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Properties

Label 14.4.254...363.1
Degree $14$
Signature $[4, 5]$
Discriminant $-2.547\times 10^{19}$
Root discriminant \(24.33\)
Ramified primes $7,13,109$
Class number $1$
Class group trivial
Galois group $C_2^4.\PSL(2,7)$ (as 14T42)

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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^14 - 3*x^13 - 2*x^12 + x^11 + 30*x^10 + 45*x^9 - 200*x^8 - 45*x^7 + 257*x^6 - 121*x^5 + 614*x^4 + 323*x^3 - 1565*x^2 - 559*x + 523)
 
Copy content gp:K = bnfinit(y^14 - 3*y^13 - 2*y^12 + y^11 + 30*y^10 + 45*y^9 - 200*y^8 - 45*y^7 + 257*y^6 - 121*y^5 + 614*y^4 + 323*y^3 - 1565*y^2 - 559*y + 523, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 3*x^13 - 2*x^12 + x^11 + 30*x^10 + 45*x^9 - 200*x^8 - 45*x^7 + 257*x^6 - 121*x^5 + 614*x^4 + 323*x^3 - 1565*x^2 - 559*x + 523);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^14 - 3*x^13 - 2*x^12 + x^11 + 30*x^10 + 45*x^9 - 200*x^8 - 45*x^7 + 257*x^6 - 121*x^5 + 614*x^4 + 323*x^3 - 1565*x^2 - 559*x + 523)
 

