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Properties

Label 14.0.143...448.2
Degree $14$
Signature $[0, 7]$
Discriminant $-1.430\times 10^{32}$
Root discriminant \(198.07\)
Ramified primes $2,3,7$
Class number $4$ (GRH)
Class group [2, 2] (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 + 9072)
 
gp: K = bnfinit(y^14 + 9072, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 + 9072);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 + 9072)
 

\( x^{14} + 9072 \) 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:   \(-143041509649772631538232817512448\) \(\medspace = -\,2^{12}\cdot 3^{12}\cdot 7^{27}\) 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:  \(198.07\)
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:  $2^{6/7}3^{6/7}7^{83/42}\approx 217.30087954309144$
Ramified primes:   \(2\), \(3\), \(7\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{6}a^{4}$, $\frac{1}{6}a^{5}$, $\frac{1}{6}a^{6}$, $\frac{1}{72}a^{7}-\frac{1}{2}$, $\frac{1}{72}a^{8}-\frac{1}{2}a$, $\frac{1}{72}a^{9}-\frac{1}{2}a^{2}$, $\frac{1}{72}a^{10}-\frac{1}{2}a^{3}$, $\frac{1}{432}a^{11}-\frac{1}{12}a^{4}$, $\frac{1}{432}a^{12}-\frac{1}{12}a^{5}$, $\frac{1}{432}a^{13}-\frac{1}{12}a^{6}$ 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

$C_{2}\times C_{2}$, which has order $4$ (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:   $\frac{1387}{27}a^{13}+\frac{7889}{72}a^{12}+\frac{431}{2}a^{11}+\frac{10283}{24}a^{10}+\frac{28369}{36}a^{9}+\frac{5257}{4}a^{8}+\frac{6566}{3}a^{7}+\frac{19579}{6}a^{6}+\frac{10927}{3}a^{5}+2758a^{4}-\frac{6223}{2}a^{3}-25823a^{2}-77945a-192865$, $\frac{3895}{216}a^{13}-\frac{2705}{216}a^{12}-\frac{28843}{432}a^{11}+\frac{835}{18}a^{10}+\frac{1487}{6}a^{9}-\frac{4351}{24}a^{8}-\frac{10619}{12}a^{7}+\frac{3731}{6}a^{6}+\frac{6643}{2}a^{5}-\frac{29197}{12}a^{4}-11830a^{3}+8134a^{2}+\frac{89551}{2}a-31310$, $\frac{3895}{216}a^{13}+\frac{2705}{216}a^{12}-\frac{28843}{432}a^{11}-\frac{835}{18}a^{10}+\frac{1487}{6}a^{9}+\frac{4351}{24}a^{8}-\frac{10619}{12}a^{7}-\frac{3731}{6}a^{6}+\frac{6643}{2}a^{5}+\frac{29197}{12}a^{4}-11830a^{3}-8134a^{2}+\frac{89551}{2}a+31310$, $\frac{24305}{432}a^{13}+\frac{6811}{432}a^{12}-\frac{1727}{9}a^{11}-\frac{25879}{72}a^{10}+\frac{381}{4}a^{9}+\frac{52283}{36}a^{8}+\frac{6566}{3}a^{7}-\frac{23429}{12}a^{6}-\frac{132055}{12}a^{5}-11011a^{4}+\frac{41209}{2}a^{3}+79093a^{2}+48125a-202859$, $\frac{341837539}{72}a^{13}+\frac{3785289995}{432}a^{12}+\frac{1652880133}{108}a^{11}+\frac{1800340903}{72}a^{10}+\frac{669948935}{18}a^{9}+\frac{850037869}{18}a^{8}+\frac{476699069}{12}a^{7}-\frac{150548965}{6}a^{6}-\frac{2878067255}{12}a^{5}-804396740a^{4}-\frac{4251099923}{2}a^{3}-4989280590a^{2}-10838593871a-22178820364$, $\frac{341837539}{72}a^{13}-\frac{3785289995}{432}a^{12}+\frac{1652880133}{108}a^{11}-\frac{1800340903}{72}a^{10}+\frac{669948935}{18}a^{9}-\frac{850037869}{18}a^{8}+\frac{476699069}{12}a^{7}+\frac{150548965}{6}a^{6}-\frac{2878067255}{12}a^{5}+804396740a^{4}-\frac{4251099923}{2}a^{3}+4989280590a^{2}-10838593871a+22178820364$ 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:  \( 24868111328.92647 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 24868111328.92647 \cdot 4}{2\cdot\sqrt{143041509649772631538232817512448}}\cr\approx \mathstrut & 1.60768453683240 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^14 + 9072)
 
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 + 9072, 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 + 9072);
 
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 + 9072);
 
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

\(\Q(\sqrt{-7}) \), 7.1.4520453669548992.28

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: 7.1.4520453669548992.28
Degree 21 sibling: deg 21
Minimal sibling: 7.1.4520453669548992.28

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 ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{7}$ ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.3.0.1}{3} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.7.0.1}{7} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.3.0.1}{3} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{7}$ ${\href{/padicField/43.1.0.1}{1} }^{14}$ ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$

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 7.14.27.40$x^{14} + 42 x^{7} + 196 x^{2} + 294 x + 105$$14$$1$$27$$F_7$$[13/6]_{6}$