Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 + 7*x^12 - 4*x^11 - 8*x^10 + 24*x^9 - 30*x^8 + 16*x^7 + 13*x^6 - 38*x^5 + 46*x^4 - 36*x^3 + 19*x^2 - 6*x + 1)
gp: K = bnfinit(y^14 - 4*y^13 + 7*y^12 - 4*y^11 - 8*y^10 + 24*y^9 - 30*y^8 + 16*y^7 + 13*y^6 - 38*y^5 + 46*y^4 - 36*y^3 + 19*y^2 - 6*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 4*x^13 + 7*x^12 - 4*x^11 - 8*x^10 + 24*x^9 - 30*x^8 + 16*x^7 + 13*x^6 - 38*x^5 + 46*x^4 - 36*x^3 + 19*x^2 - 6*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 4*x^13 + 7*x^12 - 4*x^11 - 8*x^10 + 24*x^9 - 30*x^8 + 16*x^7 + 13*x^6 - 38*x^5 + 46*x^4 - 36*x^3 + 19*x^2 - 6*x + 1)
\( x^{14} - 4 x^{13} + 7 x^{12} - 4 x^{11} - 8 x^{10} + 24 x^{9} - 30 x^{8} + 16 x^{7} + 13 x^{6} + \cdots + 1 \)
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: |
\(-9745585291264\)
\(\medspace = -\,2^{14}\cdot 29^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(8.47\) | 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: | $2\cdot 29^{6/7}\approx 35.8520500359704$ | ||
| Ramified primes: |
\(2\), \(29\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $7$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: |
\( -a^{11} + 2 a^{10} - 2 a^{9} - 2 a^{8} + 7 a^{7} - 10 a^{6} + 4 a^{5} + 5 a^{4} - 13 a^{3} + 12 a^{2} - 8 a + 2 \)
(order $4$)
| 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: |
$a^{12}-2a^{11}+2a^{10}+2a^{9}-7a^{8}+10a^{7}-4a^{6}-5a^{5}+13a^{4}-12a^{3}+8a^{2}-2a$, $4a^{13}-10a^{12}+10a^{11}+6a^{10}-30a^{9}+47a^{8}-29a^{7}-13a^{6}+54a^{5}-64a^{4}+51a^{3}-22a^{2}+5a+1$, $a^{13}-5a^{12}+8a^{11}-2a^{10}-14a^{9}+29a^{8}-28a^{7}+3a^{6}+29a^{5}-44a^{4}+38a^{3}-21a^{2}+5a+1$, $5a^{13}-16a^{12}+22a^{11}-3a^{10}-41a^{9}+86a^{8}-83a^{7}+18a^{6}+74a^{5}-131a^{4}+130a^{3}-83a^{2}+33a-6$, $3a^{13}-9a^{12}+10a^{11}+4a^{10}-27a^{9}+43a^{8}-29a^{7}-11a^{6}+49a^{5}-59a^{4}+46a^{3}-21a^{2}+3a+1$, $a^{12}-3a^{11}+3a^{10}+2a^{9}-9a^{8}+13a^{7}-8a^{6}-5a^{5}+16a^{4}-18a^{3}+13a^{2}-5a$
| 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: | \( 5.342840135225723 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 5.342840135225723 \cdot 1}{4\cdot\sqrt{9745585291264}}\cr\approx \mathstrut & 0.165412110762191 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 + 7*x^12 - 4*x^11 - 8*x^10 + 24*x^9 - 30*x^8 + 16*x^7 + 13*x^6 - 38*x^5 + 46*x^4 - 36*x^3 + 19*x^2 - 6*x + 1)
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 - 4*x^13 + 7*x^12 - 4*x^11 - 8*x^10 + 24*x^9 - 30*x^8 + 16*x^7 + 13*x^6 - 38*x^5 + 46*x^4 - 36*x^3 + 19*x^2 - 6*x + 1, 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 - 4*x^13 + 7*x^12 - 4*x^11 - 8*x^10 + 24*x^9 - 30*x^8 + 16*x^7 + 13*x^6 - 38*x^5 + 46*x^4 - 36*x^3 + 19*x^2 - 6*x + 1);
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 - 4*x^13 + 7*x^12 - 4*x^11 - 8*x^10 + 24*x^9 - 30*x^8 + 16*x^7 + 13*x^6 - 38*x^5 + 46*x^4 - 36*x^3 + 19*x^2 - 6*x + 1);
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
$C_7\times D_7$ (as 14T8):
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 98 |
| The 35 conjugacy class representatives for $C_7 \wr C_2$ |
| Character table for $C_7 \wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \) |
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
| Degree 14 siblings: | 14.0.5796901408038404767744.4, deg 14 |
| 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 | ${\href{/padicField/3.14.0.1}{14} }$ | ${\href{/padicField/5.7.0.1}{7} }^{2}$ | ${\href{/padicField/7.14.0.1}{14} }$ | ${\href{/padicField/11.14.0.1}{14} }$ | ${\href{/padicField/13.7.0.1}{7} }^{2}$ | ${\href{/padicField/17.7.0.1}{7} }^{2}$ | ${\href{/padicField/19.14.0.1}{14} }$ | ${\href{/padicField/23.14.0.1}{14} }$ | R | ${\href{/padicField/31.14.0.1}{14} }$ | ${\href{/padicField/37.7.0.1}{7} }^{2}$ | ${\href{/padicField/41.7.0.1}{7} }^{2}$ | ${\href{/padicField/43.14.0.1}{14} }$ | ${\href{/padicField/47.14.0.1}{14} }$ | ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{7}$ | ${\href{/padicField/59.2.0.1}{2} }^{7}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.14.14.38 | $x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1948 x^{10} + 8392 x^{9} + 30520 x^{8} + 84992 x^{7} + 178608 x^{6} + 284064 x^{5} + 325984 x^{4} + 242688 x^{3} + 97600 x^{2} + 11648 x - 5504$ | $2$ | $7$ | $14$ | $C_{14}$ | $[2]^{7}$ |
