↑

Properties

Label 14.0.309...463.1
Degree $14$
Signature $[0, 7]$
Discriminant $-3.095\times 10^{30}$
Root discriminant \(150.63\)
Ramified primes $3,7,43$
Class number $115712$ (GRH)
Class group [2, 2, 2, 2, 2, 2, 2, 904] (GRH)
Galois group $C_{14}$ (as 14T1)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 217*x^12 + 239*x^11 + 15033*x^10 + 42167*x^9 + 447988*x^8 + 1691539*x^7 + 7469613*x^6 + 26563459*x^5 + 71615373*x^4 + 183273867*x^3 + 308477505*x^2 + 377213368*x + 356178109)
 
gp: K = bnfinit(y^14 - y^13 + 217*y^12 + 239*y^11 + 15033*y^10 + 42167*y^9 + 447988*y^8 + 1691539*y^7 + 7469613*y^6 + 26563459*y^5 + 71615373*y^4 + 183273867*y^3 + 308477505*y^2 + 377213368*y + 356178109, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - x^13 + 217*x^12 + 239*x^11 + 15033*x^10 + 42167*x^9 + 447988*x^8 + 1691539*x^7 + 7469613*x^6 + 26563459*x^5 + 71615373*x^4 + 183273867*x^3 + 308477505*x^2 + 377213368*x + 356178109);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - x^13 + 217*x^12 + 239*x^11 + 15033*x^10 + 42167*x^9 + 447988*x^8 + 1691539*x^7 + 7469613*x^6 + 26563459*x^5 + 71615373*x^4 + 183273867*x^3 + 308477505*x^2 + 377213368*x + 356178109)
 

\( x^{14} - x^{13} + 217 x^{12} + 239 x^{11} + 15033 x^{10} + 42167 x^{9} + 447988 x^{8} + 1691539 x^{7} + \cdots + 356178109 \) 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:   \(-3094745824656233782531059200463\) \(\medspace = -\,3^{7}\cdot 7^{7}\cdot 43^{13}\) 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:  \(150.63\)
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:  $3^{1/2}7^{1/2}43^{13/14}\approx 150.626638686815$
Ramified primes:   \(3\), \(7\), \(43\) 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:  \(\Q(\sqrt{-903}) \)
$\card{ \Gal(K/\Q) }$:  $14$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(903=3\cdot 7\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{903}(64,·)$, $\chi_{903}(1,·)$, $\chi_{903}(419,·)$, $\chi_{903}(484,·)$, $\chi_{903}(902,·)$, $\chi_{903}(839,·)$, $\chi_{903}(776,·)$, $\chi_{903}(778,·)$, $\chi_{903}(524,·)$, $\chi_{903}(274,·)$, $\chi_{903}(629,·)$, $\chi_{903}(379,·)$, $\chi_{903}(125,·)$, $\chi_{903}(127,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{64}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7}a^{10}-\frac{2}{7}a^{9}-\frac{2}{7}a^{7}-\frac{1}{7}a^{6}-\frac{3}{7}a^{5}+\frac{2}{7}a^{3}+\frac{2}{7}a^{2}$, $\frac{1}{7}a^{11}+\frac{3}{7}a^{9}-\frac{2}{7}a^{8}+\frac{2}{7}a^{7}+\frac{2}{7}a^{6}+\frac{1}{7}a^{5}+\frac{2}{7}a^{4}-\frac{1}{7}a^{3}-\frac{3}{7}a^{2}$, $\frac{1}{553}a^{12}+\frac{2}{553}a^{11}+\frac{8}{553}a^{10}-\frac{167}{553}a^{9}-\frac{65}{553}a^{8}+\frac{220}{553}a^{7}-\frac{34}{79}a^{6}-\frac{228}{553}a^{5}+\frac{10}{553}a^{4}-\frac{142}{553}a^{3}-\frac{150}{553}a^{2}+\frac{24}{79}a+\frac{13}{79}$, $\frac{1}{18\!\cdots\!77}a^{13}+\frac{10\!\cdots\!16}{18\!\cdots\!77}a^{12}-\frac{44\!\cdots\!60}{18\!\cdots\!77}a^{11}+\frac{65\!\cdots\!94}{18\!\cdots\!77}a^{10}+\frac{84\!\cdots\!24}{18\!\cdots\!77}a^{9}-\frac{53\!\cdots\!49}{26\!\cdots\!11}a^{8}+\frac{23\!\cdots\!75}{18\!\cdots\!77}a^{7}-\frac{43\!\cdots\!42}{26\!\cdots\!11}a^{6}+\frac{34\!\cdots\!88}{18\!\cdots\!77}a^{5}+\frac{77\!\cdots\!30}{18\!\cdots\!77}a^{4}-\frac{75\!\cdots\!96}{23\!\cdots\!63}a^{3}+\frac{41\!\cdots\!61}{18\!\cdots\!77}a^{2}-\frac{22\!\cdots\!76}{26\!\cdots\!11}a+\frac{82\!\cdots\!09}{26\!\cdots\!11}$ 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

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{904}$, which has order $115712$ (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:   $\frac{35\!\cdots\!58}{26\!\cdots\!11}a^{13}-\frac{13\!\cdots\!41}{26\!\cdots\!11}a^{12}+\frac{77\!\cdots\!75}{26\!\cdots\!11}a^{11}-\frac{11\!\cdots\!71}{26\!\cdots\!11}a^{10}+\frac{50\!\cdots\!51}{26\!\cdots\!11}a^{9}+\frac{18\!\cdots\!86}{26\!\cdots\!11}a^{8}+\frac{12\!\cdots\!43}{26\!\cdots\!11}a^{7}+\frac{25\!\cdots\!79}{26\!\cdots\!11}a^{6}+\frac{13\!\cdots\!63}{26\!\cdots\!11}a^{5}+\frac{46\!\cdots\!17}{26\!\cdots\!11}a^{4}+\frac{93\!\cdots\!20}{26\!\cdots\!11}a^{3}+\frac{31\!\cdots\!80}{26\!\cdots\!11}a^{2}+\frac{25\!\cdots\!11}{26\!\cdots\!11}a+\frac{12\!\cdots\!91}{26\!\cdots\!11}$, $\frac{24\!\cdots\!44}{18\!\cdots\!77}a^{13}-\frac{89\!\cdots\!20}{18\!\cdots\!77}a^{12}+\frac{50\!\cdots\!96}{18\!\cdots\!77}a^{11}-\frac{73\!\cdots\!98}{18\!\cdots\!77}a^{10}+\frac{29\!\cdots\!77}{18\!\cdots\!77}a^{9}+\frac{12\!\cdots\!62}{18\!\cdots\!77}a^{8}+\frac{50\!\cdots\!22}{18\!\cdots\!77}a^{7}+\frac{16\!\cdots\!04}{26\!\cdots\!11}a^{6}+\frac{41\!\cdots\!94}{26\!\cdots\!11}a^{5}+\frac{11\!\cdots\!29}{18\!\cdots\!77}a^{4}+\frac{73\!\cdots\!92}{18\!\cdots\!77}a^{3}+\frac{97\!\cdots\!63}{18\!\cdots\!77}a^{2}-\frac{19\!\cdots\!55}{26\!\cdots\!11}a-\frac{53\!\cdots\!20}{26\!\cdots\!11}$, $\frac{31\!\cdots\!66}{18\!\cdots\!77}a^{13}-\frac{68\!\cdots\!58}{18\!\cdots\!77}a^{12}+\frac{69\!\cdots\!61}{18\!\cdots\!77}a^{11}-\frac{33\!\cdots\!29}{18\!\cdots\!77}a^{10}+\frac{47\!\cdots\!00}{18\!\cdots\!77}a^{9}+\frac{79\!\cdots\!02}{18\!\cdots\!77}a^{8}+\frac{12\!\cdots\!91}{18\!\cdots\!77}a^{7}+\frac{53\!\cdots\!66}{26\!\cdots\!11}a^{6}+\frac{17\!\cdots\!50}{18\!\cdots\!77}a^{5}+\frac{58\!\cdots\!53}{18\!\cdots\!77}a^{4}+\frac{12\!\cdots\!22}{18\!\cdots\!77}a^{3}+\frac{34\!\cdots\!26}{18\!\cdots\!77}a^{2}+\frac{44\!\cdots\!95}{26\!\cdots\!11}a+\frac{60\!\cdots\!31}{26\!\cdots\!11}$, $\frac{40\!\cdots\!52}{18\!\cdots\!77}a^{13}-\frac{15\!\cdots\!57}{26\!\cdots\!11}a^{12}+\frac{12\!\cdots\!73}{26\!\cdots\!11}a^{11}-\frac{43\!\cdots\!99}{18\!\cdots\!77}a^{10}+\frac{55\!\cdots\!85}{18\!\cdots\!77}a^{9}+\frac{75\!\cdots\!87}{18\!\cdots\!77}a^{8}+\frac{13\!\cdots\!22}{18\!\cdots\!77}a^{7}+\frac{54\!\cdots\!51}{26\!\cdots\!11}a^{6}+\frac{15\!\cdots\!40}{18\!\cdots\!77}a^{5}+\frac{53\!\cdots\!14}{18\!\cdots\!77}a^{4}+\frac{10\!\cdots\!18}{18\!\cdots\!77}a^{3}+\frac{25\!\cdots\!55}{18\!\cdots\!77}a^{2}+\frac{31\!\cdots\!99}{26\!\cdots\!11}a-\frac{49\!\cdots\!09}{26\!\cdots\!11}$, $\frac{26\!\cdots\!75}{18\!\cdots\!77}a^{13}-\frac{51\!\cdots\!84}{18\!\cdots\!77}a^{12}+\frac{55\!\cdots\!76}{18\!\cdots\!77}a^{11}+\frac{13\!\cdots\!61}{18\!\cdots\!77}a^{10}+\frac{35\!\cdots\!55}{18\!\cdots\!77}a^{9}+\frac{80\!\cdots\!03}{18\!\cdots\!77}a^{8}+\frac{84\!\cdots\!98}{18\!\cdots\!77}a^{7}+\frac{48\!\cdots\!25}{26\!\cdots\!11}a^{6}+\frac{98\!\cdots\!51}{18\!\cdots\!77}a^{5}+\frac{38\!\cdots\!18}{18\!\cdots\!77}a^{4}+\frac{71\!\cdots\!03}{18\!\cdots\!77}a^{3}+\frac{60\!\cdots\!77}{18\!\cdots\!77}a^{2}+\frac{12\!\cdots\!14}{26\!\cdots\!11}a-\frac{75\!\cdots\!12}{26\!\cdots\!11}$, $\frac{75\!\cdots\!37}{18\!\cdots\!77}a^{13}+\frac{46\!\cdots\!91}{18\!\cdots\!77}a^{12}+\frac{16\!\cdots\!58}{18\!\cdots\!77}a^{11}+\frac{12\!\cdots\!42}{18\!\cdots\!77}a^{10}+\frac{14\!\cdots\!26}{18\!\cdots\!77}a^{9}+\frac{96\!\cdots\!95}{18\!\cdots\!77}a^{8}+\frac{65\!\cdots\!22}{18\!\cdots\!77}a^{7}+\frac{41\!\cdots\!69}{26\!\cdots\!11}a^{6}+\frac{13\!\cdots\!81}{18\!\cdots\!77}a^{5}+\frac{42\!\cdots\!39}{18\!\cdots\!77}a^{4}+\frac{10\!\cdots\!60}{18\!\cdots\!77}a^{3}+\frac{24\!\cdots\!82}{18\!\cdots\!77}a^{2}+\frac{36\!\cdots\!32}{26\!\cdots\!11}a+\frac{71\!\cdots\!92}{26\!\cdots\!11}$ 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:  \( 35991.64185055774 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 35991.64185055774 \cdot 115712}{2\cdot\sqrt{3094745824656233782531059200463}}\cr\approx \mathstrut & 0.457611203630373 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 217*x^12 + 239*x^11 + 15033*x^10 + 42167*x^9 + 447988*x^8 + 1691539*x^7 + 7469613*x^6 + 26563459*x^5 + 71615373*x^4 + 183273867*x^3 + 308477505*x^2 + 377213368*x + 356178109)
 
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 - x^13 + 217*x^12 + 239*x^11 + 15033*x^10 + 42167*x^9 + 447988*x^8 + 1691539*x^7 + 7469613*x^6 + 26563459*x^5 + 71615373*x^4 + 183273867*x^3 + 308477505*x^2 + 377213368*x + 356178109, 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 - x^13 + 217*x^12 + 239*x^11 + 15033*x^10 + 42167*x^9 + 447988*x^8 + 1691539*x^7 + 7469613*x^6 + 26563459*x^5 + 71615373*x^4 + 183273867*x^3 + 308477505*x^2 + 377213368*x + 356178109);
 
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 - x^13 + 217*x^12 + 239*x^11 + 15033*x^10 + 42167*x^9 + 447988*x^8 + 1691539*x^7 + 7469613*x^6 + 26563459*x^5 + 71615373*x^4 + 183273867*x^3 + 308477505*x^2 + 377213368*x + 356178109);
 
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

\(\Q(\sqrt{-903}) \), 7.7.6321363049.1

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 ${\href{/padicField/2.7.0.1}{7} }^{2}$ R ${\href{/padicField/5.14.0.1}{14} }$ R ${\href{/padicField/11.14.0.1}{14} }$ ${\href{/padicField/13.14.0.1}{14} }$ ${\href{/padicField/17.7.0.1}{7} }^{2}$ ${\href{/padicField/19.7.0.1}{7} }^{2}$ ${\href{/padicField/23.14.0.1}{14} }$ ${\href{/padicField/29.7.0.1}{7} }^{2}$ ${\href{/padicField/31.14.0.1}{14} }$ ${\href{/padicField/37.2.0.1}{2} }^{7}$ ${\href{/padicField/41.7.0.1}{7} }^{2}$ R ${\href{/padicField/47.7.0.1}{7} }^{2}$ ${\href{/padicField/53.14.0.1}{14} }$ ${\href{/padicField/59.7.0.1}{7} }^{2}$

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
\(3\) Copy content Toggle raw display 3.14.7.2$x^{14} + 21 x^{12} + 189 x^{10} + 4 x^{9} + 945 x^{8} - 94 x^{7} + 2835 x^{6} - 630 x^{5} + 5107 x^{4} + 630 x^{3} + 5131 x^{2} + 1242 x + 2212$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
\(7\) Copy content Toggle raw display 7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
\(43\) Copy content Toggle raw display 43.14.13.11$x^{14} + 43$$14$$1$$13$$C_{14}$$[\ ]_{14}$