Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 - 3*x^6 + 8*x^5 + 3*x^4 - 10*x^3 + 16*x^2 + 28*x + 9)
gp: K = bnfinit(y^8 - y^7 - 3*y^6 + 8*y^5 + 3*y^4 - 10*y^3 + 16*y^2 + 28*y + 9, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - x^7 - 3*x^6 + 8*x^5 + 3*x^4 - 10*x^3 + 16*x^2 + 28*x + 9);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^8 - x^7 - 3*x^6 + 8*x^5 + 3*x^4 - 10*x^3 + 16*x^2 + 28*x + 9)
\( x^{8} - x^{7} - 3x^{6} + 8x^{5} + 3x^{4} - 10x^{3} + 16x^{2} + 28x + 9 \)
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: | $[4, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(6885514441\)
\(\medspace = 13^{4}\cdot 491^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(16.97\) | 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: | $13^{1/2}491^{1/2}\approx 79.89367934949547$ | ||
| Ramified primes: |
\(13\), \(491\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{37}a^{7}+\frac{10}{37}a^{6}-\frac{4}{37}a^{5}+\frac{1}{37}a^{4}+\frac{14}{37}a^{3}-\frac{4}{37}a^{2}+\frac{9}{37}a+\frac{16}{37}$
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: | $5$ | 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: |
$\frac{26}{37}a^{7}-\frac{36}{37}a^{6}-\frac{67}{37}a^{5}+\frac{248}{37}a^{4}-\frac{43}{37}a^{3}-\frac{252}{37}a^{2}+\frac{604}{37}a+\frac{416}{37}$, $a+1$, $\frac{27}{37}a^{7}-\frac{63}{37}a^{6}+\frac{3}{37}a^{5}+\frac{212}{37}a^{4}-\frac{214}{37}a^{3}+\frac{3}{37}a^{2}+\frac{428}{37}a+\frac{136}{37}$, $\frac{9}{37}a^{7}-\frac{21}{37}a^{6}+\frac{1}{37}a^{5}+\frac{83}{37}a^{4}-\frac{96}{37}a^{3}+\frac{1}{37}a^{2}+\frac{192}{37}a+\frac{70}{37}$, $\frac{10}{37}a^{7}-\frac{11}{37}a^{6}-\frac{40}{37}a^{5}+\frac{121}{37}a^{4}-\frac{8}{37}a^{3}-\frac{151}{37}a^{2}+\frac{312}{37}a+\frac{197}{37}$
| 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: | \( 138.734267611 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 138.734267611 \cdot 1}{2\cdot\sqrt{6885514441}}\cr\approx \mathstrut & 0.528038115936 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 - 3*x^6 + 8*x^5 + 3*x^4 - 10*x^3 + 16*x^2 + 28*x + 9)
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 - x^7 - 3*x^6 + 8*x^5 + 3*x^4 - 10*x^3 + 16*x^2 + 28*x + 9, 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 - x^7 - 3*x^6 + 8*x^5 + 3*x^4 - 10*x^3 + 16*x^2 + 28*x + 9);
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 - x^7 - 3*x^6 + 8*x^5 + 3*x^4 - 10*x^3 + 16*x^2 + 28*x + 9);
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
$C_2^3:S_4$ (as 8T41):
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 192 |
| The 14 conjugacy class representatives for $V_4^2:(S_3\times C_2)$ |
| Character table for $V_4^2:(S_3\times C_2)$ |
Intermediate fields
| \(\Q(\sqrt{13}) \) |
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 8 sibling: | data not computed |
| Degree 12 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 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 | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | ${\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | R | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(13\)
| 13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 13.6.3.1 | $x^{6} + 390 x^{5} + 50743 x^{4} + 2208202 x^{3} + 765301 x^{2} + 5017316 x + 24555184$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(491\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.13.2t1.a.a | $1$ | $ 13 $ | \(\Q(\sqrt{13}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.6383.2t1.a.a | $1$ | $ 13 \cdot 491 $ | \(\Q(\sqrt{-6383}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.491.2t1.a.a | $1$ | $ 491 $ | \(\Q(\sqrt{-491}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 2.491.3t2.a.a | $2$ | $ 491 $ | 3.1.491.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.82979.6t3.b.a | $2$ | $ 13^{2} \cdot 491 $ | 6.2.529654957.3 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| 3.241081.6t8.d.a | $3$ | $ 491^{2}$ | 4.2.491.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
| 3.1078727.6t11.a.a | $3$ | $ 13^{3} \cdot 491 $ | 6.0.260060583887.3 | $S_4\times C_2$ (as 6T11) | $1$ | $1$ | |
| 3.491.4t5.b.a | $3$ | $ 491 $ | 4.2.491.1 | $S_4$ (as 4T5) | $1$ | $1$ | |
| 3.529654957.6t11.a.a | $3$ | $ 13^{3} \cdot 491^{2}$ | 6.0.260060583887.3 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ | |
| 6.260060583887.8t41.a.a | $6$ | $ 13^{3} \cdot 491^{3}$ | 8.4.6885514441.1 | $V_4^2:(S_3\times C_2)$ (as 8T41) | $1$ | $0$ | |
| * | 6.529654957.8t41.a.a | $6$ | $ 13^{3} \cdot 491^{2}$ | 8.4.6885514441.1 | $V_4^2:(S_3\times C_2)$ (as 8T41) | $1$ | $2$ |
| 6.127...517.12t108.a.a | $6$ | $ 13^{3} \cdot 491^{4}$ | 8.4.6885514441.1 | $V_4^2:(S_3\times C_2)$ (as 8T41) | $1$ | $-2$ | |
| 6.260060583887.12t108.a.a | $6$ | $ 13^{3} \cdot 491^{3}$ | 8.4.6885514441.1 | $V_4^2:(S_3\times C_2)$ (as 8T41) | $1$ | $0$ |
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 *.