Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 3*x^13 + 8*x^12 - 18*x^11 + 32*x^10 - 52*x^9 + 70*x^8 - 81*x^7 + 82*x^6 - 70*x^5 + 52*x^4 - 31*x^3 + 16*x^2 - 6*x + 1)
gp: K = bnfinit(y^14 - 3*y^13 + 8*y^12 - 18*y^11 + 32*y^10 - 52*y^9 + 70*y^8 - 81*y^7 + 82*y^6 - 70*y^5 + 52*y^4 - 31*y^3 + 16*y^2 - 6*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 3*x^13 + 8*x^12 - 18*x^11 + 32*x^10 - 52*x^9 + 70*x^8 - 81*x^7 + 82*x^6 - 70*x^5 + 52*x^4 - 31*x^3 + 16*x^2 - 6*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 3*x^13 + 8*x^12 - 18*x^11 + 32*x^10 - 52*x^9 + 70*x^8 - 81*x^7 + 82*x^6 - 70*x^5 + 52*x^4 - 31*x^3 + 16*x^2 - 6*x + 1)
\( x^{14} - 3 x^{13} + 8 x^{12} - 18 x^{11} + 32 x^{10} - 52 x^{9} + 70 x^{8} - 81 x^{7} + 82 x^{6} + \cdots + 1 \)
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: |
\(-13824820988163\)
\(\medspace = -\,3^{7}\cdot 43^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(8.68\) | 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: | $3^{1/2}43^{6/7}\approx 43.518753397811835$ | ||
| Ramified primes: |
\(3\), \(43\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $7$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{179}a^{13}+\frac{64}{179}a^{12}-\frac{18}{179}a^{10}+\frac{79}{179}a^{9}+\frac{50}{179}a^{8}+\frac{19}{179}a^{7}-\frac{61}{179}a^{6}-\frac{67}{179}a^{5}-\frac{84}{179}a^{4}-\frac{27}{179}a^{3}-\frac{50}{179}a^{2}+\frac{67}{179}a+\frac{8}{179}$
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
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: |
\( \frac{533}{179} a^{13} - \frac{1330}{179} a^{12} + 20 a^{11} - \frac{7804}{179} a^{10} + \frac{13109}{179} a^{9} - \frac{21143}{179} a^{8} + \frac{26774}{179} a^{7} - \frac{29828}{179} a^{6} + \frac{29087}{179} a^{5} - \frac{23113}{179} a^{4} + \frac{16576}{179} a^{3} - \frac{8750}{179} a^{2} + \frac{4565}{179} a - \frac{1106}{179} \)
(order $6$)
| 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: |
$a^{13}-3a^{12}+8a^{11}-18a^{10}+32a^{9}-52a^{8}+70a^{7}-81a^{6}+82a^{5}-70a^{4}+52a^{3}-31a^{2}+16a-6$, $\frac{409}{179}a^{13}-\frac{1032}{179}a^{12}+16a^{11}-\frac{6288}{179}a^{10}+\frac{10831}{179}a^{9}-\frac{17677}{179}a^{8}+\frac{22986}{179}a^{7}-\frac{26381}{179}a^{6}+\frac{26297}{179}a^{5}-\frac{21647}{179}a^{4}+\frac{15628}{179}a^{3}-\frac{8457}{179}a^{2}+\frac{4312}{179}a-\frac{1024}{179}$, $\frac{709}{179}a^{13}-\frac{1880}{179}a^{12}+28a^{11}-\frac{10972}{179}a^{10}+\frac{18779}{179}a^{9}-\frac{30064}{179}a^{8}+\frac{38710}{179}a^{7}-\frac{43249}{179}a^{6}+\frac{41997}{179}a^{5}-\frac{33959}{179}a^{4}+\frac{23996}{179}a^{3}-\frac{12896}{179}a^{2}+\frac{6512}{179}a-\frac{1667}{179}$, $\frac{80}{179}a^{13}-\frac{71}{179}a^{12}+a^{11}-\frac{187}{179}a^{10}-\frac{124}{179}a^{9}+\frac{420}{179}a^{8}-\frac{1702}{179}a^{7}+\frac{2638}{179}a^{6}-\frac{3391}{179}a^{5}+\frac{3483}{179}a^{4}-\frac{2518}{179}a^{3}+\frac{1907}{179}a^{2}-\frac{547}{179}a+\frac{282}{179}$, $\frac{489}{179}a^{13}-\frac{1103}{179}a^{12}+17a^{11}-\frac{6475}{179}a^{10}+\frac{10707}{179}a^{9}-\frac{17257}{179}a^{8}+\frac{21284}{179}a^{7}-\frac{23743}{179}a^{6}+\frac{22906}{179}a^{5}-\frac{18164}{179}a^{4}+\frac{13110}{179}a^{3}-\frac{6550}{179}a^{2}+\frac{3765}{179}a-\frac{742}{179}$, $\frac{1383}{179}a^{13}-\frac{3494}{179}a^{12}+52a^{11}-\frac{20240}{179}a^{10}+\frac{34077}{179}a^{9}-\frac{54539}{179}a^{8}+\frac{69058}{179}a^{7}-\frac{76487}{179}a^{6}+\frac{73988}{179}a^{5}-\frac{58892}{179}a^{4}+\frac{41777}{179}a^{3}-\frac{21894}{179}a^{2}+\frac{11216}{179}a-\frac{2898}{179}$
