Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 - 2*x^4 + 3*x^3 - x^2 - 1)
gp: K = bnfinit(y^6 - y^5 - 2*y^4 + 3*y^3 - y^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^6 - x^5 - 2*x^4 + 3*x^3 - x^2 - 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^6 - x^5 - 2*x^4 + 3*x^3 - x^2 - 1)
\( x^{6} - x^{5} - 2x^{4} + 3x^{3} - x^{2} - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $6$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[2, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(306341\)
\(\medspace = 7\cdot 107\cdot 409\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(8.21\) | 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: | $7^{1/2}107^{1/2}409^{1/2}\approx 553.4808036418247$ | ||
| Ramified primes: |
\(7\), \(107\), \(409\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{306341}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$
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: | $3$ | 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: |
$a$, $a-1$, $a^{4}-2a^{2}+2a$
| 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: | \( 2.54378008071 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{2}\cdot 2.54378008071 \cdot 1}{2\cdot\sqrt{306341}}\cr\approx \mathstrut & 0.362883090648 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 - 2*x^4 + 3*x^3 - x^2 - 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^6 - x^5 - 2*x^4 + 3*x^3 - x^2 - 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^6 - x^5 - 2*x^4 + 3*x^3 - x^2 - 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^6 - x^5 - 2*x^4 + 3*x^3 - x^2 - 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
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 non-solvable group of order 720 |
| The 11 conjugacy class representatives for $S_6$ |
| Character table for $S_6$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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 algebras
| Twin sextic algebra: | 6.2.28748512413609821.1 |
| Degree 6 sibling: | 6.2.28748512413609821.1 |
| Degree 10 sibling: | 10.2.28748512413609821.1 |
| Degree 12 siblings: | deg 12, deg 12 |
| Degree 15 siblings: | deg 15, deg 15 |
| Degree 20 siblings: | deg 20, deg 20, deg 20 |
| Degree 30 siblings: | deg 30, deg 30, deg 30, deg 30, deg 30, deg 30 |
| Degree 36 sibling: | deg 36 |
| Degree 40 siblings: | deg 40, deg 40, deg 40 |
| Degree 45 sibling: | deg 45 |
| 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 | ${\href{/padicField/2.6.0.1}{6} }$ | ${\href{/padicField/3.6.0.1}{6} }$ | ${\href{/padicField/5.2.0.1}{2} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 7.4.0.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(107\)
| 107.2.1.2 | $x^{2} + 107$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 107.4.0.1 | $x^{4} + 13 x^{2} + 79 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(409\)
| $\Q_{409}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{409}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| 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 | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.306341.2t1.a.a | $1$ | $ 7 \cdot 107 \cdot 409 $ | \(\Q(\sqrt{306341}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| * | 5.306341.6t16.a.a | $5$ | $ 7 \cdot 107 \cdot 409 $ | 6.2.306341.1 | $S_6$ (as 6T16) | $1$ | $1$ |
| 5.93844808281.12t183.a.a | $5$ | $ 7^{2} \cdot 107^{2} \cdot 409^{2}$ | 6.2.306341.1 | $S_6$ (as 6T16) | $1$ | $1$ | |
| 5.880...961.12t183.a.a | $5$ | $ 7^{4} \cdot 107^{4} \cdot 409^{4}$ | 6.2.306341.1 | $S_6$ (as 6T16) | $1$ | $1$ | |
| 5.287...821.6t16.a.a | $5$ | $ 7^{3} \cdot 107^{3} \cdot 409^{3}$ | 6.2.306341.1 | $S_6$ (as 6T16) | $1$ | $1$ | |
| 9.287...821.10t32.a.a | $9$ | $ 7^{3} \cdot 107^{3} \cdot 409^{3}$ | 6.2.306341.1 | $S_6$ (as 6T16) | $1$ | $1$ | |
| 9.826...041.20t145.a.a | $9$ | $ 7^{6} \cdot 107^{6} \cdot 409^{6}$ | 6.2.306341.1 | $S_6$ (as 6T16) | $1$ | $1$ | |
| 10.826...041.30t164.a.a | $10$ | $ 7^{6} \cdot 107^{6} \cdot 409^{6}$ | 6.2.306341.1 | $S_6$ (as 6T16) | $1$ | $-2$ | |
| 10.880...961.30t164.a.a | $10$ | $ 7^{4} \cdot 107^{4} \cdot 409^{4}$ | 6.2.306341.1 | $S_6$ (as 6T16) | $1$ | $-2$ | |
| 16.775...521.36t1252.a.a | $16$ | $ 7^{8} \cdot 107^{8} \cdot 409^{8}$ | 6.2.306341.1 | $S_6$ (as 6T16) | $1$ | $0$ |
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 *.