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Properties

Label 14.2.750...000.3
Degree $14$
Signature $[2, 6]$
Discriminant $7.506\times 10^{26}$
Root discriminant \(83.11\)
Ramified primes $2,3,5,7$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $F_7 \times C_2$ (as 14T7)

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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 - 870*x^7 - 900)
 
Copy content gp:K = bnfinit(y^14 - 870*y^7 - 900, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 870*x^7 - 900);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^14 - 870*x^7 - 900)
 

\( x^{14} - 870x^{7} - 900 \) 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:   \(750594644019045000000000000\) \(\medspace = 2^{12}\cdot 3^{12}\cdot 5^{13}\cdot 7^{10}\) 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:  \(83.11\)
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:  $2^{6/7}3^{6/7}5^{13/14}7^{5/6}\approx 104.78008059808623$
Ramified primes:   \(2\), \(3\), \(5\), \(7\) 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:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{390}a^{7}+\frac{5}{13}$, $\frac{1}{390}a^{8}+\frac{5}{13}a$, $\frac{1}{390}a^{9}+\frac{5}{13}a^{2}$, $\frac{1}{390}a^{10}+\frac{5}{13}a^{3}$, $\frac{1}{1950}a^{11}+\frac{18}{65}a^{4}$, $\frac{1}{13650}a^{12}+\frac{1}{6825}a^{11}+\frac{1}{910}a^{10}+\frac{1}{1365}a^{9}+\frac{1}{1365}a^{8}-\frac{1}{1365}a^{7}-\frac{2}{7}a^{6}+\frac{83}{455}a^{5}+\frac{101}{455}a^{4}+\frac{41}{91}a^{3}-\frac{29}{91}a^{2}+\frac{36}{91}a-\frac{23}{91}$, $\frac{1}{13650}a^{13}-\frac{1}{4550}a^{11}+\frac{1}{910}a^{10}-\frac{1}{1365}a^{9}+\frac{1}{2730}a^{8}+\frac{1}{2730}a^{7}-\frac{16}{65}a^{6}-\frac{1}{7}a^{5}+\frac{206}{455}a^{4}+\frac{15}{91}a^{3}+\frac{3}{91}a^{2}+\frac{31}{91}a+\frac{18}{91}$ 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 $1$ (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:  $7$
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:   $\frac{1}{390}a^{7}-\frac{8}{13}$, $\frac{1}{195}a^{13}+\frac{2}{2275}a^{12}-\frac{67}{13650}a^{11}+\frac{11}{1365}a^{10}-\frac{1}{70}a^{9}+\frac{59}{2730}a^{8}+\frac{11}{2730}a^{7}-\frac{424}{91}a^{6}+\frac{86}{455}a^{5}+\frac{484}{455}a^{4}+\frac{149}{91}a^{3}-\frac{44}{7}a^{2}+\frac{880}{91}a-\frac{829}{91}$, $\frac{23}{2275}a^{13}-\frac{8}{91}a^{12}-\frac{2}{75}a^{11}+\frac{223}{2730}a^{10}+\frac{257}{2730}a^{9}+\frac{101}{1365}a^{8}+\frac{487}{2730}a^{7}-\frac{3691}{455}a^{6}+\frac{1019}{13}a^{5}+\frac{981}{35}a^{4}-\frac{5333}{91}a^{3}-\frac{4448}{91}a^{2}+\frac{2063}{91}a+\frac{6634}{91}$, $\frac{22}{325}a^{13}+\frac{37}{910}a^{12}-\frac{24}{175}a^{11}+\frac{307}{2730}a^{10}-\frac{73}{2730}a^{9}-\frac{283}{2730}a^{8}+\frac{183}{910}a^{7}-\frac{26853}{455}a^{6}-\frac{3254}{91}a^{5}+\frac{4208}{35}a^{4}-\frac{8787}{91}a^{3}+\frac{1468}{91}a^{2}+\frac{8101}{91}a-\frac{11204}{91}$, $\frac{46}{2275}a^{13}-\frac{131}{2730}a^{12}-\frac{88}{2275}a^{11}+\frac{19}{546}a^{10}+\frac{23}{390}a^{9}-\frac{3}{70}a^{8}-\frac{183}{910}a^{7}-\frac{8097}{455}a^{6}+\frac{3830}{91}a^{5}+\frac{16496}{455}a^{4}-\frac{324}{13}a^{3}-\frac{4460}{91}a^{2}+\frac{23}{7}a+\frac{4717}{91}$, $\frac{1643}{13650}a^{13}+\frac{1426}{6825}a^{12}-\frac{68}{2275}a^{11}-\frac{209}{910}a^{10}-\frac{151}{2730}a^{9}+\frac{809}{2730}a^{8}-\frac{337}{2730}a^{7}-\frac{47906}{455}a^{6}-\frac{81654}{455}a^{5}+\frac{11896}{455}a^{4}+\frac{16599}{91}a^{3}+\frac{6200}{91}a^{2}-\frac{14493}{91}a-\frac{1901}{13}$, $\frac{108739}{4550}a^{13}+\frac{311462}{6825}a^{12}+\frac{57008}{975}a^{11}+\frac{163147}{2730}a^{10}+\frac{13462}{273}a^{9}+\frac{39493}{1365}a^{8}+\frac{2426}{1365}a^{7}-\frac{9471669}{455}a^{6}-\frac{2583434}{65}a^{5}-\frac{23174279}{455}a^{4}-\frac{4739698}{91}a^{3}-\frac{3902744}{91}a^{2}-\frac{2282004}{91}a-\frac{191657}{91}$ 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:  \( 145796753.1376601 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 145796753.1376601 \cdot 1}{2\cdot\sqrt{750594644019045000000000000}}\cr\approx \mathstrut & 0.654868842111669 \end{aligned}\] (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 - 870*x^7 - 900) 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 - 870*x^7 - 900, 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 - 870*x^7 - 900); 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 - 870*x^7 - 900); 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

