 16.0.739537035580145664.1 $\SL(2,3):C_2$ (as 16T60) $0$ $0$
2.2592.24t21.a.b$2$ $ 2^{5} \cdot 3^{4}$ 16.0.739537035580145664.1 $\SL(2,3):C_2$ (as 16T60) $0$ $0$
*48 2.288.16t60.a.a$2$ $ 2^{5} \cdot 3^{2}$ 16.0.739537035580145664.1 $\SL(2,3):C_2$ (as 16T60) $0$ $0$
*48 2.288.16t60.a.b$2$ $ 2^{5} \cdot 3^{2}$ 16.0.739537035580145664.1 $\SL(2,3):C_2$ (as 16T60) $0$ $0$
*48 2.288.16t60.a.c$2$ $ 2^{5} \cdot 3^{2}$ 16.0.739537035580145664.1 $\SL(2,3):C_2$ (as 16T60) $0$ $0$
*48 2.288.16t60.a.d$2$ $ 2^{5} \cdot 3^{2}$ 16.0.739537035580145664.1 $\SL(2,3):C_2$ (as 16T60) $0$ $0$
*48 3.5184.6t6.a.a$3$ $ 2^{6} \cdot 3^{4}$ 6.4.419904.1 $A_4\times C_2$ (as 6T6) $1$ $1$
*48 3.5184.4t4.b.a$3$ $ 2^{6} \cdot 3^{4}$ 4.0.5184.1 $A_4$ (as 4T4) $1$ $-1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)