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Properties

Label 14.0.100...688.1
Degree $14$
Signature $[0, 7]$
Discriminant $-1.003\times 10^{32}$
Root discriminant \(193.11\)
Ramified primes $2,3,7,13$
Class number $1372$ (GRH)
Class group [7, 14, 14] (GRH)
Galois group $F_7$ (as 14T4)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 35*x^12 - 119*x^11 + 329*x^10 - 721*x^9 + 1337*x^8 + 2011*x^7 - 11522*x^6 - 130648*x^5 + 357532*x^4 + 69076*x^3 - 467348*x^2 + 127316*x + 4139276)
 
gp: K = bnfinit(y^14 - 7*y^13 + 35*y^12 - 119*y^11 + 329*y^10 - 721*y^9 + 1337*y^8 + 2011*y^7 - 11522*y^6 - 130648*y^5 + 357532*y^4 + 69076*y^3 - 467348*y^2 + 127316*y + 4139276, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 7*x^13 + 35*x^12 - 119*x^11 + 329*x^10 - 721*x^9 + 1337*x^8 + 2011*x^7 - 11522*x^6 - 130648*x^5 + 357532*x^4 + 69076*x^3 - 467348*x^2 + 127316*x + 4139276);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 7*x^13 + 35*x^12 - 119*x^11 + 329*x^10 - 721*x^9 + 1337*x^8 + 2011*x^7 - 11522*x^6 - 130648*x^5 + 357532*x^4 + 69076*x^3 - 467348*x^2 + 127316*x + 4139276)
 

\( x^{14} - 7 x^{13} + 35 x^{12} - 119 x^{11} + 329 x^{10} - 721 x^{9} + 1337 x^{8} + 2011 x^{7} + \cdots + 4139276 \) 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:   \(-100279849256417888141681913458688\) \(\medspace = -\,2^{12}\cdot 3^{12}\cdot 7^{11}\cdot 13^{12}\) 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:  \(193.11\)
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:  $2^{6/7}3^{6/7}7^{5/6}13^{6/7}\approx 211.8574498902768$
Ramified primes:   \(2\), \(3\), \(7\), \(13\) 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:  \(\Q(\sqrt{-7}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7}a^{6}-\frac{1}{7}a^{5}+\frac{1}{7}a^{4}-\frac{1}{7}a^{3}+\frac{1}{7}a^{2}-\frac{1}{7}a+\frac{1}{7}$, $\frac{1}{7}a^{7}+\frac{1}{7}$, $\frac{1}{182}a^{8}-\frac{2}{91}a^{7}-\frac{6}{91}a^{6}-\frac{1}{91}a^{5}+\frac{23}{182}a^{4}-\frac{15}{91}a^{3}+\frac{15}{91}a^{2}+\frac{10}{91}a+\frac{34}{91}$, $\frac{1}{182}a^{9}-\frac{1}{91}a^{7}+\frac{1}{91}a^{6}-\frac{37}{182}a^{5}-\frac{34}{91}a^{4}+\frac{20}{91}a^{3}+\frac{5}{91}a^{2}-\frac{43}{91}a-\frac{1}{13}$, $\frac{1}{1274}a^{10}+\frac{3}{1274}a^{9}+\frac{3}{1274}a^{8}-\frac{38}{637}a^{7}-\frac{1}{14}a^{6}+\frac{11}{26}a^{5}-\frac{85}{182}a^{4}+\frac{81}{637}a^{3}+\frac{138}{637}a^{2}+\frac{187}{637}a+\frac{214}{637}$, $\frac{1}{1274}a^{11}+\frac{1}{1274}a^{9}-\frac{1}{1274}a^{8}-\frac{31}{1274}a^{7}-\frac{3}{14}a^{5}-\frac{237}{1274}a^{4}-\frac{6}{91}a^{3}-\frac{115}{637}a^{2}+\frac{101}{637}a-\frac{201}{637}$, $\frac{1}{1274}a^{12}+\frac{3}{1274}a^{9}+\frac{1}{1274}a^{8}-\frac{3}{49}a^{7}-\frac{3}{91}a^{6}-\frac{29}{98}a^{5}+\frac{8}{91}a^{4}-\frac{31}{91}a^{3}+\frac{159}{637}a^{2}+\frac{25}{637}a-\frac{74}{637}$, $\frac{1}{40\!\cdots\!46}a^{13}+\frac{13\!\cdots\!61}{31\!\cdots\!42}a^{12}+\frac{63\!\cdots\!74}{20\!\cdots\!73}a^{11}+\frac{16\!\cdots\!02}{20\!\cdots\!73}a^{10}+\frac{40\!\cdots\!21}{31\!\cdots\!42}a^{9}+\frac{41\!\cdots\!00}{20\!\cdots\!73}a^{8}-\frac{11\!\cdots\!35}{20\!\cdots\!73}a^{7}+\frac{12\!\cdots\!06}{20\!\cdots\!73}a^{6}+\frac{94\!\cdots\!30}{20\!\cdots\!73}a^{5}+\frac{12\!\cdots\!67}{40\!\cdots\!46}a^{4}+\frac{46\!\cdots\!72}{20\!\cdots\!73}a^{3}-\frac{44\!\cdots\!11}{20\!\cdots\!73}a^{2}-\frac{84\!\cdots\!30}{20\!\cdots\!73}a-\frac{31\!\cdots\!47}{20\!\cdots\!73}$ 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

