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Properties

Label 14.2.378...000.2
Degree $14$
Signature $[2, 6]$
Discriminant $3.785\times 10^{31}$
Root discriminant \(180.12\)
Ramified primes $2,3,5,7$
Class number $4$ (GRH)
Class group [2, 2] (GRH)
Galois group $F_7 \times C_2$ (as 14T7)

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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 - 7*x^13 + 49*x^12 + 7*x^11 + 161*x^10 - 3941*x^9 + 31647*x^8 - 59709*x^7 + 107898*x^6 - 16856*x^5 + 233954*x^4 + 737212*x^3 + 106876*x^2 + 70028*x + 205354)
 
Copy content gp:K = bnfinit(y^14 - 7*y^13 + 49*y^12 + 7*y^11 + 161*y^10 - 3941*y^9 + 31647*y^8 - 59709*y^7 + 107898*y^6 - 16856*y^5 + 233954*y^4 + 737212*y^3 + 106876*y^2 + 70028*y + 205354, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 7*x^13 + 49*x^12 + 7*x^11 + 161*x^10 - 3941*x^9 + 31647*x^8 - 59709*x^7 + 107898*x^6 - 16856*x^5 + 233954*x^4 + 737212*x^3 + 106876*x^2 + 70028*x + 205354);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^14 - 7*x^13 + 49*x^12 + 7*x^11 + 161*x^10 - 3941*x^9 + 31647*x^8 - 59709*x^7 + 107898*x^6 - 16856*x^5 + 233954*x^4 + 737212*x^3 + 106876*x^2 + 70028*x + 205354)
 

