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Properties

Label 13.1.480...449.1
Degree $13$
Signature $[1, 6]$
Discriminant $4.810\times 10^{19}$
Root discriminant \(32.66\)
Ramified prime $1907$
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 - x^12 + 15*x^11 - 34*x^10 + 93*x^9 - 251*x^8 + 394*x^7 - 405*x^6 + 245*x^5 + 62*x^4 - 509*x^3 + 543*x^2 - 271*x + 374)
 
Copy content gp:K = bnfinit(y^13 - y^12 + 15*y^11 - 34*y^10 + 93*y^9 - 251*y^8 + 394*y^7 - 405*y^6 + 245*y^5 + 62*y^4 - 509*y^3 + 543*y^2 - 271*y + 374, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^13 - x^12 + 15*x^11 - 34*x^10 + 93*x^9 - 251*x^8 + 394*x^7 - 405*x^6 + 245*x^5 + 62*x^4 - 509*x^3 + 543*x^2 - 271*x + 374);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^13 - x^12 + 15*x^11 - 34*x^10 + 93*x^9 - 251*x^8 + 394*x^7 - 405*x^6 + 245*x^5 + 62*x^4 - 509*x^3 + 543*x^2 - 271*x + 374)
 

\( x^{13} - x^{12} + 15 x^{11} - 34 x^{10} + 93 x^{9} - 251 x^{8} + 394 x^{7} - 405 x^{6} + 245 x^{5} + \cdots + 374 \) 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:   \(48095468356445867449\) \(\medspace = 1907^{6}\) 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:  \(32.66\)
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:  $1907^{1/2}\approx 43.669211121796096$
Ramified primes:   \(1907\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{7}-\frac{1}{2}a^{3}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{8}a^{8}-\frac{1}{8}a^{7}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{8}a^{2}-\frac{1}{8}a+\frac{1}{4}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{7}-\frac{1}{4}a^{5}+\frac{1}{8}a^{3}-\frac{1}{4}a^{2}+\frac{1}{8}a+\frac{1}{4}$, $\frac{1}{16}a^{10}+\frac{1}{16}a^{7}-\frac{1}{8}a^{6}-\frac{1}{16}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{7}{16}a+\frac{3}{8}$, $\frac{1}{800}a^{11}+\frac{3}{800}a^{10}+\frac{9}{200}a^{9}+\frac{37}{800}a^{8}+\frac{13}{160}a^{7}+\frac{21}{400}a^{6}+\frac{47}{800}a^{5}-\frac{39}{800}a^{4}+\frac{39}{100}a^{3}+\frac{59}{800}a^{2}+\frac{47}{160}a-\frac{143}{400}$, $\frac{1}{6335420800}a^{12}-\frac{476131}{791927600}a^{11}+\frac{32328983}{6335420800}a^{10}+\frac{35401901}{6335420800}a^{9}-\frac{97718111}{3167710400}a^{8}-\frac{95148473}{6335420800}a^{7}+\frac{311378581}{1267084160}a^{6}+\frac{283402191}{1583855200}a^{5}+\frac{845594001}{6335420800}a^{4}-\frac{67292787}{333443200}a^{3}+\frac{1119002513}{3167710400}a^{2}+\frac{2746542529}{6335420800}a+\frac{123143293}{3167710400}$ 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:  $2$