\( x^{14} - 3 x^{13} - 2 x^{12} + x^{11} + 30 x^{10} + 45 x^{9} - 200 x^{8} - 45 x^{7} + 257 x^{6} + \cdots + 523 \) 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:  $[4, 5]$
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:   \(-25473388759225315363\) \(\medspace = -\,7^{3}\cdot 13^{6}\cdot 109^{5}\) 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:  \(24.33\)
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:  $7^{1/2}13^{3/4}109^{1/2}\approx 189.11230288198414$
Ramified primes:   \(7\), \(13\), \(109\) 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:  \(\Q(\sqrt{-763}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{20\cdots 73}a^{13}-\frac{45\cdots 70}{20\cdots 73}a^{12}-\frac{209067191004083}{15\cdots 21}a^{11}+\frac{58\cdots 70}{20\cdots 73}a^{10}-\frac{46973154904851}{15\cdots 21}a^{9}-\frac{44\cdots 62}{20\cdots 73}a^{8}-\frac{30\cdots 21}{20\cdots 73}a^{7}-\frac{55\cdots 55}{20\cdots 73}a^{6}+\frac{55\cdots 80}{20\cdots 73}a^{5}-\frac{286416571035642}{15\cdots 21}a^{4}+\frac{22\cdots 84}{20\cdots 73}a^{3}+\frac{56\cdots 80}{20\cdots 73}a^{2}+\frac{22\cdots 26}{20\cdots 73}a-\frac{80\cdots 43}{20\cdots 73}$ 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:  $8$
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:   $\frac{305125172505341}{20\cdots 73}a^{13}-\frac{11\cdots 71}{20\cdots 73}a^{12}+\frac{26537635373607}{15\cdots 21}a^{11}-\frac{340469965838867}{20\cdots 73}a^{10}+\frac{681866938083748}{15\cdots 21}a^{9}+\frac{69\cdots 14}{20\cdots 73}a^{8}-\frac{61\cdots 99}{20\cdots 73}a^{7}+\frac{39\cdots 11}{20\cdots 73}a^{6}+\frac{29\cdots 21}{20\cdots 73}a^{5}-\frac{50\cdots 27}{15\cdots 21}a^{4}+\frac{24\cdots 84}{20\cdots 73}a^{3}-\frac{11\cdots 19}{20\cdots 73}a^{2}-\frac{30\cdots 95}{20\cdots 73}a+\frac{12\cdots 94}{20\cdots 73}$, $\frac{334914768653626}{20\cdots 73}a^{13}-\frac{13\cdots 58}{20\cdots 73}a^{12}+\frac{59071740598136}{15\cdots 21}a^{11}-\frac{746542798170008}{20\cdots 73}a^{10}+\frac{838381570414618}{15\cdots 21}a^{9}+\frac{38\cdots 16}{20\cdots 73}a^{8}-\frac{67\cdots 04}{20\cdots 73}a^{7}+\frac{55\cdots 89}{20\cdots 73}a^{6}+\frac{14\cdots 62}{20\cdots 73}a^{5}-\frac{38\cdots 89}{15\cdots 21}a^{4}+\frac{25\cdots 28}{20\cdots 73}a^{3}-\frac{15\cdots 42}{20\cdots 73}a^{2}-\frac{30\cdots 37}{20\cdots 73}a+\frac{11\cdots 82}{20\cdots 73}$, $\frac{305125172505341}{20\cdots 73}a^{13}-\frac{11\cdots 71}{20\cdots 73}a^{12}+\frac{26537635373607}{15\cdots 21}a^{11}-\frac{340469965838867}{20\cdots 73}a^{10}+\frac{681866938083748}{15\cdots 21}a^{9}+\frac{69\cdots 14}{20\cdots 73}a^{8}-\frac{61\cdots 99}{20\cdots 73}a^{7}+\frac{39\cdots 11}{20\cdots 73}a^{6}+\frac{29\cdots 21}{20\cdots 73}a^{5}-\frac{50\cdots 27}{15\cdots 21}a^{4}+\frac{24\cdots 84}{20\cdots 73}a^{3}-\frac{11\cdots 19}{20\cdots 73}a^{2}-\frac{30\cdots 95}{20\cdots 73}a+\frac{10\cdots 21}{20\cdots 73}$, $\frac{131742704307580}{20\cdots 73}a^{13}-\frac{583199986958369}{20\cdots 73}a^{12}+\frac{43480562044631}{15\cdots 21}a^{11}-\frac{592957105708953}{20\cdots 73}a^{10}+\frac{345145464015694}{15\cdots 21}a^{9}-\frac{519412717727333}{20\cdots 73}a^{8}-\frac{25\cdots 78}{20\cdots 73}a^{7}+\frac{33\cdots 49}{20\cdots 73}a^{6}-\frac{10\cdots 89}{20\cdots 73}a^{5}-\frac{10\cdots 76}{15\cdots 21}a^{4}+\frac{10\cdots 64}{20\cdots 73}a^{3}-\frac{10\cdots 57}{20\cdots 73}a^{2}-\frac{89\cdots 37}{20\cdots 73}a+\frac{68\cdots 80}{20\cdots 73}$, $\frac{30101918962212}{20\cdots 73}a^{13}-\frac{65640405867650}{20\cdots 73}a^{12}-\frac{15175246516715}{15\cdots 21}a^{11}+\frac{394554479783782}{20\cdots 73}a^{10}+\frac{20321670499865}{15\cdots 21}a^{9}+\frac{25\cdots 12}{20\cdots 73}a^{8}-\frac{89\cdots 38}{20\cdots 73}a^{7}-\frac{449875322800023}{20\cdots 73}a^{6}+\frac{21\cdots 92}{20\cdots 73}a^{5}-\frac{16\cdots 95}{15\cdots 21}a^{4}+\frac{11\cdots 58}{20\cdots 73}a^{3}+\frac{68\cdots 02}{20\cdots 73}a^{2}-\frac{44\cdots 87}{20\cdots 73}a+\frac{14\cdots 05}{20\cdots 73}$, $\frac{495569380862893}{20\cdots 73}a^{13}-\frac{19\cdots 88}{20\cdots 73}a^{12}+\frac{106736636665165}{15\cdots 21}a^{11}-\frac{19\cdots 69}{20\cdots 73}a^{10}+\frac{12\cdots 28}{15\cdots 21}a^{9}+\frac{35\cdots 54}{20\cdots 73}a^{8}-\frac{91\cdots 74}{20\cdots 73}a^{7}+\frac{87\cdots 99}{20\cdots 73}a^{6}-\frac{95\cdots 47}{20\cdots 73}a^{5}-\frac{32\cdots 63}{15\cdots 21}a^{4}+\frac{32\cdots 75}{20\cdots 73}a^{3}-\frac{22\cdots 54}{20\cdots 73}a^{2}-\frac{30\cdots 68}{20\cdots 73}a+\frac{13\cdots 11}{20\cdots 73}$, $\frac{226945372805063}{20\cdots 73}a^{13}-\frac{367428463638985}{20\cdots 73}a^{12}-\frac{71284357915722}{15\cdots 21}a^{11}-\frac{677275590608107}{20\cdots 73}a^{10}+\frac{393145960557015}{15\cdots 21}a^{9}+\frac{15\cdots 55}{20\cdots 73}a^{8}-\frac{26\cdots 76}{20\cdots 73}a^{7}-\frac{36\cdots 08}{20\cdots 73}a^{6}+\frac{36\cdots 42}{20\cdots 73}a^{5}-\frac{756861199265041}{15\cdots 21}a^{4}+\frac{85\cdots 92}{20\cdots 73}a^{3}+\frac{16\cdots 69}{20\cdots 73}a^{2}-\frac{22\cdots 51}{20\cdots 73}a-\frac{27\cdots 04}{20\cdots 73}$, $\frac{26366802082206}{15\cdots 21}a^{13}-\frac{85423539808312}{15\cdots 21}a^{12}+\frac{15463546432407}{15\cdots 21}a^{11}-\frac{90030537945866}{15\cdots 21}a^{10}+\frac{733882951882169}{15\cdots 21}a^{9}+\frac{829139785214816}{15\cdots 21}a^{8}-\frac{43\cdots 62}{15\cdots 21}a^{7}+\frac{24\cdots 14}{15\cdots 21}a^{6}-\frac{60071427885786}{15\cdots 21}a^{5}-\frac{38\cdots 31}{15\cdots 21}a^{4}+\frac{17\cdots 44}{15\cdots 21}a^{3}-\frac{29\cdots 31}{15\cdots 21}a^{2}-\frac{13\cdots 60}{15\cdots 21}a+\frac{53\cdots 15}{15\cdots 21}$ 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:  \( 11131.4204258 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{5}\cdot 11131.4204258 \cdot 1}{2\cdot\sqrt{25473388759225315363}}\cr\approx \mathstrut & 0.172781226696 \end{aligned}\]