\( x^{14} - 7 x^{13} + 49 x^{12} + 7 x^{11} + 161 x^{10} - 3941 x^{9} + 31647 x^{8} - 59709 x^{7} + \cdots + 205354 \) 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:   \(37845732546084267945000000000000\) \(\medspace = 2^{12}\cdot 3^{13}\cdot 5^{13}\cdot 7^{15}\) 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:  \(180.12\)
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:  $2^{6/7}3^{13/14}5^{13/14}7^{47/42}\approx 197.61294964896845$
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:  \(\Q(\sqrt{105}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{16\cdots 47}a^{13}-\frac{31\cdots 52}{16\cdots 47}a^{12}-\frac{62\cdots 55}{16\cdots 47}a^{11}+\frac{97\cdots 98}{16\cdots 47}a^{10}+\frac{37\cdots 89}{16\cdots 47}a^{9}-\frac{87\cdots 62}{16\cdots 47}a^{8}+\frac{36\cdots 51}{16\cdots 47}a^{7}-\frac{77\cdots 54}{16\cdots 47}a^{6}-\frac{48\cdots 98}{12\cdots 19}a^{5}-\frac{76\cdots 69}{16\cdots 47}a^{4}-\frac{31\cdots 76}{16\cdots 47}a^{3}+\frac{26\cdots 12}{16\cdots 47}a^{2}-\frac{57\cdots 16}{16\cdots 47}a+\frac{41\cdots 38}{16\cdots 47}$ 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:  $C_{2}\times C_{2}$, which has order $4$ (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:  $C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:   $\frac{32\cdots 10}{16\cdots 47}a^{13}+\frac{24\cdots 40}{16\cdots 47}a^{12}-\frac{17\cdots 85}{16\cdots 47}a^{11}+\frac{94\cdots 13}{16\cdots 47}a^{10}-\frac{60\cdots 22}{16\cdots 47}a^{9}+\frac{13\cdots 91}{16\cdots 47}a^{8}-\frac{11\cdots 56}{16\cdots 47}a^{7}+\frac{26\cdots 91}{16\cdots 47}a^{6}-\frac{41\cdots 86}{12\cdots 19}a^{5}+\frac{46\cdots 28}{16\cdots 47}a^{4}-\frac{11\cdots 89}{16\cdots 47}a^{3}-\frac{14\cdots 50}{16\cdots 47}a^{2}+\frac{78\cdots 45}{16\cdots 47}a-\frac{90\cdots 87}{16\cdots 47}$, $\frac{12\cdots 52}{16\cdots 57}a^{13}+\frac{10\cdots 72}{16\cdots 57}a^{12}-\frac{76\cdots 24}{16\cdots 57}a^{11}+\frac{91\cdots 24}{16\cdots 57}a^{10}-\frac{34\cdots 12}{16\cdots 57}a^{9}+\frac{52\cdots 76}{16\cdots 57}a^{8}-\frac{47\cdots 52}{16\cdots 57}a^{7}+\frac{13\cdots 92}{16\cdots 57}a^{6}-\frac{32\cdots 44}{16\cdots 57}a^{5}+\frac{40\cdots 96}{16\cdots 57}a^{4}-\frac{72\cdots 40}{16\cdots 57}a^{3}-\frac{31\cdots 12}{16\cdots 57}a^{2}+\frac{18\cdots 80}{16\cdots 57}a-\frac{28\cdots 27}{16\cdots 57}$, $\frac{61\cdots 57}{16\cdots 47}a^{13}+\frac{13\cdots 90}{16\cdots 47}a^{12}-\frac{98\cdots 50}{16\cdots 47}a^{11}+\frac{44\cdots 37}{16\cdots 47}a^{10}+\frac{16\cdots 29}{16\cdots 47}a^{9}+\frac{19\cdots 91}{16\cdots 47}a^{8}-\frac{56\cdots 90}{16\cdots 47}a^{7}+\frac{34\cdots 22}{16\cdots 47}a^{6}-\frac{44\cdots 89}{12\cdots 19}a^{5}+\frac{35\cdots 24}{16\cdots 47}a^{4}+\frac{18\cdots 02}{16\cdots 47}a^{3}+\frac{43\cdots 15}{16\cdots 47}a^{2}+\frac{96\cdots 44}{16\cdots 47}a+\frac{42\cdots 65}{16\cdots 47}$, $\frac{24\cdots 55}{16\cdots 47}a^{13}+\frac{17\cdots 50}{16\cdots 47}a^{12}-\frac{10\cdots 30}{16\cdots 47}a^{11}-\frac{51\cdots 28}{16\cdots 47}a^{10}+\frac{22\cdots 43}{16\cdots 47}a^{9}+\frac{12\cdots 06}{16\cdots 47}a^{8}-\frac{69\cdots 87}{16\cdots 47}a^{7}+\frac{98\cdots 06}{16\cdots 47}a^{6}+\frac{38\cdots 03}{12\cdots 19}a^{5}+\frac{55\cdots 84}{16\cdots 47}a^{4}+\frac{99\cdots 39}{16\cdots 47}a^{3}+\frac{16\cdots 43}{16\cdots 47}a^{2}+\frac{16\cdots 82}{16\cdots 47}a+\frac{27\cdots 83}{16\cdots 47}$, $\frac{27\cdots 46}{16\cdots 47}a^{13}-\frac{10\cdots 46}{16\cdots 47}a^{12}+\frac{83\cdots 66}{16\cdots 47}a^{11}+\frac{34\cdots 20}{16\cdots 47}a^{10}+\frac{10\cdots 