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Properties

Label 13.1.489...249.1
Degree $13$
Signature $[1, 6]$
Discriminant $4.892\times 10^{20}$
Root discriminant \(39.04\)
Ramified primes $7,401$
Class number $1$
Class group trivial
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 + 8*x^11 - 23*x^10 - 10*x^9 + 14*x^8 + 98*x^7 + 269*x^6 + 309*x^5 + 121*x^4 + 175*x^3 - 28*x^2 + 245*x + 49)
 
Copy content gp:K = bnfinit(y^13 - 3*y^12 + 8*y^11 - 23*y^10 - 10*y^9 + 14*y^8 + 98*y^7 + 269*y^6 + 309*y^5 + 121*y^4 + 175*y^3 - 28*y^2 + 245*y + 49, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^13 - 3*x^12 + 8*x^11 - 23*x^10 - 10*x^9 + 14*x^8 + 98*x^7 + 269*x^6 + 309*x^5 + 121*x^4 + 175*x^3 - 28*x^2 + 245*x + 49);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^13 - 3*x^12 + 8*x^11 - 23*x^10 - 10*x^9 + 14*x^8 + 98*x^7 + 269*x^6 + 309*x^5 + 121*x^4 + 175*x^3 - 28*x^2 + 245*x + 49)
 

\( x^{13} - 3 x^{12} + 8 x^{11} - 23 x^{10} - 10 x^{9} + 14 x^{8} + 98 x^{7} + 269 x^{6} + 309 x^{5} + \cdots + 49 \) 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:   \(489163986649360075249\) \(\medspace = 7^{6}\cdot 401^{6}\) 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:  \(39.04\)
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:  $7^{1/2}401^{1/2}\approx 52.98112871579842$
Ramified primes:   \(7\), \(401\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{7}a^{9}-\frac{1}{7}a^{8}-\frac{2}{7}a^{7}+\frac{2}{7}a^{6}+\frac{3}{7}a^{5}-\frac{3}{7}a^{4}-\frac{2}{7}a^{3}+\frac{2}{7}a^{2}$, $\frac{1}{7}a^{10}-\frac{3}{7}a^{8}-\frac{2}{7}a^{6}+\frac{2}{7}a^{4}+\frac{2}{7}a^{2}$, $\frac{1}{427}a^{11}-\frac{29}{427}a^{10}-\frac{16}{427}a^{9}-\frac{103}{427}a^{8}+\frac{171}{427}a^{7}+\frac{123}{427}a^{6}+\frac{75}{427}a^{5}-\frac{19}{427}a^{4}+\frac{14}{61}a^{3}-\frac{4}{61}a^{2}-\frac{27}{61}a+\frac{28}{61}$, $\frac{1}{675202881395}a^{12}-\frac{11723827}{11068899695}a^{11}-\frac{5377361124}{675202881395}a^{10}+\frac{6819782229}{96457554485}a^{9}-\frac{142788197757}{675202881395}a^{8}+\frac{21593619867}{675202881395}a^{7}+\frac{7545472259}{135040576279}a^{6}+\frac{56710326179}{675202881395}a^{5}-\frac{6154245252}{675202881395}a^{4}+\frac{182041008699}{675202881395}a^{3}+\frac{310567263429}{675202881395}a^{2}+\frac{25170688678}{96457554485}a+\frac{9180320618}{96457554485}$ 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:  $6$
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:   $\frac{891455154}{135040576279}a^{12}-\frac{3695655077}{135040576279}a^{11}+\frac{9704245729}{135040576279}a^{10}-\frac{25456654394}{135040576279}a^{9}+\frac{3598843541}{135040576279}a^{8}+\frac{52722139079}{135040576279}a^{7}+\frac{28245553339}{135040576279}a^{6}+\frac{139888866884}{135040576279}a^{5}+\frac{5929856411}{19291510897}a^{4}-\frac{39719962448}{19291510897}a^{3}+\frac{31939281675}{19291510897}a^{2}-\frac{38051808218}{19291510897}a+\frac{8806762064}{19291510897}$, $\frac{2086741886}{96457554485}a^{12}-\frac{44674217509}{675202881395}a^{11}+\frac{106099209507}{675202881395}a^{10}-\frac{297509607124}{675202881395}a^{9}-\frac{245666695199}{675202881395}a^{8}+\frac{531283845619}{675202881395}a^{7}+\frac{44023894055}{19291510897}a^{6}+\frac{54518650108}{11068899695}a^{5}+\frac{3522895265711}{675202881395}a^{4}-\frac{1381062536137}{675202881395}a^{3}-\frac{1500239995637}{675202881395}a^{2}-\frac{67729049404}{96457554485}a+\frac{275490411451}{96457554485}$, $\frac{13681497061}{675202881395}a^{12}-\frac{44072363257}{675202881395}a^{11}+\frac{14612849648}{96457554485}a^{10}-\frac{278529739312}{675202881395}a^{9}-\frac{220588775387}{675202881395}a^{8}+\frac{644503264262}{675202881395}a^{7}+\frac{261364033257}{135040576279}a^{6}+\frac{385724500217}{96457554485}a^{5}+\frac{2068779394923}{675202881395}a^{4}-\frac{32384661586}{11068899695}a^{3}+\frac{207098360787}{96457554485}a^{2}+\frac{276592087958}{96457554485}a-\frac{34585685662}{96457554485}$, $\frac{8454489937}{675202881395}a^{12}-\frac{25227590644}{675202881395}a^{11}+\frac{63729384142}{675202881395}a^{10}-\frac{180443817179}{675202881395}a^{9}-\frac{118783113144}{675202881395}a^{8}+\frac{203827809069}{675202881395}a^{7}+\frac{170991624154}{135040576279}a^{6}+\frac{2113635577168}{675202881395}a^{5}+\frac{2327914974716}{675202881395}a^{4}+\frac{220960128908}{675202881395}a^{3}+\frac{569008799493}{675202881395}a^{2}-\frac{41301999189}{96457554485}a+\frac{155032519946}{96457554485}$, $\frac{31918681281}{675202881395}a^{12}-\frac{139699543632}{675202881395}a^{11}+\frac{384361175801}{675202881395}a^{10}-\frac{1024394768857}{675202881395}a^{9}+\frac{474584255818}{675202881395}a^{8}+\frac{1445901405117}{675202881395}a^{7}+\frac{244400338288}{135040576279}a^{6}+\frac{573214672702}{96457554485}a^{5}+\frac{4105263668}{11068899695}a^{4}-\frac{6911404108026}{675202881395}a^{3}+\frac{1147245289212}{96457554485}a^{2}-\frac{240365907102}{96457554485}a-\frac{90960130792}{96457554485}$, $\frac{76055023696}{675202881395}a^{12}-\frac{259279467182}{675202881395}a^{11}+\frac{718220812646}{675202881395}a^{10}-\frac{2038547955202}{675202881395}a^{9}+\frac{35240303713}{675202881395}a^{8}+\frac{163839753796}{96457554485}a^{7}+\frac{1293735605227}{135040576279}a^{6}+\frac{18031189495089}{675202881395}a^{5}+\frac{2468971081079}{96457554485}a^{4}+\frac{3822646370679}{675202881395}a^{3}+\frac{14390393186459}{675202881395}a^{2}-\frac{968825843762}{96457554485}a+\frac{2661038614618}{96457554485}$ 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:  \( 297132.5632297882 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{6}\cdot 297132.5632297882 \cdot 1}{2\cdot\sqrt{489163986649360075249}}\cr\approx \mathstrut & 0.826612982119755 \end{aligned}\]

