↑

Properties

Label 14.0.337...704.1
Degree $14$
Signature $[0, 7]$
Discriminant $-3.373\times 10^{32}$
Root discriminant \(210.59\)
Ramified primes $2,7,31$
Class number $21$ (GRH)
Class group [21] (GRH)
Galois group $C_7^2:S_3$ (as 14T15)

Related objects

Downloads

Learn more

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 - 105*x^12 - 147*x^11 + 5271*x^10 + 19838*x^9 - 94150*x^8 - 607634*x^7 + 570164*x^6 + 12260920*x^5 + 42847770*x^4 + 95169270*x^3 + 197804880*x^2 + 348280352*x + 336238208)
 
Copy content gp:K = bnfinit(y^14 - 105*y^12 - 147*y^11 + 5271*y^10 + 19838*y^9 - 94150*y^8 - 607634*y^7 + 570164*y^6 + 12260920*y^5 + 42847770*y^4 + 95169270*y^3 + 197804880*y^2 + 348280352*y + 336238208, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 105*x^12 - 147*x^11 + 5271*x^10 + 19838*x^9 - 94150*x^8 - 607634*x^7 + 570164*x^6 + 12260920*x^5 + 42847770*x^4 + 95169270*x^3 + 197804880*x^2 + 348280352*x + 336238208);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^14 - 105*x^12 - 147*x^11 + 5271*x^10 + 19838*x^9 - 94150*x^8 - 607634*x^7 + 570164*x^6 + 12260920*x^5 + 42847770*x^4 + 95169270*x^3 + 197804880*x^2 + 348280352*x + 336238208)
 

