Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 2*x^9 + 6*x^8 - 6*x^7 - 21*x^6 + 42*x^5 - 154*x^4 + 116*x^3 - 183*x^2 - 32*x + 4)
gp: K = bnfinit(y^10 - 2*y^9 + 6*y^8 - 6*y^7 - 21*y^6 + 42*y^5 - 154*y^4 + 116*y^3 - 183*y^2 - 32*y + 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 2*x^9 + 6*x^8 - 6*x^7 - 21*x^6 + 42*x^5 - 154*x^4 + 116*x^3 - 183*x^2 - 32*x + 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - 2*x^9 + 6*x^8 - 6*x^7 - 21*x^6 + 42*x^5 - 154*x^4 + 116*x^3 - 183*x^2 - 32*x + 4)
\( x^{10} - 2x^{9} + 6x^{8} - 6x^{7} - 21x^{6} + 42x^{5} - 154x^{4} + 116x^{3} - 183x^{2} - 32x + 4 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[4, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-196073702927424\)
\(\medspace = -\,2^{6}\cdot 3^{12}\cdot 7^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(26.87\) | 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: | $2\cdot 3^{4/3}7^{4/5}\approx 41.045930044873074$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{14}a^{5}+\frac{3}{7}a^{4}-\frac{1}{14}a^{3}-\frac{2}{7}a^{2}-\frac{1}{14}a+\frac{1}{7}$, $\frac{1}{14}a^{6}+\frac{5}{14}a^{4}+\frac{1}{7}a^{3}-\frac{5}{14}a^{2}-\frac{3}{7}a+\frac{1}{7}$, $\frac{1}{14}a^{7}-\frac{1}{2}a+\frac{2}{7}$, $\frac{1}{28}a^{8}-\frac{1}{28}a^{7}-\frac{1}{28}a^{5}-\frac{3}{14}a^{4}+\frac{1}{28}a^{3}-\frac{3}{28}a^{2}+\frac{3}{7}a+\frac{2}{7}$, $\frac{1}{532}a^{9}+\frac{1}{133}a^{8}+\frac{11}{532}a^{7}+\frac{3}{532}a^{6}-\frac{3}{532}a^{5}-\frac{185}{532}a^{4}+\frac{71}{266}a^{3}+\frac{113}{532}a^{2}-\frac{33}{133}a+\frac{60}{133}$
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
$C_{2}$, which has order $2$
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: |
\( -1 \)
(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: |
$\frac{3}{532}a^{9}-\frac{13}{266}a^{8}+\frac{33}{532}a^{7}-\frac{29}{532}a^{6}-\frac{123}{532}a^{5}+\frac{699}{532}a^{4}-\frac{150}{133}a^{3}+\frac{719}{532}a^{2}+\frac{429}{266}a-\frac{29}{133}$, $\frac{29}{532}a^{9}-\frac{55}{532}a^{8}+\frac{37}{133}a^{7}-\frac{141}{532}a^{6}-\frac{169}{133}a^{5}+\frac{1133}{532}a^{4}-\frac{3729}{532}a^{3}+\frac{1325}{266}a^{2}-\frac{249}{38}a-\frac{236}{133}$, $\frac{13}{532}a^{9}-\frac{6}{133}a^{8}+\frac{29}{532}a^{7}-\frac{37}{532}a^{6}-\frac{381}{532}a^{5}+\frac{69}{76}a^{4}-\frac{156}{133}a^{3}+\frac{1089}{532}a^{2}+\frac{160}{133}a+\frac{58}{133}$, $\frac{45}{532}a^{9}-\frac{43}{266}a^{8}+\frac{267}{532}a^{7}-\frac{35}{76}a^{6}-\frac{971}{532}a^{5}+\frac{1783}{532}a^{4}-\frac{3417}{266}a^{3}+\frac{4743}{532}a^{2}-\frac{2036}{133}a-\frac{530}{133}$, $\frac{4}{133}a^{9}-\frac{3}{133}a^{8}+\frac{31}{266}a^{7}-\frac{1}{19}a^{6}-\frac{88}{133}a^{5}+\frac{39}{133}a^{4}-\frac{458}{133}a^{3}+\frac{53}{133}a^{2}-\frac{961}{266}a-\frac{142}{133}$, $\frac{11}{532}a^{9}-\frac{8}{133}a^{8}+\frac{83}{532}a^{7}-\frac{81}{532}a^{6}-\frac{261}{532}a^{5}+\frac{815}{532}a^{4}-\frac{1081}{266}a^{3}+\frac{2193}{532}a^{2}-\frac{1353}{266}a-\frac{81}{133}$
| 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: | \( 4651.46956596 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 4651.46956596 \cdot 2}{2\cdot\sqrt{196073702927424}}\cr\approx \mathstrut & 1.31837845189 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^10 - 2*x^9 + 6*x^8 - 6*x^7 - 21*x^6 + 42*x^5 - 154*x^4 + 116*x^3 - 183*x^2 - 32*x + 4)
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^10 - 2*x^9 + 6*x^8 - 6*x^7 - 21*x^6 + 42*x^5 - 154*x^4 + 116*x^3 - 183*x^2 - 32*x + 4, 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^10 - 2*x^9 + 6*x^8 - 6*x^7 - 21*x^6 + 42*x^5 - 154*x^4 + 116*x^3 - 183*x^2 - 32*x + 4);
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^10 - 2*x^9 + 6*x^8 - 6*x^7 - 21*x^6 + 42*x^5 - 154*x^4 + 116*x^3 - 183*x^2 - 32*x + 4);
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 fields
| Degree 6 siblings: | 6.0.12446784.1, 6.4.63011844.2 |
| Degree 12 siblings: | deg 12, deg 12 |
| Degree 15 siblings: | deg 15, deg 15 |
| Degree 20 siblings: | deg 20, deg 20, 20.0.9841893626795960030057286598656.1 |
| Degree 30 siblings: | data not computed |
| Degree 36 sibling: | data not computed |
| Degree 40 siblings: | data not computed |
| Degree 45 sibling: | data not computed |
| Minimal sibling: | 6.0.12446784.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.5.0.1}{5} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
| 2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 2.4.4.2 | $x^{4} + 4 x^{3} + 4 x^{2} + 12$ | $2$ | $2$ | $4$ | $C_4$ | $[2]^{2}$ | |
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 3.9.12.16 | $x^{9} + 9 x^{8} + 27 x^{7} + 36 x^{6} + 54 x^{5} + 81 x^{4} + 675 x^{3} + 2025 x^{2} - 3861$ | $3$ | $3$ | $12$ | $S_3\times C_3$ | $[2]^{6}$ | |
|
\(7\)
| 7.10.8.1 | $x^{10} + 30 x^{9} + 375 x^{8} + 2520 x^{7} + 9810 x^{6} + 22370 x^{5} + 29640 x^{4} + 24780 x^{3} + 21465 x^{2} + 33300 x + 33934$ | $5$ | $2$ | $8$ | $F_5$ | $[\ ]_{5}^{4}$ |