$C_2\times F_7$ (as 14T7):

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 84
The 14 conjugacy class representatives for $F_7 \times C_2$
Character table for $F_7 \times C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), 7.1.12252303000000.6

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 sibling: 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 R R R ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{7}$ ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.1.0.1}{1} }^{14}$ ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.14.0.1}{14} }$ ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 2.2.7.12a1.1$x^{14} + 7 x^{13} + 28 x^{12} + 77 x^{11} + 161 x^{10} + 266 x^{9} + 357 x^{8} + 393 x^{7} + 357 x^{6} + 266 x^{5} + 161 x^{4} + 77 x^{3} + 28 x^{2} + 7 x + 3$$7$$2$$12$$(C_7:C_3) \times C_2$$$[\ ]_{7}^{6}$$
\(3\) Copy content Toggle raw display 3.2.7.12a1.1$x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1484 x^{10} + 3752 x^{9} + 7448 x^{8} + 11776 x^{7} + 14896 x^{6} + 15008 x^{5} + 11872 x^{4} + 7168 x^{3} + 3136 x^{2} + 896 x + 131$$7$$2$$12$$F_7$$$[\ ]_{7}^{6}$$
\(5\) Copy content Toggle raw display 5.1.14.13a1.1$x^{14} + 5$$14$$1$$13$$F_7 \times C_2$$$[\ ]_{14}^{6}$$
\(7\) Copy content Toggle raw display 7.2.1.0a1.1$x^{2} + 6 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
7.2.6.10a1.2$x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 192996 x^{6} + 266328 x^{5} + 234495 x^{4} + 131220 x^{3} + 45198 x^{2} + 8748 x + 736$$6$$2$$10$$C_6\times C_2$$$[\ ]_{6}^{2}$$

Spectrum of ring of integers

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