\( x^{14} - 105 x^{12} - 147 x^{11} + 5271 x^{10} + 19838 x^{9} - 94150 x^{8} - 607634 x^{7} + \cdots + 336238208 \) 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:  $[0, 7]$
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:   \(-337337631352558562817384077600704\) \(\medspace = -\,2^{6}\cdot 7^{24}\cdot 31^{7}\) 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:  \(210.59\)
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:  $2^{6/7}7^{96/49}31^{1/2}\approx 456.46642613478804$
Ramified primes:   \(2\), \(7\), \(31\) 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:  \(\Q(\sqrt{-31}) \)
$\Aut(K/\Q)$:   $C_1$
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.
Maximal CM subfield:  \(\Q(\sqrt{-31}) \)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{10}-\frac{3}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{4}a^{7}+\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{2}-\frac{1}{4}a$, $\frac{1}{33\cdots 44}a^{13}+\frac{41\cdots 11}{84\cdots 36}a^{12}+\frac{26\cdots 67}{33\cdots 44}a^{11}-\frac{14\cdots 07}{33\cdots 44}a^{10}-\frac{64\cdots 93}{33\cdots 44}a^{9}+\frac{19\cdots 01}{16\cdots 72}a^{8}+\frac{71\cdots 89}{16\cdots 72}a^{7}-\frac{31\cdots 77}{16\cdots 72}a^{6}-\frac{79\cdots 73}{84\cdots 36}a^{5}+\frac{76\cdots 01}{42\cdots 68}a^{4}-\frac{33\cdots 71}{16\cdots 72}a^{3}+\frac{68\cdots 99}{16\cdots 72}a^{2}+\frac{18\cdots 79}{42\cdots 68}a-\frac{30\cdots 52}{10\cdots 67}$ 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:  $C_{21}$, which has order $21$ (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:  $C_{21}$, which has order $21$ (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:   $\frac{13\cdots 81}{33\cdots 44}a^{13}-\frac{16\cdots 19}{84\cdots 36}a^{12}-\frac{12\cdots 33}{33\cdots 44}a^{11}+\frac{50\cdots 85}{33\cdots 44}a^{10}+\frac{55\cdots 03}{33\cdots 44}a^{9}-\frac{44\cdots 55}{16\cdots 72}a^{8}-\frac{61\cdots 03}{16\cdots 72}a^{7}-\frac{65\cdots 33}{16\cdots 72}a^{6}+\frac{43\cdots 99}{84\cdots 36}a^{5}+\frac{75\cdots 97}{42\cdots 68}a^{4}+\frac{66\cdots 09}{16\cdots 72}a^{3}+\frac{14\cdots 55}{16\cdots 72}a^{2}+\frac{67\cdots 61}{42\cdots 68}a+\frac{14\cdots 87}{10\cdots 67}$, $\frac{13\cdots 39}{33\cdots 44}a^{13}-\frac{12\cdots 97}{21\cdots 34}a^{12}-\frac{13\cdots 75}{33\cdots 44}a^{11}-\frac{24\cdots 97}{33\cdots 44}a^{10}+\frac{51\cdots 69}{33\cdots 44}a^{9}+\frac{11\cdots 33}{16\cdots 72}a^{8}-\frac{32\cdots 21}{16\cdots 72}a^{7}-\frac{33\cdots 87}{16\cdots 72}a^{6}-\frac{55\cdots 13}{84\cdots 36}a^{5}-\frac{74\cdots 55}{42\cdots 68}a^{4}-\frac{73\cdots 65}{16\cdots 72}a^{3}-\frac{10\cdots 11}{16\cdots 72}a^{2}-\frac{34\cdots 57}{21\cdots 34}a+\frac{85\cdots 81}{10\cdots 67}$, $\frac{79\cdots 61}{16\cdots 72}a^{13}-\frac{14\cdots 39}{84\cdots 36}a^{12}-\frac{86\cdots 25}{16\cdots 72}a^{11}-\frac{73\cdots 21}{16\cdots 72}a^{10}+\frac{45\cdots 21}{16\cdots 72}a^{9}+\frac{33\cdots 97}{42\cdots 68}a^{8}-\frac{49\cdots 45}{84\cdots 36}a^{7}-\frac{22\cdots 75}{84\cdots 36}a^{6}+\frac{67\cdots 59}{10\cdots 67}a^{5}+\frac{62\cdots 31}{10\cdots 67}a^{4}+\frac{11\cdots 49}{84\cdots 36}a^{3}+\frac{20\cdots 89}{84\cdots 36}a^{2}+\frac{27\cdots 11}{42\cdots 68}a+\frac{87\cdots 61}{10\cdots 67}$, $\frac{11\cdots 03}{33\cdots 44}a^{13}-\frac{10\cdots 45}{84\cdots 36}a^{12}-\frac{10\cdots 35}{33\cdots 44}a^{11}+\frac{23\cdots 79}{33\cdots 44}a^{10}+\frac{53\cdots 45}{33\cdots 44}a^{9}+\frac{15\cdots 15}{16\cdots 72}a^{8}-\frac{61\cdots 85}{16\cdots 72}a^{7}-\frac{12\cdots 47}{16\cdots 72}a^{6}+\frac{41\cdots 05}{84\cdots 36}a^{5}+\frac{10\cdots 75}{42\cdots 68}a^{4}+\frac{94\cdots 87}{16\cdots 72}a^{3}+\frac{20\cdots 25}{16\cdots 72}a^{2}+\frac{10\cdots 31}{42\cdots 68}a+\frac{32\cdots 63}{10\cdots 67}$, $\frac{69\cdots 27}{33\cdots 44}a^{13}-\frac{37\cdots 89}{42\cdots 68}a^{12}-\frac{60\cdots 87}{33\cdots 44}a^{11}+\frac{15\cdots 07}{33\cdots 44}a^{10}+\frac{29\cdots 61}{33\cdots 44}a^{9}+\frac{46\cdots 85}{16\cdots 72}a^{8}-\frac{34\cdots 17}{16\cdots 72}a^{7}-\frac{62\cdots 47}{16\cdots 72}a^{6}+\frac{23\cdots 11}{84\cdots 36}a^{5}+\frac{56\cdots 57}{42\cdots 68}a^{4}+\frac{51\cdots 07}{16\cdots 72}a^{3}+\frac{10\cdots 21}{16\cdots 72}a^{2}+\frac{13\cdots 86}{10\cdots 67}a+\frac{17\cdots 65}{10\cdots 67}$, $\frac{15\cdots 17}{16\cdots 72}a^{13}-\frac{73\cdots 33}{84\cdots 36}a^{12}-\frac{16\cdots 61}{16\cdots 72}a^{11}-\frac{21\cdots 89}{16\cdots 72}a^{10}+\frac{83\cdots 37}{16\cdots 72}a^{9}+\frac{18\cdots 10}{10\cdots 67}a^{8}-\frac{84\cdots 61}{84\cdots 36}a^{7}-\frac{48\cdots 63}{84\cdots 36}a^{6}+\frac{19\cdots 35}{21\cdots 34}a^{5}+\frac{12\cdots 91}{10\cdots 67}a^{4}+\frac{26\cdots 97}{84\cdots 36}a^{3}+\frac{45\cdots 33}{84\cdots 36}a^{2}+\frac{58\cdots 29}{42\cdots 68}a+\frac{22\cdots 58}{10\cdots 67}$ 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:  \( 8415941567.41 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 8415941567.41 \cdot 21}{2\cdot\sqrt{337337631352558562817384077600704}}\cr\approx \mathstrut & 1.86002462311 \end{aligned}\] (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^14 - 105*x^12 - 147*x^11 + 5271*x^10 + 19838*x^9 - 94150*x^8 - 607634*x^7 + 570164*x^6 + 12260920*x^5 + 42847770*x^4 + 95169270*x^3 + 197804880*x^2 + 348280352*x + 336238208) 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 - 105*x^12 - 147*x^11 + 5271*x^10 + 19838*x^9 - 94150*x^8 - 607634*x^7 + 570164*x^6 + 12260920*x^5 + 42847770*x^4 + 95169270*x^3 + 197804880*x^2 + 348280352*x + 336238208, 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 - 105*x^12 - 147*x^11 + 5271*x^10 + 19838*x^9 - 94150*x^8 - 607634*x^7 + 570164*x^6 + 12260920*x^5 + 42847770*x^4 + 95169270*x^3 + 197804880*x^2 + 348280352*x + 336238208); 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 - 105*x^12 - 147*x^11 + 5271*x^10 + 19838*x^9 - 94150*x^8 - 607634*x^7 + 570164*x^6 + 12260920*x^5 + 42847770*x^4 + 95169270*x^3 + 197804880*x^2 + 348280352*x + 336238208); 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