| 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: | \( 9.923787361493524 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 9.923787361493524 \cdot 1}{6\cdot\sqrt{13824820988163}}\cr\approx \mathstrut & 0.171971120063312 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^14 - 3*x^13 + 8*x^12 - 18*x^11 + 32*x^10 - 52*x^9 + 70*x^8 - 81*x^7 + 82*x^6 - 70*x^5 + 52*x^4 - 31*x^3 + 16*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 - 3*x^13 + 8*x^12 - 18*x^11 + 32*x^10 - 52*x^9 + 70*x^8 - 81*x^7 + 82*x^6 - 70*x^5 + 52*x^4 - 31*x^3 + 16*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 - 3*x^13 + 8*x^12 - 18*x^11 + 32*x^10 - 52*x^9 + 70*x^8 - 81*x^7 + 82*x^6 - 70*x^5 + 52*x^4 - 31*x^3 + 16*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 - 3*x^13 + 8*x^12 - 18*x^11 + 32*x^10 - 52*x^9 + 70*x^8 - 81*x^7 + 82*x^6 - 70*x^5 + 52*x^4 - 31*x^3 + 16*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
| \(\Q(\sqrt{-3}) \) |
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: | deg 14, 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 | ${\href{/padicField/2.14.0.1}{14} }$ | R | ${\href{/padicField/5.14.0.1}{14} }$ | ${\href{/padicField/7.7.0.1}{7} }^{2}$ | ${\href{/padicField/11.14.0.1}{14} }$ | ${\href{/padicField/13.7.0.1}{7} }^{2}$ | ${\href{/padicField/17.14.0.1}{14} }$ | ${\href{/padicField/19.7.0.1}{7} }^{2}$ | ${\href{/padicField/23.14.0.1}{14} }$ | ${\href{/padicField/29.14.0.1}{14} }$ | ${\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{7}$ | ${\href{/padicField/37.7.0.1}{7} }^{2}$ | ${\href{/padicField/41.14.0.1}{14} }$ | R | ${\href{/padicField/47.14.0.1}{14} }$ | ${\href{/padicField/53.14.0.1}{14} }$ | ${\href{/padicField/59.14.0.1}{14} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.14.7.2 | $x^{14} + 21 x^{12} + 189 x^{10} + 4 x^{9} + 945 x^{8} - 94 x^{7} + 2835 x^{6} - 630 x^{5} + 5107 x^{4} + 630 x^{3} + 5131 x^{2} + 1242 x + 2212$ | $2$ | $7$ | $7$ | $C_{14}$ | $[\ ]_{2}^{7}$ |
|
\(43\)
| 43.7.6.6 | $x^{7} + 430$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ |
| 43.7.0.1 | $x^{7} + 42 x^{2} + 7 x + 40$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.129.14t1.a.d | $1$ | $ 3 \cdot 43 $ | 14.0.87391712553613254588987.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| 1.129.14t1.a.c | $1$ | $ 3 \cdot 43 $ | 14.0.87391712553613254588987.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| 1.43.7t1.a.c | $1$ | $ 43 $ | 7.7.6321363049.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
| 1.43.7t1.a.f | $1$ | $ 43 $ | 7.7.6321363049.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
| 1.129.14t1.a.b | $1$ | $ 3 \cdot 43 $ | 14.0.87391712553613254588987.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| 1.43.7t1.a.b | $1$ | $ 43 $ | 7.7.6321363049.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
