Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 + 35*x^4 - 14*x^3 - 14*x^2 - 60*x - 152)
gp: K = bnfinit(y^8 - 4*y^7 + 35*y^4 - 14*y^3 - 14*y^2 - 60*y - 152, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - 4*x^7 + 35*x^4 - 14*x^3 - 14*x^2 - 60*x - 152);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^8 - 4*x^7 + 35*x^4 - 14*x^3 - 14*x^2 - 60*x - 152)
\( x^{8} - 4x^{7} + 35x^{4} - 14x^{3} - 14x^{2} - 60x - 152 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[2, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-314437188128\)
\(\medspace = -\,2^{5}\cdot 7^{6}\cdot 17^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(27.36\) | 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: | $2^{11/6}7^{3/4}17^{1/2}\approx 63.23191227961974$ | ||
| Ramified primes: |
\(2\), \(7\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-2}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{7}a^{4}-\frac{2}{7}a^{3}-\frac{2}{7}a^{2}+\frac{3}{7}a-\frac{3}{7}$, $\frac{1}{14}a^{5}-\frac{1}{14}a^{4}+\frac{3}{14}a^{3}+\frac{1}{14}a^{2}+\frac{2}{7}$, $\frac{1}{14}a^{6}-\frac{3}{7}a^{3}+\frac{5}{14}a^{2}-\frac{1}{7}a-\frac{2}{7}$, $\frac{1}{364}a^{7}-\frac{3}{182}a^{6}+\frac{3}{91}a^{5}-\frac{6}{91}a^{4}-\frac{177}{364}a^{3}+\frac{33}{91}a^{2}-\frac{61}{182}a+\frac{33}{91}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: | $4$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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: |
$\frac{1}{364}a^{7}+\frac{5}{91}a^{6}-\frac{10}{91}a^{5}-\frac{19}{91}a^{4}+\frac{83}{364}a^{3}+\frac{27}{182}a^{2}+\frac{121}{182}a+\frac{124}{91}$, $\frac{2}{7}a^{4}-\frac{4}{7}a^{3}-\frac{4}{7}a^{2}-\frac{8}{7}a-\frac{13}{7}$, $\frac{2}{91}a^{7}-\frac{38}{91}a^{6}+\frac{22}{13}a^{5}-\frac{100}{91}a^{4}-\frac{237}{91}a^{3}-\frac{165}{91}a^{2}+\frac{58}{13}a+\frac{927}{91}$, $\frac{6}{91}a^{7}+\frac{3}{91}a^{6}-\frac{32}{91}a^{5}-\frac{157}{91}a^{4}-\frac{100}{91}a^{3}+\frac{298}{91}a^{2}+\frac{932}{91}a+\frac{753}{91}$
| 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: | \( 3149.67293746 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 3149.67293746 \cdot 1}{2\cdot\sqrt{314437188128}}\cr\approx \mathstrut & 2.78655813294 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 + 35*x^4 - 14*x^3 - 14*x^2 - 60*x - 152)
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^8 - 4*x^7 + 35*x^4 - 14*x^3 - 14*x^2 - 60*x - 152, 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^8 - 4*x^7 + 35*x^4 - 14*x^3 - 14*x^2 - 60*x - 152);
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^8 - 4*x^7 + 35*x^4 - 14*x^3 - 14*x^2 - 60*x - 152);
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
$S_4\wr C_2$ (as 8T47):
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 1152 |
| The 20 conjugacy class representatives for $S_4\wr C_2$ |
| Character table for $S_4\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{17}) \) |
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 12 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Degree 36 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 | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.8.0.1}{8} }$ | R | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | R | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{6}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 2.2.3.3 | $x^{2} + 2$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
| 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(7\)
| 7.8.6.3 | $x^{8} - 154 x^{4} - 1421$ | $4$ | $2$ | $6$ | $C_8:C_2$ | $[\ ]_{4}^{4}$ |
|
\(17\)
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 17.6.3.1 | $x^{6} + 459 x^{5} + 70280 x^{4} + 3597823 x^{3} + 1271380 x^{2} + 4696159 x + 50437479$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.136.2t1.b.a | $1$ | $ 2^{3} \cdot 17 $ | \(\Q(\sqrt{-34}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 1.17.2t1.a.a | $1$ | $ 17 $ | \(\Q(\sqrt{17}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 2.6664.4t3.a.a | $2$ | $ 2^{3} \cdot 7^{2} \cdot 17 $ | 4.0.53312.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
| 4.61628672.12t34.b.a | $4$ | $ 2^{8} \cdot 7^{2} \cdot 17^{3}$ | 6.0.1705984.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
| 4.7250432.12t34.b.a | $4$ | $ 2^{9} \cdot 7^{2} \cdot 17^{2}$ | 6.0.1705984.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $-2$ | |
| 4.213248.6t13.b.a | $4$ | $ 2^{8} \cdot 7^{2} \cdot 17 $ | 6.0.1705984.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
| 4.453152.6t13.b.a | $4$ | $ 2^{5} \cdot 7^{2} \cdot 17^{2}$ | 6.0.1705984.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $2$ | |
| 6.48316878848.12t201.a.a | $6$ | $ 2^{12} \cdot 7^{4} \cdot 17^{3}$ | 8.2.314437188128.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
| 6.189...416.12t202.a.a | $6$ | $ 2^{15} \cdot 7^{6} \cdot 17^{3}$ | 8.2.314437188128.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ | |
| * | 6.18496305184.8t47.a.a | $6$ | $ 2^{5} \cdot 7^{6} \cdot 17^{3}$ | 8.2.314437188128.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |
| 6.3019804928.12t200.a.a | $6$ | $ 2^{8} \cdot 7^{4} \cdot 17^{3}$ | 8.2.314437188128.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
| 9.473...104.16t1294.a.a | $9$ | $ 2^{13} \cdot 7^{6} \cdot 17^{3}$ | 8.2.314437188128.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
| 9.606...312.18t272.a.a | $9$ | $ 2^{20} \cdot 7^{6} \cdot 17^{3}$ | 8.2.314437188128.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
| 9.297...856.18t273.a.a | $9$ | $ 2^{20} \cdot 7^{6} \cdot 17^{6}$ | 8.2.314437188128.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
| 9.232...952.18t274.a.a | $9$ | $ 2^{13} \cdot 7^{6} \cdot 17^{6}$ | 8.2.314437188128.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
| 12.228...192.36t1763.a.a | $12$ | $ 2^{25} \cdot 7^{10} \cdot 17^{6}$ | 8.2.314437188128.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
| 12.142...512.24t2821.a.a | $12$ | $ 2^{21} \cdot 7^{10} \cdot 17^{6}$ | 8.2.314437188128.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
| 18.690...176.36t1758.a.a | $18$ | $ 2^{33} \cdot 7^{14} \cdot 17^{9}$ | 8.2.314437188128.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |
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 *.