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:   $\frac{31}{10112}a^{12}-\frac{3}{3160}a^{11}+\frac{2141}{50560}a^{10}-\frac{3633}{50560}a^{9}+\frac{4767}{25280}a^{8}-\frac{5575}{10112}a^{7}+\frac{27699}{50560}a^{6}-\frac{2569}{12640}a^{5}-\frac{18533}{50560}a^{4}+\frac{20029}{50560}a^{3}-\frac{36681}{25280}a^{2}+\frac{3391}{10112}a+\frac{2719}{25280}$, $\frac{11520159}{6335420800}a^{12}-\frac{446777}{791927600}a^{11}+\frac{33910669}{1267084160}a^{10}-\frac{49658393}{1267084160}a^{9}+\frac{439497447}{3167710400}a^{8}-\frac{1995120367}{6335420800}a^{7}+\frac{2863837767}{6335420800}a^{6}-\frac{530047743}{1583855200}a^{5}+\frac{94380107}{1267084160}a^{4}+\frac{5745447}{66688640}a^{3}-\frac{707162761}{3167710400}a^{2}-\frac{1303019129}{6335420800}a+\frac{212892299}{3167710400}$, $\frac{72784009}{6335420800}a^{12}-\frac{10809559}{791927600}a^{11}+\frac{923303927}{6335420800}a^{10}-\frac{2867889931}{6335420800}a^{9}+\frac{2218821561}{3167710400}a^{8}-\frac{18644534457}{6335420800}a^{7}+\frac{4385751213}{1267084160}a^{6}-\frac{2652416801}{1583855200}a^{5}+\frac{17928325169}{6335420800}a^{4}+\frac{1281545397}{333443200}a^{3}-\frac{18208608663}{3167710400}a^{2}+\frac{4190808961}{6335420800}a-\frac{15882856643}{3167710400}$, $\frac{99172937}{6335420800}a^{12}-\frac{40490323}{791927600}a^{11}+\frac{1519290447}{6335420800}a^{10}-\frac{6182631051}{6335420800}a^{9}+\frac{1389306629}{633542080}a^{8}-\frac{34598445121}{6335420800}a^{7}+\frac{69499755249}{6335420800}a^{6}-\frac{15848335177}{1583855200}a^{5}-\frac{13524518951}{6335420800}a^{4}+\frac{3953754197}{333443200}a^{3}-\frac{6427042731}{633542080}a^{2}+\frac{64830865993}{6335420800}a-\frac{5114179423}{633542080}$, $\frac{87297}{98990950}a^{12}-\frac{5026337}{1583855200}a^{11}+\frac{24236193}{1583855200}a^{10}-\frac{5828489}{98990950}a^{9}+\frac{236745439}{1583855200}a^{8}-\frac{569149861}{1583855200}a^{7}+\frac{597123319}{791927600}a^{6}-\frac{1287217799}{1583855200}a^{5}+\frac{173401051}{1583855200}a^{4}+\frac{11026627}{20840200}a^{3}-\frac{1590696567}{1583855200}a^{2}+\frac{2166101873}{1583855200}a-\frac{535978301}{791927600}$, $\frac{11969851}{3167710400}a^{12}-\frac{546253}{791927600}a^{11}+\frac{126888541}{3167710400}a^{10}-\frac{258784753}{3167710400}a^{9}+\frac{24554181}{316771040}a^{8}-\frac{1258048883}{3167710400}a^{7}+\frac{488018867}{3167710400}a^{6}+\frac{144974283}{98990950}a^{5}-\frac{5159654853}{3167710400}a^{4}-\frac{220244609}{166721600}a^{3}+\frac{100832469}{63354208}a^{2}+\frac{3051297739}{3167710400}a+\frac{150804299}{316771040}$ 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:  \( 333391.286353 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{6}\cdot 333391.286353 \cdot 1}{2\cdot\sqrt{48095468356445867449}}\cr\approx \mathstrut & 2.95788561815 \end{aligned}\] (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 - x^12 + 15*x^11 - 34*x^10 + 93*x^9 - 251*x^8 + 394*x^7 - 405*x^6 + 245*x^5 + 62*x^4 - 509*x^3 + 543*x^2 - 271*x + 374) 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 - x^12 + 15*x^11 - 34*x^10 + 93*x^9 - 251*x^8 + 394*x^7 - 405*x^6 + 245*x^5 + 62*x^4 - 509*x^3 + 543*x^2 - 271*x + 374, 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 - x^12 + 15*x^11 - 34*x^10 + 93*x^9 - 251*x^8 + 394*x^7 - 405*x^6 + 245*x^5 + 62*x^4 - 509*x^3 + 543*x^2 - 271*x + 374); 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 - x^12 + 15*x^11 - 34*x^10 + 93*x^9 - 251*x^8 + 394*x^7 - 405*x^6 + 245*x^5 + 62*x^4 - 509*x^3 + 543*x^2 - 271*x + 374); 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 ${\href{/padicField/2.2.0.1}{2} }^{6}{,}\,{\href{/padicField/2.1.0.1}{1} }$ ${\href{/padicField/3.13.0.1}{13} }$ ${\href{/padicField/5.2.0.1}{2} }^{6}{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.13.0.1}{13} }$ ${\href{/padicField/11.2.0.1}{2} }^{6}{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.13.0.1}{13} }$ ${\href{/padicField/17.2.0.1}{2} }^{6}{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.2.0.1}{2} }^{6}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.13.0.1}{13} }$ ${\href{/padicField/29.13.0.1}{13} }$ ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.2.0.1}{2} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.13.0.1}{13} }$ ${\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.13.0.1}{13} }$ ${\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(1907\) Copy content Toggle raw display $\Q_{1907}$$x$$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)