| 1.129.14t1.a.a | $1$ | $ 3 \cdot 43 $ | 14.0.87391712553613254588987.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| 1.43.7t1.a.d | $1$ | $ 43 $ | 7.7.6321363049.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
| 1.129.14t1.a.e | $1$ | $ 3 \cdot 43 $ | 14.0.87391712553613254588987.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| 1.43.7t1.a.a | $1$ | $ 43 $ | 7.7.6321363049.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
| 1.43.7t1.a.e | $1$ | $ 43 $ | 7.7.6321363049.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
| 1.129.14t1.a.f | $1$ | $ 3 \cdot 43 $ | 14.0.87391712553613254588987.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| * | 2.129.14t8.a.d | $2$ | $ 3 \cdot 43 $ | 14.0.13824820988163.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ |
| * | 2.129.14t8.a.c | $2$ | $ 3 \cdot 43 $ | 14.0.13824820988163.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ |
| 2.5547.14t8.a.e | $2$ | $ 3 \cdot 43^{2}$ | 14.0.13824820988163.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| * | 2.129.14t8.a.e | $2$ | $ 3 \cdot 43 $ | 14.0.13824820988163.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ |
| * | 2.129.14t8.a.f | $2$ | $ 3 \cdot 43 $ | 14.0.13824820988163.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ |
| 2.5547.7t2.a.a | $2$ | $ 3 \cdot 43^{2}$ | 7.1.170676802323.1 | $D_{7}$ (as 7T2) | $1$ | $0$ | |
| 2.5547.14t8.a.a | $2$ | $ 3 \cdot 43^{2}$ | 14.0.13824820988163.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.5547.7t2.a.c | $2$ | $ 3 \cdot 43^{2}$ | 7.1.170676802323.1 | $D_{7}$ (as 7T2) | $1$ | $0$ | |
| 2.5547.14t8.b.c | $2$ | $ 3 \cdot 43^{2}$ | 14.0.13824820988163.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.5547.14t8.b.b | $2$ | $ 3 \cdot 43^{2}$ | 14.0.13824820988163.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.5547.14t8.a.c | $2$ | $ 3 \cdot 43^{2}$ | 14.0.13824820988163.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.5547.14t8.b.d | $2$ | $ 3 \cdot 43^{2}$ | 14.0.13824820988163.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.5547.14t8.b.e | $2$ | $ 3 \cdot 43^{2}$ | 14.0.13824820988163.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.5547.7t2.a.b | $2$ | $ 3 \cdot 43^{2}$ | 7.1.170676802323.1 | $D_{7}$ (as 7T2) | $1$ | $0$ | |
| * | 2.129.14t8.a.b | $2$ | $ 3 \cdot 43 $ | 14.0.13824820988163.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ |
| * | 2.129.14t8.a.a | $2$ | $ 3 \cdot 43 $ | 14.0.13824820988163.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ |
| 2.5547.14t8.b.f | $2$ | $ 3 \cdot 43^{2}$ | 14.0.13824820988163.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.5547.14t8.a.d | $2$ | $ 3 \cdot 43^{2}$ | 14.0.13824820988163.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.5547.14t8.a.b | $2$ | $ 3 \cdot 43^{2}$ | 14.0.13824820988163.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.5547.14t8.a.f | $2$ | $ 3 \cdot 43^{2}$ | 14.0.13824820988163.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.5547.14t8.b.a | $2$ | $ 3 \cdot 43^{2}$ | 14.0.13824820988163.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ |
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 *.