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Properties

Label 13.1.411...729.1
Degree $13$
Signature $[1, 6]$
Discriminant $4.114\times 10^{21}$
Root discriminant \(45.99\)
Ramified prime $4003$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{13}$ (as 13T2)

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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^13 - 3*x^12 + 18*x^10 - 6*x^9 - 56*x^8 + 152*x^7 + 128*x^6 + 256*x^5 - 106*x^4 - 370*x^3 - 136*x^2 - x - 5)
 
Copy content gp:K = bnfinit(y^13 - 3*y^12 + 18*y^10 - 6*y^9 - 56*y^8 + 152*y^7 + 128*y^6 + 256*y^5 - 106*y^4 - 370*y^3 - 136*y^2 - y - 5, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^13 - 3*x^12 + 18*x^10 - 6*x^9 - 56*x^8 + 152*x^7 + 128*x^6 + 256*x^5 - 106*x^4 - 370*x^3 - 136*x^2 - x - 5);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^13 - 3*x^12 + 18*x^10 - 6*x^9 - 56*x^8 + 152*x^7 + 128*x^6 + 256*x^5 - 106*x^4 - 370*x^3 - 136*x^2 - x - 5)
 

\( x^{13} - 3 x^{12} + 18 x^{10} - 6 x^{9} - 56 x^{8} + 152 x^{7} + 128 x^{6} + 256 x^{5} - 106 x^{4} + \cdots - 5 \) 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:  $13$
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:  $[1, 6]$
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:   \(4114466594579445832729\) \(\medspace = 4003^{6}\) 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:  \(45.99\)
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:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $4003^{1/2}\approx 63.26926584053272$
Ramified primes:   \(4003\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{6}-\frac{1}{4}$, $\frac{1}{12}a^{7}+\frac{1}{12}a^{6}-\frac{1}{6}a^{5}-\frac{1}{6}a^{4}-\frac{1}{6}a^{3}-\frac{1}{6}a^{2}+\frac{1}{12}a+\frac{1}{12}$, $\frac{1}{12}a^{8}+\frac{1}{4}a^{2}-\frac{1}{3}$, $\frac{1}{24}a^{9}-\frac{1}{8}a^{6}+\frac{1}{8}a^{3}+\frac{1}{3}a-\frac{3}{8}$, $\frac{1}{48}a^{10}-\frac{1}{48}a^{9}-\frac{1}{48}a^{7}+\frac{5}{48}a^{6}+\frac{1}{6}a^{5}+\frac{11}{48}a^{4}+\frac{5}{48}a^{3}-\frac{1}{6}a^{2}+\frac{7}{16}a+\frac{23}{48}$, $\frac{1}{432}a^{11}-\frac{1}{108}a^{10}-\frac{1}{432}a^{9}+\frac{1}{144}a^{8}-\frac{1}{108}a^{7}+\frac{41}{432}a^{6}+\frac{59}{432}a^{5}-\frac{25}{108}a^{4}-\frac{11}{432}a^{3}-\frac{23}{48}a^{2}-\frac{5}{12}a+\frac{179}{432}$, $\frac{1}{107136}a^{12}-\frac{19}{53568}a^{11}+\frac{5}{5952}a^{10}-\frac{101}{26784}a^{9}+\frac{2107}{53568}a^{8}+\frac{85}{17856}a^{7}-\frac{1751}{17856}a^{6}+\frac{1367}{5952}a^{5}+\frac{6127}{53568}a^{4}-\frac{2035}{26784}a^{3}+\frac{603}{1984}a^{2}+\frac{9301}{53568}a-\frac{46199}{107136}$ 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:  No
Index:  Not computed
Inessential primes:  $2$, $3$

