Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 33*x^9 - 17537553)
gp: K = bnfinit(y^11 - 33*y^9 - 17537553, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^11 - 33*x^9 - 17537553);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^11 - 33*x^9 - 17537553)
\( x^{11} - 33x^{9} - 17537553 \)
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: | $[1, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-3802786437878517362765933727\)
\(\medspace = -\,3^{14}\cdot 11^{19}\cdot 13\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(321.57\) | 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: | $3^{31/18}11^{199/110}13^{1/2}\approx 1830.8574991422927$ | ||
| Ramified primes: |
\(3\), \(11\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-143}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $\frac{1}{3}a^{2}$, $\frac{1}{9}a^{3}+\frac{1}{3}a$, $\frac{1}{27}a^{4}+\frac{1}{9}a^{2}$, $\frac{1}{243}a^{5}-\frac{2}{81}a^{3}+\frac{1}{3}a$, $\frac{1}{729}a^{6}-\frac{2}{243}a^{4}+\frac{1}{9}a^{2}$, $\frac{1}{6561}a^{7}-\frac{2}{2187}a^{5}+\frac{1}{81}a^{4}-\frac{2}{81}a^{3}-\frac{2}{27}a^{2}+\frac{1}{3}a$, $\frac{1}{59049}a^{8}-\frac{11}{19683}a^{6}-\frac{1}{729}a^{5}+\frac{11}{243}a^{3}+\frac{1}{9}a^{2}+\frac{1}{3}$, $\frac{1}{177147}a^{9}-\frac{2}{59049}a^{7}-\frac{1}{2187}a^{6}-\frac{2}{2187}a^{5}-\frac{7}{729}a^{4}+\frac{1}{81}a^{3}+\frac{4}{27}a^{2}+\frac{1}{9}a$, $\frac{1}{531441}a^{10}+\frac{1}{177147}a^{8}+\frac{10}{19683}a^{6}-\frac{1}{729}a^{5}+\frac{2}{243}a^{4}+\frac{11}{243}a^{3}-\frac{4}{27}a^{2}+\frac{1}{3}$
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 $1$ (assuming GRH)
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: |
$\frac{332835674}{177147}a^{10}-\frac{4047305188}{177147}a^{9}+\frac{2639042252}{59049}a^{8}+\frac{29586032477}{59049}a^{7}-\frac{2820704968}{729}a^{6}+\frac{11089383802}{729}a^{5}-\frac{18543590063}{729}a^{4}-\frac{3314815417}{27}a^{3}+\frac{11327275850}{9}a^{2}-\frac{50177333767}{9}a+11161167370$, $\frac{4439519305}{531441}a^{10}+\frac{8308820006}{177147}a^{9}-\frac{10611527309}{177147}a^{8}-\frac{42773777575}{59049}a^{7}-\frac{92751546478}{19683}a^{6}-\frac{17475658606}{729}a^{5}-\frac{73949895920}{729}a^{4}-\frac{84748296053}{243}a^{3}-\frac{22119772621}{27}a^{2}+\frac{2530283414}{9}a+\frac{55950746162}{3}$, $\frac{11\!\cdots\!11}{531441}a^{10}-\frac{33\!\cdots\!54}{177147}a^{9}-\frac{23\!\cdots\!56}{177147}a^{8}-\frac{22\!\cdots\!19}{59049}a^{7}-\frac{10\!\cdots\!23}{2187}a^{6}+\frac{80\!\cdots\!44}{2187}a^{5}+\frac{29\!\cdots\!42}{729}a^{4}+\frac{20\!\cdots\!52}{81}a^{3}+\frac{34\!\cdots\!01}{27}a^{2}+\frac{47\!\cdots\!27}{9}a+17\!\cdots\!07$, $\frac{13\!\cdots\!34}{531441}a^{10}-\frac{483239505081575}{59049}a^{9}-\frac{13\!\cdots\!83}{177147}a^{8}+\frac{76\!\cdots\!45}{19683}a^{7}-\frac{19\!\cdots\!41}{19683}a^{6}-\frac{623424563002537}{729}a^{5}+\frac{19\!\cdots\!03}{81}a^{4}-\frac{31\!\cdots\!07}{243}a^{3}+\frac{10\!\cdots\!59}{27}a^{2}+\frac{8333380272253}{3}a-\frac{21\!\cdots\!74}{3}$, $\frac{533805033664}{531441}a^{10}-\frac{408400948457}{177147}a^{9}-\frac{6574720353638}{177147}a^{8}+\frac{8454158571157}{59049}a^{7}-\frac{4180768835845}{19683}a^{6}-\frac{2831209330868}{2187}a^{5}+\frac{8704261164296}{729}a^{4}-\frac{12298766706740}{243}a^{3}+\frac{871194025054}{9}a^{2}+\frac{2910441609154}{9}a-\frac{11332145416270}{3}$
| 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: | \( 6934535437.05 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{5}\cdot 6934535437.05 \cdot 1}{2\cdot\sqrt{3802786437878517362765933727}}\cr\approx \mathstrut & 1.10119884051 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^11 - 33*x^9 - 17537553)
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 - 33*x^9 - 17537553, 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 - 33*x^9 - 17537553);
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 - 33*x^9 - 17537553);
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 $S_{11}$ are not computed |
| Character table for $S_{11}$ is not computed |
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 | ${\href{/padicField/2.11.0.1}{11} }$ | R | ${\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | R | R | ${\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.5.0.1}{5} }$ | ${\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.11.0.1}{11} }$ | ${\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.3.4.3 | $x^{3} + 6 x^{2} + 12$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
| 3.6.9.15 | $x^{6} + 6 x^{5} + 6 x^{4} + 3$ | $6$ | $1$ | $9$ | $S_3\times C_3$ | $[3/2, 2]_{2}$ | |
|
\(11\)
| 11.11.19.5 | $x^{11} + 110 x^{9} + 11$ | $11$ | $1$ | $19$ | $F_{11}$ | $[19/10]_{10}$ |
|
\(13\)
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 13.9.0.1 | $x^{9} + 12 x^{4} + 8 x^{3} + 12 x^{2} + 12 x + 11$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |