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Properties

Label 14.2.965463713906498833.1
Degree $14$
Signature $[2, 6]$
Discriminant $9.655\times 10^{17}$
Root discriminant \(19.26\)
Ramified primes $13,109$
Class number $1$
Class group trivial
Galois group $C_2^4.\PSL(2,7)$ (as 14T42)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 + 18*x^12 - 35*x^11 + 43*x^10 - 23*x^9 - 3*x^8 - 65*x^7 + 163*x^6 - 168*x^5 + 12*x^4 + 175*x^3 - 180*x^2 + 13*x - 13)
 
Copy content gp:K = bnfinit(y^14 - 5*y^13 + 18*y^12 - 35*y^11 + 43*y^10 - 23*y^9 - 3*y^8 - 65*y^7 + 163*y^6 - 168*y^5 + 12*y^4 + 175*y^3 - 180*y^2 + 13*y - 13, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 5*x^13 + 18*x^12 - 35*x^11 + 43*x^10 - 23*x^9 - 3*x^8 - 65*x^7 + 163*x^6 - 168*x^5 + 12*x^4 + 175*x^3 - 180*x^2 + 13*x - 13);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^14 - 5*x^13 + 18*x^12 - 35*x^11 + 43*x^10 - 23*x^9 - 3*x^8 - 65*x^7 + 163*x^6 - 168*x^5 + 12*x^4 + 175*x^3 - 180*x^2 + 13*x - 13)
 

\( x^{14} - 5 x^{13} + 18 x^{12} - 35 x^{11} + 43 x^{10} - 23 x^{9} - 3 x^{8} - 65 x^{7} + 163 x^{6} + \cdots - 13 \) 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:  $14$
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:  $[2, 6]$
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:   \(965463713906498833\) \(\medspace = 13^{7}\cdot 109^{5}\) 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:  \(19.26\)
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:  $13^{3/4}109^{1/2}\approx 71.4777318983505$
Ramified primes:   \(13\), \(109\) 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:  \(\Q(\sqrt{1417}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{238359580461007}a^{13}+\frac{67452008019111}{238359580461007}a^{12}-\frac{235903118738}{238359580461007}a^{11}-\frac{90079481428426}{238359580461007}a^{10}-\frac{50025093285771}{238359580461007}a^{9}+\frac{86018523961010}{238359580461007}a^{8}+\frac{3701225049714}{238359580461007}a^{7}-\frac{83210936486800}{238359580461007}a^{6}+\frac{110133901322719}{238359580461007}a^{5}-\frac{72670433684315}{238359580461007}a^{4}+\frac{13898741506216}{238359580461007}a^{3}-\frac{88859828916603}{238359580461007}a^{2}+\frac{103768400445579}{238359580461007}a+\frac{85802307008269}{238359580461007}$ 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:   \( -1 \)  (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:   $\frac{746252748966}{238359580461007}a^{13}-\frac{5214141004098}{238359580461007}a^{12}+\frac{22357610543750}{238359580461007}a^{11}-\frac{58737980091251}{238359580461007}a^{10}+\frac{102487986452145}{238359580461007}a^{9}-\frac{102156599992666}{238359580461007}a^{8}+\frac{38743726466366}{238359580461007}a^{7}-\frac{11216112398086}{238359580461007}a^{6}+\frac{203571878759117}{238359580461007}a^{5}-\frac{463442241195441}{238359580461007}a^{4}+\frac{340272383568515}{238359580461007}a^{3}+\frac{179954030784882}{238359580461007}a^{2}-\frac{633738282965489}{238359580461007}a+\frac{298538816624038}{238359580461007}$, $\frac{1613096339700}{238359580461007}a^{13}-\frac{9375140481580}{238359580461007}a^{12}+\frac{36074662484199}{238359580461007}a^{11}-\frac{83801106065312}{238359580461007}a^{10}+\frac{127023514544906}{238359580461007}a^{9}-\frac{117708279592227}{238359580461007}a^{8}+\frac{39804057724684}{238359580461007}a^{7}-\frac{101730011322516}{238359580461007}a^{6}+\frac{343416769043173}{238359580461007}a^{5}-\frac{589658399522470}{238359580461007}a^{4}+\frac{425010058326966}{238359580461007}a^{3}+\frac{284563124036155}{238359580461007}a^{2}-\frac{588726128103061}{238359580461007}a+\frac{205615427619208}{238359580461007}$, $\frac{1589783175325}{238359580461007}a^{13}-\frac{6884210118230}{238359580461007}a^{12}+\frac{25612547866607}{238359580461007}a^{11}-\frac{47225704506324}{238359580461007}a^{10}+\frac{63839034752406}{238359580461007}a^{9}-\frac{38118509123200}{238359580461007}a^{8}-\frac{15057189927768}{238359580461007}a^{7}-\frac{58981481323196}{238359580461007}a^{6}+\frac{151045325911951}{238359580461007}a^{5}-\frac{307445930218327}{238359580461007}a^{4}+\frac{184503240845407}{238359580461007}a^{3}+\frac{474141108310848}{238359580461007}a^{2}-\frac{486753314351291}{238359580461007}a+\frac{44825167109834}{238359580461007}$, $\frac{480228459158}{238359580461007}a^{13}-\frac{3219408117729}{238359580461007}a^{12}+\frac{12855259022986}{238359580461007}a^{11}-\frac{32074365650702}{238359580461007}a^{10}+\frac{46501586375681}{238359580461007}a^{9}-\frac{37255669341479}{238359580461007}a^{8}-\frac{13212941625245}{238359580461007}a^{7}+\frac{2073338289007}{238359580461007}a^{6}+\frac{129088587996058}{238359580461007}a^{5}-\frac{204104317914736}{238359580461007}a^{4}+\frac{161286764106365}{238359580461007}a^{3}+\frac{224690279394069}{238359580461007}a^{2}-\frac{314351293901970}{238359580461007}a+\frac{129492276831878}{238359580461007}$, $\frac{5471001514106}{238359580461007}a^{13}-\frac{20695437960298}{238359580461007}a^{12}+\frac{72340014322968}{238359580461007}a^{11}-\frac{99194747593219}{238359580461007}a^{10}+\frac{96313786388260}{238359580461007}a^{9}+\frac{27966909863832}{238359580461007}a^{8}-\frac{57288013712321}{238359580461007}a^{7}-\frac{338375572836108}{238359580461007}a^{6}+\frac{360043670505324}{238359580461007}a^{5}-\frac{337639038397111}{238359580461007}a^{4}-\frac{474826721455489}{238359580461007}a^{3}+\frac{738614403101021}{238359580461007}a^{2}-\frac{451916597013105}{238359580461007}a-\frac{32278893420379}{238359580461007}$, $\frac{877657273268}{238359580461007}a^{13}-\frac{4489969346389}{238359580461007}a^{12}+\frac{15296820774037}{238359580461007}a^{11}-\frac{26010822377680}{238359580461007}a^{10}+\frac{19607545330343}{238359580461007}a^{9}+\frac{9915901205913}{238359580461007}a^{8}-\frac{18800341344293}{238359580461007}a^{7}-\frac{86798315603443}{238359580461007}a^{6}+\frac{159934561150257}{238359580461007}a^{5}-\frac{3209010500241}{238359580461007}a^{4}-\frac{125098357043223}{238359580461007}a^{3}+\frac{218082417327471}{238359580461007}a^{2}-\frac{45024766510003}{238359580461007}a-\frac{11220902965098}{238359580461007}$, $\frac{4429868386028}{238359580461007}a^{13}-\frac{22516045126275}{238359580461007}a^{12}+\frac{82847072198857}{238359580461007}a^{11}-\frac{169850039805798}{238359580461007}a^{10}+\frac{235096209039355}{238359580461007}a^{9}-\frac{190750770937278}{238359580461007}a^{8}+\frac{105283509023056}{238359580461007}a^{7}-\frac{391242329829080}{238359580461007}a^{6}+\frac{836977918731997}{238359580461007}a^{5}-\frac{10\cdots 03}{238359580461007}a^{4}+\frac{511029674797426}{238359580461007}a^{3}+\frac{425201833145661}{238359580461007}a^{2}-\frac{774846196131562}{238359580461007}a+\frac{49045166019413}{238359580461007}$ 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:  \( 1288.44929096 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 1288.44929096 \cdot 1}{2\cdot\sqrt{965463713906498833}}\cr\approx \mathstrut & 0.161364708858 \end{aligned}\]