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
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 $1$ (assuming GRH)
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:  $6$
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:   \( -1 \)  (order $2$) 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:   $\frac{45011}{107136}a^{12}-\frac{77977}{53568}a^{11}+\frac{3551}{53568}a^{10}+\frac{29165}{2976}a^{9}-\frac{413167}{53568}a^{8}-\frac{1829131}{53568}a^{7}+\frac{5068793}{53568}a^{6}+\frac{3013421}{53568}a^{5}-\frac{3684587}{53568}a^{4}-\frac{516319}{8928}a^{3}-\frac{55501}{5952}a^{2}-\frac{64729}{53568}a-\frac{13079}{35712}$, $\frac{97}{6696}a^{12}-\frac{245}{6696}a^{11}-\frac{97}{2232}a^{10}+\frac{433}{1674}a^{9}+\frac{23}{6696}a^{8}-\frac{2513}{2232}a^{7}+\frac{1583}{1116}a^{6}+\frac{6551}{2232}a^{5}+\frac{22883}{6696}a^{4}+\frac{1625}{1674}a^{3}-\frac{5647}{744}a^{2}-\frac{21661}{6696}a-\frac{4283}{6696}$, $\frac{28873}{53568}a^{12}-\frac{16847}{8928}a^{11}+\frac{2369}{26784}a^{10}+\frac{42911}{3348}a^{9}-\frac{280283}{26784}a^{8}-\frac{1201693}{26784}a^{7}+\frac{3339617}{26784}a^{6}+\frac{1932833}{26784}a^{5}-\frac{953015}{8928}a^{4}-\frac{61085}{837}a^{3}+\frac{287}{2976}a^{2}+\frac{121873}{26784}a+\frac{10277}{53568}$, $\frac{509}{11904}a^{12}-\frac{7555}{53568}a^{11}+\frac{5389}{53568}a^{10}+\frac{19307}{26784}a^{9}-\frac{15415}{17856}a^{8}-\frac{110657}{53568}a^{7}+\frac{498175}{53568}a^{6}+\frac{276703}{53568}a^{5}+\frac{623983}{53568}a^{4}-\frac{279635}{26784}a^{3}-\frac{100207}{5952}a^{2}-\frac{15859}{5952}a-\frac{26659}{107136}$, $\frac{43}{6696}a^{12}+\frac{17}{1116}a^{11}-\frac{1121}{6696}a^{10}+\frac{1327}{3348}a^{9}+\frac{214}{837}a^{8}-\frac{11519}{6696}a^{7}+\frac{2159}{837}a^{6}+\frac{6617}{3348}a^{5}+\frac{3577}{744}a^{4}+\frac{977}{3348}a^{3}-\frac{773}{93}a^{2}-\frac{19537}{6696}a-\frac{139}{6696}$, $\frac{3247}{53568}a^{12}-\frac{641}{2976}a^{11}+\frac{2771}{26784}a^{10}+\frac{14045}{13392}a^{9}-\frac{23831}{26784}a^{8}-\frac{80935}{26784}a^{7}+\frac{290117}{26784}a^{6}+\frac{76013}{26784}a^{5}+\frac{130715}{8928}a^{4}-\frac{185189}{13392}a^{3}-\frac{47797}{2976}a^{2}-\frac{36557}{26784}a+\frac{28871}{53568}$ Copy content Toggle raw display (assuming GRH)
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:  \( 7222649.14775 \) (assuming GRH)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{6}\cdot 7222649.14775 \cdot 1}{2\cdot\sqrt{4114466594579445832729}}\cr\approx \mathstrut & 6.92817674773 \end{aligned}\] (assuming GRH)

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^13 - 3*x^12 + 18*x^10 - 6*x^9 - 56*x^8 + 152*x^7 + 128*x^6 + 256*x^5 - 106*x^4 - 370*x^3 - 136*x^2 - x - 5) 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^13 - 3*x^12 + 18*x^10 - 6*x^9 - 56*x^8 + 152*x^7 + 128*x^6 + 256*x^5 - 106*x^4 - 370*x^3 - 136*x^2 - x - 5, 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^13 - 3*x^12 + 18*x^10 - 6*x^9 - 56*x^8 + 152*x^7 + 128*x^6 + 256*x^5 - 106*x^4 - 370*x^3 - 136*x^2 - x - 5); 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 = PolynomialRing(QQ); K, a = NumberField(x^13 - 3*x^12 + 18*x^10 - 6*x^9 - 56*x^8 + 152*x^7 + 128*x^6 + 256*x^5 - 106*x^4 - 370*x^3 - 136*x^2 - x - 5); 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

$D_{13}$ (as 13T2):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 26
The 8 conjugacy class representatives for $D_{13}$
Character table for $D_{13}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Galois closure: deg 26
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 ${\href{/padicField/2.2.0.1}{2} }^{6}{,}\,{\href{/padicField/2.1.0.1}{1} }$ ${\href{/padicField/3.2.0.1}{2} }^{6}{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.2.0.1}{2} }^{6}{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.13.0.1}{13} }$ ${\href{/padicField/11.13.0.1}{13} }$ ${\href{/padicField/13.13.0.1}{13} }$ ${\href{/padicField/17.13.0.1}{13} }$ ${\href{/padicField/19.13.0.1}{13} }$ ${\href{/padicField/23.2.0.1}{2} }^{6}{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.13.0.1}{13} }$ ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.13.0.1}{13} }$ ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.13.0.1}{13} }$ ${\href{/padicField/59.13.0.1}{13} }$

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
\(4003\) Copy content Toggle raw display $\Q_{4003}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$

Spectrum of ring of integers

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