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^13 - 3*x^12 + 8*x^11 - 23*x^10 - 10*x^9 + 14*x^8 + 98*x^7 + 269*x^6 + 309*x^5 + 121*x^4 + 175*x^3 - 28*x^2 + 245*x + 49) 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 + 8*x^11 - 23*x^10 - 10*x^9 + 14*x^8 + 98*x^7 + 269*x^6 + 309*x^5 + 121*x^4 + 175*x^3 - 28*x^2 + 245*x + 49, 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 + 8*x^11 - 23*x^10 - 10*x^9 + 14*x^8 + 98*x^7 + 269*x^6 + 309*x^5 + 121*x^4 + 175*x^3 - 28*x^2 + 245*x + 49); 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 + 8*x^11 - 23*x^10 - 10*x^9 + 14*x^8 + 98*x^7 + 269*x^6 + 309*x^5 + 121*x^4 + 175*x^3 - 28*x^2 + 245*x + 49); 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 ${\href{/padicField/2.13.0.1}{13} }$ ${\href{/padicField/3.13.0.1}{13} }$ ${\href{/padicField/5.2.0.1}{2} }^{6}{,}\,{\href{/padicField/5.1.0.1}{1} }$ R ${\href{/padicField/11.13.0.1}{13} }$ ${\href{/padicField/13.13.0.1}{13} }$ ${\href{/padicField/17.13.0.1}{13} }$ ${\href{/padicField/19.13.0.1}{13} }$ ${\href{/padicField/23.2.0.1}{2} }^{6}{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.13.0.1}{13} }$ ${\href{/padicField/31.13.0.1}{13} }$ ${\href{/padicField/37.2.0.1}{2} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.13.0.1}{13} }$ ${\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.2.0.1}{2} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.13.0.1}{13} }$

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
\(7\) Copy content Toggle raw display $\Q_{7}$$x + 4$$1$$1$$0$Trivial$$[\ ]$$
7.1.2.1a1.1$x^{2} + 7$$2$$1$$1$$C_2$$$[\ ]_{2}$$
7.1.2.1a1.1$x^{2} + 7$$2$$1$$1$$C_2$$$[\ ]_{2}$$
7.1.2.1a1.1$x^{2} + 7$$2$$1$$1$$C_2$$$[\ ]_{2}$$
7.1.2.1a1.1$x^{2} + 7$$2$$1$$1$$C_2$$$[\ ]_{2}$$
7.1.2.1a1.1$x^{2} + 7$$2$$1$$1$$C_2$$$[\ ]_{2}$$
7.1.2.1a1.1$x^{2} + 7$$2$$1$$1$$C_2$$$[\ ]_{2}$$
\(401\) Copy content Toggle raw display $\Q_{401}$$x$$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$$$[\ ]_{2}$$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
*26 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.2807.2t1.a.a$1$ $ 7 \cdot 401 $ \(\Q(\sqrt{-2807}) \) $C_2$ (as 2T1) $1$ $-1$
*26 2.2807.13t2.a.f$2$ $ 7 \cdot 401 $ 13.1.489163986649360075249.1 $D_{13}$ (as 13T2) $1$ $0$
*26 2.2807.13t2.a.b$2$ $ 7 \cdot 401 $ 13.1.489163986649360075249.1 $D_{13}$ (as 13T2) $1$ $0$
*26 2.2807.13t2.a.e$2$ $ 7 \cdot 401 $ 13.1.489163986649360075249.1 $D_{13}$ (as 13T2) $1$ $0$
*26 2.2807.13t2.a.c$2$ $ 7 \cdot 401 $ 13.1.489163986649360075249.1 $D_{13}$ (as 13T2) $1$ $0$
*26 2.2807.13t2.a.a$2$ $ 7 \cdot 401 $ 13.1.489163986649360075249.1 $D_{13}$ (as 13T2) $1$ $0$
*26 2.2807.13t2.a.d$2$ $ 7 \cdot 401 $ 13.1.489163986649360075249.1 $D_{13}$ (as 13T2) $1$ $0$

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)