$C_{7}\times C_{14}\times C_{14}$, which has order $1372$ (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:   $\frac{93\!\cdots\!73}{20\!\cdots\!73}a^{13}-\frac{48\!\cdots\!14}{29\!\cdots\!39}a^{12}+\frac{21\!\cdots\!41}{20\!\cdots\!73}a^{11}-\frac{40\!\cdots\!37}{20\!\cdots\!73}a^{10}+\frac{14\!\cdots\!59}{15\!\cdots\!21}a^{9}-\frac{13\!\cdots\!42}{29\!\cdots\!39}a^{8}+\frac{10\!\cdots\!68}{20\!\cdots\!73}a^{7}+\frac{50\!\cdots\!93}{20\!\cdots\!73}a^{6}+\frac{10\!\cdots\!09}{29\!\cdots\!39}a^{5}-\frac{97\!\cdots\!27}{20\!\cdots\!73}a^{4}+\frac{10\!\cdots\!92}{20\!\cdots\!73}a^{3}-\frac{55\!\cdots\!24}{20\!\cdots\!73}a^{2}-\frac{89\!\cdots\!71}{29\!\cdots\!39}a-\frac{15\!\cdots\!33}{20\!\cdots\!73}$, $\frac{15\!\cdots\!15}{20\!\cdots\!73}a^{13}-\frac{79\!\cdots\!11}{20\!\cdots\!73}a^{12}+\frac{31\!\cdots\!59}{20\!\cdots\!73}a^{11}-\frac{14\!\cdots\!85}{31\!\cdots\!42}a^{10}+\frac{34\!\cdots\!43}{40\!\cdots\!46}a^{9}-\frac{32\!\cdots\!51}{20\!\cdots\!73}a^{8}+\frac{15\!\cdots\!81}{29\!\cdots\!39}a^{7}+\frac{13\!\cdots\!09}{40\!\cdots\!46}a^{6}-\frac{59\!\cdots\!89}{40\!\cdots\!46}a^{5}-\frac{23\!\cdots\!80}{20\!\cdots\!73}a^{4}+\frac{14\!\cdots\!30}{20\!\cdots\!73}a^{3}+\frac{97\!\cdots\!00}{20\!\cdots\!73}a^{2}+\frac{14\!\cdots\!48}{20\!\cdots\!73}a-\frac{29\!\cdots\!99}{29\!\cdots\!39}$, $\frac{13\!\cdots\!65}{31\!\cdots\!42}a^{13}-\frac{28\!\cdots\!96}{29\!\cdots\!39}a^{12}+\frac{14\!\cdots\!88}{20\!\cdots\!73}a^{11}-\frac{42\!\cdots\!67}{15\!\cdots\!21}a^{10}+\frac{79\!\cdots\!65}{20\!\cdots\!73}a^{9}+\frac{62\!\cdots\!35}{58\!\cdots\!78}a^{8}-\frac{10\!\cdots\!58}{20\!\cdots\!73}a^{7}-\frac{18\!\cdots\!75}{20\!\cdots\!73}a^{6}+\frac{46\!\cdots\!33}{58\!\cdots\!78}a^{5}-\frac{37\!\cdots\!91}{40\!\cdots\!46}a^{4}-\frac{18\!\cdots\!92}{20\!\cdots\!73}a^{3}+\frac{36\!\cdots\!32}{20\!\cdots\!73}a^{2}+\frac{11\!\cdots\!46}{29\!\cdots\!39}a-\frac{23\!\cdots\!69}{20\!\cdots\!73}$, $\frac{64\!\cdots\!22}{15\!\cdots\!21}a^{13}-\frac{75\!\cdots\!46}{20\!\cdots\!73}a^{12}+\frac{44\!\cdots\!99}{20\!\cdots\!73}a^{11}-\frac{37\!\cdots\!98}{41\!\cdots\!77}a^{10}+\frac{60\!\cdots\!45}{20\!\cdots\!73}a^{9}-\frac{14\!\cdots\!57}{20\!\cdots\!73}a^{8}+\frac{25\!\cdots\!87}{20\!\cdots\!73}a^{7}+\frac{61\!\cdots\!14}{20\!\cdots\!73}a^{6}-\frac{13\!\cdots\!51}{20\!\cdots\!73}a^{5}-\frac{96\!\cdots\!63}{20\!\cdots\!73}a^{4}+\frac{71\!\cdots\!57}{29\!\cdots\!39}a^{3}-\frac{71\!\cdots\!00}{20\!\cdots\!73}a^{2}+\frac{17\!\cdots\!63}{20\!\cdots\!73}a+\frac{11\!\cdots\!95}{20\!\cdots\!73}$, $\frac{36\!\cdots\!83}{22\!\cdots\!03}a^{13}-\frac{60\!\cdots\!01}{40\!\cdots\!46}a^{12}+\frac{27\!\cdots\!27}{40\!\cdots\!46}a^{11}-\frac{44\!\cdots\!90}{20\!\cdots\!73}a^{10}+\frac{10\!\cdots\!37}{15\!\cdots\!21}a^{9}-\frac{11\!\cdots\!45}{58\!\cdots\!78}a^{8}+\frac{87\!\cdots\!37}{58\!\cdots\!78}a^{7}+\frac{24\!\cdots\!08}{29\!\cdots\!39}a^{6}-\frac{24\!\cdots\!84}{15\!\cdots\!21}a^{5}-\frac{25\!\cdots\!94}{15\!\cdots\!21}a^{4}+\frac{67\!\cdots\!67}{15\!\cdots\!21}a^{3}+\frac{14\!\cdots\!80}{20\!\cdots\!73}a^{2}-\frac{20\!\cdots\!92}{29\!\cdots\!39}a-\frac{11\!\cdots\!55}{41\!\cdots\!77}$, $\frac{12\!\cdots\!43}{20\!\cdots\!73}a^{13}-\frac{46\!\cdots\!20}{20\!\cdots\!73}a^{12}+\frac{40\!\cdots\!55}{40\!\cdots\!46}a^{11}-\frac{13\!\cdots\!95}{58\!\cdots\!78}a^{10}+\frac{20\!\cdots\!07}{40\!\cdots\!46}a^{9}-\frac{29\!\cdots\!09}{40\!\cdots\!46}a^{8}+\frac{40\!\cdots\!01}{40\!\cdots\!46}a^{7}+\frac{94\!\cdots\!47}{40\!\cdots\!46}a^{6}-\frac{35\!\cdots\!09}{40\!\cdots\!46}a^{5}-\frac{38\!\cdots\!77}{40\!\cdots\!46}a^{4}-\frac{18\!\cdots\!75}{29\!\cdots\!39}a^{3}+\frac{85\!\cdots\!25}{20\!\cdots\!73}a^{2}+\frac{19\!\cdots\!18}{20\!\cdots\!73}a+\frac{10\!\cdots\!63}{20\!\cdots\!73}$ 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:  \( 71968673.99079628 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 71968673.99079628 \cdot 1372}{2\cdot\sqrt{100279849256417888141681913458688}}\cr\approx \mathstrut & 1.90598666997896 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 35*x^12 - 119*x^11 + 329*x^10 - 721*x^9 + 1337*x^8 + 2011*x^7 - 11522*x^6 - 130648*x^5 + 357532*x^4 + 69076*x^3 - 467348*x^2 + 127316*x + 4139276)
 
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 - 7*x^13 + 35*x^12 - 119*x^11 + 329*x^10 - 721*x^9 + 1337*x^8 + 2011*x^7 - 11522*x^6 - 130648*x^5 + 357532*x^4 + 69076*x^3 - 467348*x^2 + 127316*x + 4139276, 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 - 7*x^13 + 35*x^12 - 119*x^11 + 329*x^10 - 721*x^9 + 1337*x^8 + 2011*x^7 - 11522*x^6 - 130648*x^5 + 357532*x^4 + 69076*x^3 - 467348*x^2 + 127316*x + 4139276);
 
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 - 7*x^13 + 35*x^12 - 119*x^11 + 329*x^10 - 721*x^9 + 1337*x^8 + 2011*x^7 - 11522*x^6 - 130648*x^5 + 357532*x^4 + 69076*x^3 - 467348*x^2 + 127316*x + 4139276);
 
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

$F_7$ (as 14T4):

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 42
The 7 conjugacy class representatives for $F_7$
Character table for $F_7$

Intermediate fields

\(\Q(\sqrt{-7}) \), 7.1.3784929689032128.1

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]
 

Sibling fields

Galois closure: deg 42
Degree 7 sibling: 7.1.3784929689032128.1
Degree 21 sibling: deg 21
Minimal sibling: 7.1.3784929689032128.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ R ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ R ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.3.0.1}{3} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.7.0.1}{7} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.3.0.1}{3} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{7}$ ${\href{/padicField/43.1.0.1}{1} }^{14}$ ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$

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 2.7.6.1$x^{7} + 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.7.6.1$x^{7} + 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
\(3\) Copy content Toggle raw display 3.14.12.1$x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1484 x^{10} + 3752 x^{9} + 7448 x^{8} + 11782 x^{7} + 14938 x^{6} + 15008 x^{5} + 11452 x^{4} + 6328 x^{3} + 2632 x^{2} + 896 x + 185$$7$$2$$12$$F_7$$[\ ]_{7}^{6}$
\(7\) Copy content Toggle raw display 7.2.1.2$x^{2} + 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.6.5.5$x^{6} + 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 7$$6$$1$$5$$C_6$$[\ ]_{6}$
\(13\) Copy content Toggle raw display 13.14.12.1$x^{14} + 84 x^{13} + 3038 x^{12} + 61488 x^{11} + 756084 x^{10} + 5714352 x^{9} + 25377688 x^{8} + 58198682 x^{7} + 50756468 x^{6} + 22895628 x^{5} + 6802152 x^{4} + 9898168 x^{3} + 63376432 x^{2} + 249556496 x + 421775625$$7$$2$$12$$D_{7}$$[\ ]_{7}^{2}$