|
\(29\)
| 29.7.0.1 | $x^{7} + 2 x + 27$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |
| 29.7.6.3 | $x^{7} + 87$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.116.14t1.b.a | $1$ | $ 2^{2} \cdot 29 $ | 14.0.5796901408038404767744.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| 1.116.14t1.b.f | $1$ | $ 2^{2} \cdot 29 $ | 14.0.5796901408038404767744.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| 1.116.14t1.b.e | $1$ | $ 2^{2} \cdot 29 $ | 14.0.5796901408038404767744.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| 1.116.14t1.b.d | $1$ | $ 2^{2} \cdot 29 $ | 14.0.5796901408038404767744.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| 1.29.7t1.a.c | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
| 1.29.7t1.a.d | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
| 1.29.7t1.a.b | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
| 1.29.7t1.a.f | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
| 1.29.7t1.a.e | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
| 1.116.14t1.b.b | $1$ | $ 2^{2} \cdot 29 $ | 14.0.5796901408038404767744.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| 1.116.14t1.b.c | $1$ | $ 2^{2} \cdot 29 $ | 14.0.5796901408038404767744.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| 1.29.7t1.a.a | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
| 2.3364.14t8.c.e | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.9745585291264.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.3364.7t2.a.b | $2$ | $ 2^{2} \cdot 29^{2}$ | 7.1.38068692544.1 | $D_{7}$ (as 7T2) | $1$ | $0$ | |
| 2.3364.14t8.c.f | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.9745585291264.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.3364.14t8.c.c | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.9745585291264.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.3364.14t8.c.b | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.9745585291264.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.3364.14t8.c.d | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.9745585291264.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.3364.14t8.d.b | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.9745585291264.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| * | 2.116.14t8.b.d | $2$ | $ 2^{2} \cdot 29 $ | 14.0.9745585291264.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ |
| * | 2.116.14t8.b.f | $2$ | $ 2^{2} \cdot 29 $ | 14.0.9745585291264.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ |
| 2.3364.7t2.a.c | $2$ | $ 2^{2} \cdot 29^{2}$ | 7.1.38068692544.1 | $D_{7}$ (as 7T2) | $1$ | $0$ | |
| 2.3364.14t8.d.f | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.9745585291264.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| * | 2.116.14t8.b.a | $2$ | $ 2^{2} \cdot 29 $ | 14.0.9745585291264.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ |
| 2.3364.14t8.d.e | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.9745585291264.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.3364.14t8.d.a | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.9745585291264.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| * | 2.116.14t8.b.c | $2$ | $ 2^{2} \cdot 29 $ | 14.0.9745585291264.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ |
| 2.3364.14t8.d.c | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.9745585291264.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.3364.14t8.d.d | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.9745585291264.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| * | 2.116.14t8.b.b | $2$ | $ 2^{2} \cdot 29 $ | 14.0.9745585291264.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ |
| * | 2.116.14t8.b.e | $2$ | $ 2^{2} \cdot 29 $ | 14.0.9745585291264.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ |
| 2.3364.14t8.c.a | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.9745585291264.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.3364.7t2.a.a | $2$ | $ 2^{2} \cdot 29^{2}$ | 7.1.38068692544.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.