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 - 3*x^13 - 2*x^12 + x^11 + 30*x^10 + 45*x^9 - 200*x^8 - 45*x^7 + 257*x^6 - 121*x^5 + 614*x^4 + 323*x^3 - 1565*x^2 - 559*x + 523) 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 - 3*x^13 - 2*x^12 + x^11 + 30*x^10 + 45*x^9 - 200*x^8 - 45*x^7 + 257*x^6 - 121*x^5 + 614*x^4 + 323*x^3 - 1565*x^2 - 559*x + 523, 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 - 3*x^13 - 2*x^12 + x^11 + 30*x^10 + 45*x^9 - 200*x^8 - 45*x^7 + 257*x^6 - 121*x^5 + 614*x^4 + 323*x^3 - 1565*x^2 - 559*x + 523); 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^14 - 3*x^13 - 2*x^12 + x^11 + 30*x^10 + 45*x^9 - 200*x^8 - 45*x^7 + 257*x^6 - 121*x^5 + 614*x^4 + 323*x^3 - 1565*x^2 - 559*x + 523); 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); G, transitive_group_identification(G)
 
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(b)]
 
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 ${\href{/padicField/2.14.0.1}{14} }$ ${\href{/padicField/3.14.0.1}{14} }$ ${\href{/padicField/5.14.0.1}{14} }$ R ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ R ${\href{/padicField/17.3.0.1}{3} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ ${\href{/padicField/23.14.0.1}{14} }$ ${\href{/padicField/29.3.0.1}{3} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ ${\href{/padicField/43.7.0.1}{7} }^{2}$ ${\href{/padicField/47.7.0.1}{7} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$

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 7.2.1.0a1.1$x^{2} + 6 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
7.6.1.0a1.1$x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3$$1$$6$$0$$C_6$$$[\ ]^{6}$$
7.3.2.3a1.2$x^{6} + 12 x^{5} + 36 x^{4} + 8 x^{3} + 48 x^{2} + 23$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
\(13\) Copy content Toggle raw display 13.2.1.0a1.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
13.2.1.0a1.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
13.2.1.0a1.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
13.2.4.6a1.1$x^{8} + 48 x^{7} + 872 x^{6} + 7200 x^{5} + 24216 x^{4} + 14400 x^{3} + 3488 x^{2} + 397 x + 16$$4$$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$$[\ ]$$
109.1.2.1a1.2$x^{2} + 654$$2$$1$$1$$C_2$$$[\ ]_{2}$$
109.2.1.0a1.1$x^{2} + 108 x + 6$$1$$2$$0$$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)