91}{16\cdots 47}a^{9}-\frac{99\cdots 04}{16\cdots 47}a^{8}+\frac{48\cdots 03}{16\cdots 47}a^{7}+\frac{48\cdots 58}{16\cdots 47}a^{6}+\frac{11\cdots 61}{12\cdots 19}a^{5}+\frac{95\cdots 84}{16\cdots 47}a^{4}-\frac{28\cdots 78}{16\cdots 47}a^{3}-\frac{18\cdots 06}{16\cdots 47}a^{2}+\frac{61\cdots 29}{16\cdots 47}a+\frac{31\cdots 61}{16\cdots 47}$, $\frac{26\cdots 55}{16\cdots 47}a^{13}-\frac{21\cdots 29}{16\cdots 47}a^{12}+\frac{15\cdots 43}{16\cdots 47}a^{11}-\frac{14\cdots 70}{16\cdots 47}a^{10}+\frac{57\cdots 70}{16\cdots 47}a^{9}-\frac{10\cdots 55}{16\cdots 47}a^{8}+\frac{94\cdots 40}{16\cdots 47}a^{7}-\frac{25\cdots 89}{16\cdots 47}a^{6}+\frac{42\cdots 60}{12\cdots 19}a^{5}-\frac{63\cdots 91}{16\cdots 47}a^{4}+\frac{12\cdots 94}{16\cdots 47}a^{3}+\frac{53\cdots 63}{16\cdots 47}a^{2}-\frac{29\cdots 45}{16\cdots 47}a+\frac{49\cdots 03}{16\cdots 47}$, $\frac{37\cdots 56}{16\cdots 47}a^{13}+\frac{31\cdots 99}{16\cdots 47}a^{12}-\frac{23\cdots 99}{16\cdots 47}a^{11}+\frac{41\cdots 96}{16\cdots 47}a^{10}-\frac{20\cdots 10}{16\cdots 47}a^{9}+\frac{20\cdots 92}{16\cdots 47}a^{8}-\frac{14\cdots 29}{16\cdots 47}a^{7}+\frac{39\cdots 03}{16\cdots 47}a^{6}-\frac{64\cdots 46}{12\cdots 19}a^{5}+\frac{97\cdots 76}{16\cdots 47}a^{4}-\frac{18\cdots 52}{16\cdots 47}a^{3}-\frac{80\cdots 85}{16\cdots 47}a^{2}+\frac{45\cdots 17}{16\cdots 47}a-\frac{73\cdots 35}{16\cdots 47}$ 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:  \( 22259005136.311165 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 22259005136.311165 \cdot 4}{2\cdot\sqrt{37845732546084267945000000000000}}\cr\approx \mathstrut & 1.78101053695517 \end{aligned}\] (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 - 7*x^13 + 49*x^12 + 7*x^11 + 161*x^10 - 3941*x^9 + 31647*x^8 - 59709*x^7 + 107898*x^6 - 16856*x^5 + 233954*x^4 + 737212*x^3 + 106876*x^2 + 70028*x + 205354) 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 - 7*x^13 + 49*x^12 + 7*x^11 + 161*x^10 - 3941*x^9 + 31647*x^8 - 59709*x^7 + 107898*x^6 - 16856*x^5 + 233954*x^4 + 737212*x^3 + 106876*x^2 + 70028*x + 205354, 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 - 7*x^13 + 49*x^12 + 7*x^11 + 161*x^10 - 3941*x^9 + 31647*x^8 - 59709*x^7 + 107898*x^6 - 16856*x^5 + 233954*x^4 + 737212*x^3 + 106876*x^2 + 70028*x + 205354); 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 - 7*x^13 + 49*x^12 + 7*x^11 + 161*x^10 - 3941*x^9 + 31647*x^8 - 59709*x^7 + 107898*x^6 - 16856*x^5 + 233954*x^4 + 737212*x^3 + 106876*x^2 + 70028*x + 205354); 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

\(\Q(\sqrt{105}) \), 7.1.600362847000000.29

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: 14.0.5406533220869181135000000000000.5

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 ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.2.0.1}{2} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.3.0.1}{3} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.14.0.1}{14} }$ ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{7}$ ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 2.1.7.6a1.1$x^{7} + 2$$7$$1$$6$$C_7:C_3$$$[\ ]_{7}^{3}$$
2.1.7.6a1.1$x^{7} + 2$$7$$1$$6$$C_7:C_3$$$[\ ]_{7}^{3}$$
\(3\) Copy content Toggle raw display 3.1.14.13a1.1$x^{14} + 3$$14$$1$$13$$F_7 \times C_2$$$[\ ]_{14}^{6}$$
\(5\) 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 7.1.14.15a1.4$x^{14} + 7 x^{2} + 21$$14$$1$$15$$F_7 \times C_2$$$[\frac{7}{6}]_{6}^{2}$$

Spectrum of ring of integers

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