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^14 - 5*x^13 + 18*x^12 - 35*x^11 + 43*x^10 - 23*x^9 - 3*x^8 - 65*x^7 + 163*x^6 - 168*x^5 + 12*x^4 + 175*x^3 - 180*x^2 + 13*x - 13) 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^14 - 5*x^13 + 18*x^12 - 35*x^11 + 43*x^10 - 23*x^9 - 3*x^8 - 65*x^7 + 163*x^6 - 168*x^5 + 12*x^4 + 175*x^3 - 180*x^2 + 13*x - 13, 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^14 - 5*x^13 + 18*x^12 - 35*x^11 + 43*x^10 - 23*x^9 - 3*x^8 - 65*x^7 + 163*x^6 - 168*x^5 + 12*x^4 + 175*x^3 - 180*x^2 + 13*x - 13); 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^14 - 5*x^13 + 18*x^12 - 35*x^11 + 43*x^10 - 23*x^9 - 3*x^8 - 65*x^7 + 163*x^6 - 168*x^5 + 12*x^4 + 175*x^3 - 180*x^2 + 13*x - 13); 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

$C_2^4.\PSL(2,7)$ (as 14T42):

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); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A non-solvable group of order 2688
The 22 conjugacy class representatives for $C_2^4.\PSL(2,7)$
Character table for $C_2^4.\PSL(2,7)$

Intermediate fields

7.3.2007889.2

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 14 sibling: data not computed
Degree 28 siblings: data not computed
Degree 42 siblings: 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 ${\href{/padicField/2.7.0.1}{7} }^{2}$ ${\href{/padicField/3.7.0.1}{7} }^{2}$ ${\href{/padicField/5.14.0.1}{14} }$ ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ R ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ ${\href{/padicField/23.14.0.1}{14} }$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ ${\href{/padicField/43.7.0.1}{7} }^{2}$ ${\href{/padicField/47.7.0.1}{7} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
\(13\) Copy content Toggle raw display 13.1.2.1a1.2$x^{2} + 26$$2$$1$$1$$C_2$$$[\ ]_{2}$$
13.4.1.0a1.1$x^{4} + 3 x^{2} + 12 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
13.2.4.6a1.4$x^{8} + 48 x^{7} + 872 x^{6} + 7200 x^{5} + 24216 x^{4} + 14400 x^{3} + 3488 x^{2} + 540 x + 159$$4$$2$$6$$C_8$$$[\ ]_{4}^{2}$$
\(109\) Copy content Toggle raw display $\Q_{109}$$x + 103$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{109}$$x + 103$$1$$1$$0$Trivial$$[\ ]$$
109.2.1.0a1.1$x^{2} + 108 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
109.1.2.1a1.1$x^{2} + 109$$2$$1$$1$$C_2$$$[\ ]_{2}$$
109.2.2.2a1.2$x^{4} + 216 x^{3} + 11676 x^{2} + 1296 x + 145$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
109.2.2.2a1.2$x^{4} + 216 x^{3} + 11676 x^{2} + 1296 x + 145$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

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