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Properties

Label 14.0.102...056.1
Degree $14$
Signature $[0, 7]$
Discriminant $-1.027\times 10^{33}$
Root discriminant \(228.02\)
Ramified primes $2,113$
Class number $131776$ (GRH)
Class group [2, 2, 2, 16472] (GRH)
Galois group $C_{14}$ (as 14T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 226*x^12 + 15368*x^10 + 396856*x^8 + 4805664*x^6 + 28530240*x^4 + 76811072*x^2 + 72913024)
 
gp: K = bnfinit(y^14 + 226*y^12 + 15368*y^10 + 396856*y^8 + 4805664*y^6 + 28530240*y^4 + 76811072*y^2 + 72913024, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 + 226*x^12 + 15368*x^10 + 396856*x^8 + 4805664*x^6 + 28530240*x^4 + 76811072*x^2 + 72913024);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 + 226*x^12 + 15368*x^10 + 396856*x^8 + 4805664*x^6 + 28530240*x^4 + 76811072*x^2 + 72913024)
 

\( x^{14} + 226x^{12} + 15368x^{10} + 396856x^{8} + 4805664x^{6} + 28530240x^{4} + 76811072x^{2} + 72913024 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $14$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 7]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-1027187378113291173607353987629056\) \(\medspace = -\,2^{21}\cdot 113^{13}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(228.02\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{3/2}113^{13/14}\approx 228.02106865636645$
Ramified primes:   \(2\), \(113\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-226}) \)
$\card{ \Gal(K/\Q) }$:  $14$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(904=2^{3}\cdot 113\)
Dirichlet character group:    $\lbrace$$\chi_{904}(897,·)$, $\chi_{904}(451,·)$, $\chi_{904}(1,·)$, $\chi_{904}(129,·)$, $\chi_{904}(795,·)$, $\chi_{904}(323,·)$, $\chi_{904}(459,·)$, $\chi_{904}(403,·)$, $\chi_{904}(369,·)$, $\chi_{904}(49,·)$, $\chi_{904}(83,·)$, $\chi_{904}(561,·)$, $\chi_{904}(593,·)$, $\chi_{904}(763,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{64}$

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

$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64\!\cdots\!76}a^{12}+\frac{37194421858289}{32\!\cdots\!88}a^{10}+\frac{23876330484727}{16\!\cdots\!44}a^{8}-\frac{1509416692538}{100547837853209}a^{6}-\frac{40402370965415}{402191351412836}a^{4}-\frac{39690005429053}{201095675706418}a^{2}+\frac{5594120455079}{100547837853209}$, $\frac{1}{45\!\cdots\!96}a^{13}+\frac{18\!\cdots\!51}{22\!\cdots\!48}a^{11}-\frac{15\!\cdots\!17}{11\!\cdots\!24}a^{9}+\frac{289568180019323}{57\!\cdots\!12}a^{7}-\frac{336881065764283}{71\!\cdots\!39}a^{5}-\frac{844072708254725}{14\!\cdots\!78}a^{3}+\frac{26\!\cdots\!13}{71\!\cdots\!39}a$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{16472}$, which has order $131776$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $6$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{33809390983}{16\!\cdots\!44}a^{12}+\frac{14586561443565}{32\!\cdots\!88}a^{10}+\frac{222337102501507}{804382702825672}a^{8}+\frac{44\!\cdots\!75}{804382702825672}a^{6}+\frac{45\!\cdots\!72}{100547837853209}a^{4}+\frac{14\!\cdots\!29}{100547837853209}a^{2}+\frac{15\!\cdots\!03}{100547837853209}$, $\frac{7431663749}{32\!\cdots\!88}a^{12}+\frac{414166414923}{804382702825672}a^{10}+\frac{54322744292015}{16\!\cdots\!44}a^{8}+\frac{78760808485024}{100547837853209}a^{6}+\frac{717437409706533}{100547837853209}a^{4}+\frac{24\!\cdots\!27}{100547837853209}a^{2}+\frac{25\!\cdots\!98}{100547837853209}$, $\frac{56711990573}{32\!\cdots\!88}a^{12}+\frac{763771463537}{201095675706418}a^{10}+\frac{46430796283335}{201095675706418}a^{8}+\frac{18\!\cdots\!33}{402191351412836}a^{6}+\frac{72\!\cdots\!33}{201095675706418}a^{4}+\frac{22\!\cdots\!45}{201095675706418}a^{2}+\frac{10\!\cdots\!52}{100547837853209}$, $\frac{216587626357}{64\!\cdots\!76}a^{12}+\frac{23241946719591}{32\!\cdots\!88}a^{10}+\frac{699735741317001}{16\!\cdots\!44}a^{8}+\frac{67\!\cdots\!11}{804382702825672}a^{6}+\frac{13\!\cdots\!73}{201095675706418}a^{4}+\frac{20\!\cdots\!88}{100547837853209}a^{2}+\frac{19\!\cdots\!20}{100547837853209}$, $\frac{27623188239}{16\!\cdots\!44}a^{12}+\frac{11909675040791}{32\!\cdots\!88}a^{10}+\frac{362547420271571}{16\!\cdots\!44}a^{8}+\frac{18\!\cdots\!15}{402191351412836}a^{6}+\frac{73\!\cdots\!43}{201095675706418}a^{4}+\frac{24\!\cdots\!75}{201095675706418}a^{2}+\frac{14\!\cdots\!93}{100547837853209}$, $\frac{18424353753}{16\!\cdots\!44}a^{12}+\frac{3991837152771}{16\!\cdots\!44}a^{10}+\frac{122980545682323}{804382702825672}a^{8}+\frac{12\!\cdots\!17}{402191351412836}a^{6}+\frac{10\!\cdots\!83}{402191351412836}a^{4}+\frac{92\!\cdots\!57}{100547837853209}a^{2}+\frac{10\!\cdots\!56}{100547837853209}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 222748.97284811488 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 222748.97284811488 \cdot 131776}{2\cdot\sqrt{1027187378113291173607353987629056}}\cr\approx \mathstrut & 0.177033834603184 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^14 + 226*x^12 + 15368*x^10 + 396856*x^8 + 4805664*x^6 + 28530240*x^4 + 76811072*x^2 + 72913024)
 
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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^14 + 226*x^12 + 15368*x^10 + 396856*x^8 + 4805664*x^6 + 28530240*x^4 + 76811072*x^2 + 72913024, 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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^14 + 226*x^12 + 15368*x^10 + 396856*x^8 + 4805664*x^6 + 28530240*x^4 + 76811072*x^2 + 72913024);
 
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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 + 226*x^12 + 15368*x^10 + 396856*x^8 + 4805664*x^6 + 28530240*x^4 + 76811072*x^2 + 72913024);
 
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

$C_{14}$ (as 14T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A cyclic group of order 14
The 14 conjugacy class representatives for $C_{14}$
Character table for $C_{14}$

Intermediate fields

\(\Q(\sqrt{-226}) \), 7.7.2081951752609.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.14.0.1}{14} }$ ${\href{/padicField/5.7.0.1}{7} }^{2}$ ${\href{/padicField/7.14.0.1}{14} }$ ${\href{/padicField/11.7.0.1}{7} }^{2}$ ${\href{/padicField/13.14.0.1}{14} }$ ${\href{/padicField/17.14.0.1}{14} }$ ${\href{/padicField/19.14.0.1}{14} }$ ${\href{/padicField/23.7.0.1}{7} }^{2}$ ${\href{/padicField/29.7.0.1}{7} }^{2}$ ${\href{/padicField/31.14.0.1}{14} }$ ${\href{/padicField/37.7.0.1}{7} }^{2}$ ${\href{/padicField/41.7.0.1}{7} }^{2}$ ${\href{/padicField/43.14.0.1}{14} }$ ${\href{/padicField/47.7.0.1}{7} }^{2}$ ${\href{/padicField/53.14.0.1}{14} }$ ${\href{/padicField/59.14.0.1}{14} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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 2.14.21.6$x^{14} + 158 x^{12} + 1568 x^{11} + 1332 x^{10} - 32384 x^{9} + 95128 x^{8} + 683008 x^{7} + 3694128 x^{6} + 18752000 x^{5} + 49924512 x^{4} + 51764736 x^{3} - 10775616 x^{2} + 27525120 x + 167639168$$2$$7$$21$$C_{14}$$[3]^{7}$
\(113\) Copy content Toggle raw display 113.14.13.1$x^{14} + 113$$14$$1$$13$$C_{14}$$[\ ]_{14}$