Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 4*x^10 + 20*x^9 - 40*x^8 + 90*x^7 - 48*x^6 - 8*x^5 + 300*x^4 - 360*x^3 + 240*x^2 + 128*x - 592)
gp: K = bnfinit(y^11 - 4*y^10 + 20*y^9 - 40*y^8 + 90*y^7 - 48*y^6 - 8*y^5 + 300*y^4 - 360*y^3 + 240*y^2 + 128*y - 592, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^11 - 4*x^10 + 20*x^9 - 40*x^8 + 90*x^7 - 48*x^6 - 8*x^5 + 300*x^4 - 360*x^3 + 240*x^2 + 128*x - 592);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^11 - 4*x^10 + 20*x^9 - 40*x^8 + 90*x^7 - 48*x^6 - 8*x^5 + 300*x^4 - 360*x^3 + 240*x^2 + 128*x - 592)
\( x^{11} - 4 x^{10} + 20 x^{9} - 40 x^{8} + 90 x^{7} - 48 x^{6} - 8 x^{5} + 300 x^{4} - 360 x^{3} + \cdots - 592 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $11$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[3, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(29393280000000000\)
\(\medspace = 2^{16}\cdot 3^{8}\cdot 5^{10}\cdot 7\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(31.41\) | 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: | $2^{139/48}3^{7/6}5^{123/100}7^{1/2}\approx 513.6369813514681$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{7}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{4}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{8}$, $\frac{1}{16}a^{9}-\frac{1}{8}a^{7}-\frac{1}{8}a^{5}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{1184}a^{10}-\frac{9}{592}a^{9}+\frac{25}{592}a^{8}-\frac{1}{8}a^{7}-\frac{29}{592}a^{6}+\frac{43}{296}a^{5}-\frac{49}{296}a^{4}+\frac{21}{296}a^{3}+\frac{67}{148}a^{2}-\frac{57}{148}a-\frac{1}{2}$
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: | $6$ | 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: |
$\frac{11}{592}a^{10}+\frac{3}{74}a^{9}-\frac{21}{296}a^{8}+a^{7}-\frac{319}{296}a^{6}+\frac{181}{74}a^{5}+\frac{349}{148}a^{4}-\frac{731}{148}a^{3}+\frac{276}{37}a^{2}-\frac{183}{74}a-13$, $\frac{29}{592}a^{10}-\frac{19}{74}a^{9}+\frac{355}{296}a^{8}-3a^{7}+\frac{1971}{296}a^{6}-\frac{271}{37}a^{5}+\frac{873}{148}a^{4}+\frac{1423}{148}a^{3}-\frac{638}{37}a^{2}+\frac{2047}{74}a-9$, $\frac{15}{296}a^{10}+\frac{15}{592}a^{9}+\frac{21}{74}a^{8}+\frac{11}{8}a^{7}-\frac{65}{148}a^{6}+\frac{1877}{296}a^{5}+\frac{227}{74}a^{4}-\frac{369}{148}a^{3}+\frac{2281}{148}a^{2}-\frac{485}{37}a-\frac{53}{2}$, $\frac{63}{1184}a^{10}-\frac{10}{37}a^{9}+\frac{687}{592}a^{8}-\frac{11}{4}a^{7}+\frac{2909}{592}a^{6}-\frac{699}{148}a^{5}+\frac{21}{296}a^{4}+\frac{1841}{296}a^{3}-\frac{757}{74}a^{2}+\frac{257}{148}a+4$, $\frac{23}{37}a^{10}-\frac{759}{296}a^{9}+\frac{373}{37}a^{8}-\frac{73}{4}a^{7}+\frac{886}{37}a^{6}+\frac{913}{148}a^{5}-\frac{2399}{37}a^{4}+\frac{7527}{74}a^{3}-\frac{4759}{74}a^{2}-\frac{3162}{37}a+98$, $\frac{1}{74}a^{10}-\frac{33}{592}a^{9}+\frac{13}{74}a^{8}-\frac{1}{8}a^{7}-\frac{21}{74}a^{6}+\frac{577}{296}a^{5}-\frac{233}{74}a^{4}+\frac{649}{148}a^{3}+\frac{73}{148}a^{2}-\frac{80}{37}a+\frac{19}{2}$
| 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: | \( 131601.956099 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{4}\cdot 131601.956099 \cdot 1}{2\cdot\sqrt{29393280000000000}}\cr\approx \mathstrut & 4.78539491153 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^11 - 4*x^10 + 20*x^9 - 40*x^8 + 90*x^7 - 48*x^6 - 8*x^5 + 300*x^4 - 360*x^3 + 240*x^2 + 128*x - 592)
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^11 - 4*x^10 + 20*x^9 - 40*x^8 + 90*x^7 - 48*x^6 - 8*x^5 + 300*x^4 - 360*x^3 + 240*x^2 + 128*x - 592, 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^11 - 4*x^10 + 20*x^9 - 40*x^8 + 90*x^7 - 48*x^6 - 8*x^5 + 300*x^4 - 360*x^3 + 240*x^2 + 128*x - 592);
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^11 - 4*x^10 + 20*x^9 - 40*x^8 + 90*x^7 - 48*x^6 - 8*x^5 + 300*x^4 - 360*x^3 + 240*x^2 + 128*x - 592);
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 39916800 |
| The 56 conjugacy class representatives for $S_{11}$ |
| Character table for $S_{11}$ |
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 22 sibling: | 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 | R | R | R | ${\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.4.0.1}{4} }$ | ${\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.11.0.1}{11} }$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.11.0.1}{11} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.11.0.1}{11} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.4.10.4 | $x^{4} + 4 x^{3} + 8 x^{2} + 10$ | $4$ | $1$ | $10$ | $D_{4}$ | $[2, 3, 7/2]$ | |
| 2.4.4.5 | $x^{4} + 2 x + 2$ | $4$ | $1$ | $4$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ | |
|
\(3\)
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 3.6.7.2 | $x^{6} + 3 x^{2} + 6$ | $6$ | $1$ | $7$ | $D_{6}$ | $[3/2]_{2}^{2}$ | |
|
\(5\)
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 5.10.10.12 | $x^{10} - 30 x^{6} + 10 x^{5} + 25 x^{2} - 150 x + 25$ | $5$ | $2$ | $10$ | $(C_5^2 : C_4) : C_2$ | $[5/4, 5/4]_{4}^{2}$ | |
|
\(7\)
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.7.0.1 | $x^{7} + 6 x + 4$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |