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)

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Normalized defining polynomial

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)
 
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)
 
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);
 
oscar: Qx, x = PolynomialRing(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)
 

\( x^{14} - 105 x^{12} - 147 x^{11} + 5271 x^{10} + 19838 x^{9} - 94150 x^{8} - 607634 x^{7} + 570164 x^{6} + 12260920 x^{5} + 42847770 x^{4} + 95169270 x^{3} + \cdots + 336238208 \) 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:   \(-337337631352558562817384077600704\) \(\medspace = -\,2^{6}\cdot 7^{24}\cdot 31^{7}\) 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:  \(210.59\)
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:  $2^{6/7}7^{96/49}31^{1/2}\approx 456.46642613478804$
Ramified primes:   \(2\), \(7\), \(31\) 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:  \(\Q(\sqrt{-31}) \)
$\card{ \Aut(K/\Q) }$:  $1$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

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

$C_{21}$, which has order $21$ (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:   $\frac{13\!\cdots\!81}{33\!\cdots\!44}a^{13}-\frac{16\!\cdots\!19}{84\!\cdots\!36}a^{12}-\frac{12\!\cdots\!33}{33\!\cdots\!44}a^{11}+\frac{50\!\cdots\!85}{33\!\cdots\!44}a^{10}+\frac{55\!\cdots\!03}{33\!\cdots\!44}a^{9}-\frac{44\!\cdots\!55}{16\!\cdots\!72}a^{8}-\frac{61\!\cdots\!03}{16\!\cdots\!72}a^{7}-\frac{65\!\cdots\!33}{16\!\cdots\!72}a^{6}+\frac{43\!\cdots\!99}{84\!\cdots\!36}a^{5}+\frac{75\!\cdots\!97}{42\!\cdots\!68}a^{4}+\frac{66\!\cdots\!09}{16\!\cdots\!72}a^{3}+\frac{14\!\cdots\!55}{16\!\cdots\!72}a^{2}+\frac{67\!\cdots\!61}{42\!\cdots\!68}a+\frac{14\!\cdots\!87}{10\!\cdots\!67}$, $\frac{13\!\cdots\!39}{33\!\cdots\!44}a^{13}-\frac{12\!\cdots\!97}{21\!\cdots\!34}a^{12}-\frac{13\!\cdots\!75}{33\!\cdots\!44}a^{11}-\frac{24\!\cdots\!97}{33\!\cdots\!44}a^{10}+\frac{51\!\cdots\!69}{33\!\cdots\!44}a^{9}+\frac{11\!\cdots\!33}{16\!\cdots\!72}a^{8}-\frac{32\!\cdots\!21}{16\!\cdots\!72}a^{7}-\frac{33\!\cdots\!87}{16\!\cdots\!72}a^{6}-\frac{55\!\cdots\!13}{84\!\cdots\!36}a^{5}-\frac{74\!\cdots\!55}{42\!\cdots\!68}a^{4}-\frac{73\!\cdots\!65}{16\!\cdots\!72}a^{3}-\frac{10\!\cdots\!11}{16\!\cdots\!72}a^{2}-\frac{34\!\cdots\!57}{21\!\cdots\!34}a+\frac{85\!\cdots\!81}{10\!\cdots\!67}$, $\frac{79\!\cdots\!61}{16\!\cdots\!72}a^{13}-\frac{14\!\cdots\!39}{84\!\cdots\!36}a^{12}-\frac{86\!\cdots\!25}{16\!\cdots\!72}a^{11}-\frac{73\!\cdots\!21}{16\!\cdots\!72}a^{10}+\frac{45\!\cdots\!21}{16\!\cdots\!72}a^{9}+\frac{33\!\cdots\!97}{42\!\cdots\!68}a^{8}-\frac{49\!\cdots\!45}{84\!\cdots\!36}a^{7}-\frac{22\!\cdots\!75}{84\!\cdots\!36}a^{6}+\frac{67\!\cdots\!59}{10\!\cdots\!67}a^{5}+\frac{62\!\cdots\!31}{10\!\cdots\!67}a^{4}+\frac{11\!\cdots\!49}{84\!\cdots\!36}a^{3}+\frac{20\!\cdots\!89}{84\!\cdots\!36}a^{2}+\frac{27\!\cdots\!11}{42\!\cdots\!68}a+\frac{87\!\cdots\!61}{10\!\cdots\!67}$, $\frac{11\!\cdots\!03}{33\!\cdots\!44}a^{13}-\frac{10\!\cdots\!45}{84\!\cdots\!36}a^{12}-\frac{10\!\cdots\!35}{33\!\cdots\!44}a^{11}+\frac{23\!\cdots\!79}{33\!\cdots\!44}a^{10}+\frac{53\!\cdots\!45}{33\!\cdots\!44}a^{9}+\frac{15\!\cdots\!15}{16\!\cdots\!72}a^{8}-\frac{61\!\cdots\!85}{16\!\cdots\!72}a^{7}-\frac{12\!\cdots\!47}{16\!\cdots\!72}a^{6}+\frac{41\!\cdots\!05}{84\!\cdots\!36}a^{5}+\frac{10\!\cdots\!75}{42\!\cdots\!68}a^{4}+\frac{94\!\cdots\!87}{16\!\cdots\!72}a^{3}+\frac{20\!\cdots\!25}{16\!\cdots\!72}a^{2}+\frac{10\!\cdots\!31}{42\!\cdots\!68}a+\frac{32\!\cdots\!63}{10\!\cdots\!67}$, $\frac{69\!\cdots\!27}{33\!\cdots\!44}a^{13}-\frac{37\!\cdots\!89}{42\!\cdots\!68}a^{12}-\frac{60\!\cdots\!87}{33\!\cdots\!44}a^{11}+\frac{15\!\cdots\!07}{33\!\cdots\!44}a^{10}+\frac{29\!\cdots\!61}{33\!\cdots\!44}a^{9}+\frac{46\!\cdots\!85}{16\!\cdots\!72}a^{8}-\frac{34\!\cdots\!17}{16\!\cdots\!72}a^{7}-\frac{62\!\cdots\!47}{16\!\cdots\!72}a^{6}+\frac{23\!\cdots\!11}{84\!\cdots\!36}a^{5}+\frac{56\!\cdots\!57}{42\!\cdots\!68}a^{4}+\frac{51\!\cdots\!07}{16\!\cdots\!72}a^{3}+\frac{10\!\cdots\!21}{16\!\cdots\!72}a^{2}+\frac{13\!\cdots\!86}{10\!\cdots\!67}a+\frac{17\!\cdots\!65}{10\!\cdots\!67}$, $\frac{15\!\cdots\!17}{16\!\cdots\!72}a^{13}-\frac{73\!\cdots\!33}{84\!\cdots\!36}a^{12}-\frac{16\!\cdots\!61}{16\!\cdots\!72}a^{11}-\frac{21\!\cdots\!89}{16\!\cdots\!72}a^{10}+\frac{83\!\cdots\!37}{16\!\cdots\!72}a^{9}+\frac{18\!\cdots\!10}{10\!\cdots\!67}a^{8}-\frac{84\!\cdots\!61}{84\!\cdots\!36}a^{7}-\frac{48\!\cdots\!63}{84\!\cdots\!36}a^{6}+\frac{19\!\cdots\!35}{21\!\cdots\!34}a^{5}+\frac{12\!\cdots\!91}{10\!\cdots\!67}a^{4}+\frac{26\!\cdots\!97}{84\!\cdots\!36}a^{3}+\frac{45\!\cdots\!33}{84\!\cdots\!36}a^{2}+\frac{58\!\cdots\!29}{42\!\cdots\!68}a+\frac{22\!\cdots\!58}{10\!\cdots\!67}$ Copy content Toggle raw display (assuming GRH)
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:  \( 8415941567.41 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 8415941567.41 \cdot 21}{2\cdot\sqrt{337337631352558562817384077600704}}\cr\approx \mathstrut & 1.86002462311 \end{aligned}\] (assuming GRH)

# 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))))
 
# 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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(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);
 
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 - 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):

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 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.

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]
 

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.

# 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 $\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.3.0.1$x^{3} + x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} + x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.7.6.1$x^{7} + 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
\(7\) Copy content Toggle raw display 7.7.12.8$x^{7} + 21 x^{6} + 7$$7$$1$$12$$C_7:C_3$$[2]^{3}$
7.7.12.9$x^{7} + 35 x^{6} + 7$$7$$1$$12$$C_7:C_3$$[2]^{3}$
\(31\) Copy content Toggle raw display 31.14.7.2$x^{14} + 217 x^{12} + 20181 x^{10} + 1042687 x^{8} + 56 x^{7} + 32322367 x^{6} - 36456 x^{5} + 601212171 x^{4} + 1883560 x^{3} + 6213359916 x^{2} - 11678016 x + 27510767884$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$