$C_7^2:S_3$ (as 14T15):

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 solvable group of order 294
The 20 conjugacy class representatives for $C_7^2:S_3$
Character table for $C_7^2:S_3$

Intermediate fields

\(\Q(\sqrt{-31}) \)

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 21 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 R ${\href{/padicField/3.14.0.1}{14} }$ ${\href{/padicField/5.3.0.1}{3} }^{4}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ R ${\href{/padicField/11.14.0.1}{14} }$ ${\href{/padicField/13.14.0.1}{14} }$ ${\href{/padicField/17.14.0.1}{14} }$ ${\href{/padicField/19.3.0.1}{3} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.14.0.1}{14} }$ ${\href{/padicField/29.14.0.1}{14} }$ R ${\href{/padicField/37.14.0.1}{14} }$ ${\href{/padicField/41.3.0.1}{3} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.14.0.1}{14} }$ ${\href{/padicField/47.7.0.1}{7} }^{2}$ ${\href{/padicField/53.14.0.1}{14} }$ ${\href{/padicField/59.3.0.1}{3} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
\(2\) Copy content Toggle raw display $\Q_{2}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
2.3.1.0a1.1$x^{3} + x + 1$$1$$3$$0$$C_3$$$[\ ]^{3}$$
2.3.1.0a1.1$x^{3} + x + 1$$1$$3$$0$$C_3$$$[\ ]^{3}$$
2.1.7.6a1.1$x^{7} + 2$$7$$1$$6$$C_7:C_3$$$[\ ]_{7}^{3}$$
\(7\) Copy content Toggle raw display 7.1.7.12a3.1$x^{7} + 21 x^{6} + 7$$7$$1$$12$$C_7:C_3$$$[2]^{3}$$
7.1.7.12a5.1$x^{7} + 35 x^{6} + 7$$7$$1$$12$$C_7:C_3$$$[2]^{3}$$
\(31\) Copy content Toggle raw display 31.7.2.7a1.2$x^{14} + 2 x^{8} + 56 x^{7} + x^{2} + 56 x + 815$$2$$7$$7$$C_{14}$$$[\ ]_{2}^{7}$$

Spectrum of ring of integers

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