Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 484*x^7 + 58564*x^3 - 1362944)
gp: K = bnfinit(y^11 - 484*y^7 + 58564*y^3 - 1362944, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^11 - 484*x^7 + 58564*x^3 - 1362944);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^11 - 484*x^7 + 58564*x^3 - 1362944)
\( x^{11} - 484x^{7} + 58564x^{3} - 1362944 \)
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: |
\(-75125419892107285091969606352896\)
\(\medspace = -\,2^{33}\cdot 11^{20}\cdot 13\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(790.31\) | 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: | $2^{97/24}11^{20/11}13^{1/2}\approx 4645.9637941866895$ | ||
| Ramified primes: |
\(2\), \(11\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-26}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{22}a^{4}$, $\frac{1}{88}a^{5}+\frac{1}{4}a$, $\frac{1}{352}a^{6}+\frac{5}{16}a^{2}$, $\frac{1}{352}a^{7}+\frac{5}{16}a^{3}$, $\frac{1}{3872}a^{8}-\frac{3}{176}a^{4}$, $\frac{1}{3872}a^{9}-\frac{1}{176}a^{5}+\frac{1}{4}a$, $\frac{1}{77440}a^{10}+\frac{1}{19360}a^{9}-\frac{1}{19360}a^{8}+\frac{1}{1760}a^{7}-\frac{1}{880}a^{6}+\frac{1}{880}a^{5}+\frac{19}{880}a^{4}+\frac{21}{80}a^{3}+\frac{19}{160}a^{2}+\frac{1}{10}a+\frac{2}{5}$
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: |
$\frac{24\!\cdots\!57}{19360}a^{10}+\frac{17\!\cdots\!69}{9680}a^{9}-\frac{15\!\cdots\!99}{9680}a^{8}-\frac{23\!\cdots\!21}{5}a^{7}-\frac{96\!\cdots\!17}{220}a^{6}+\frac{20\!\cdots\!59}{440}a^{5}+\frac{25\!\cdots\!71}{40}a^{4}+\frac{38\!\cdots\!73}{5}a^{3}-\frac{39\!\cdots\!37}{40}a^{2}-\frac{11\!\cdots\!87}{10}a-\frac{59\!\cdots\!49}{5}$, $\frac{90\!\cdots\!89}{38720}a^{10}+\frac{84\!\cdots\!57}{4840}a^{9}+\frac{79\!\cdots\!11}{9680}a^{8}+\frac{25\!\cdots\!89}{80}a^{7}+\frac{14\!\cdots\!47}{880}a^{6}-\frac{20\!\cdots\!71}{440}a^{5}-\frac{11\!\cdots\!59}{40}a^{4}-\frac{49\!\cdots\!81}{40}a^{3}-\frac{26\!\cdots\!59}{80}a^{2}-\frac{12\!\cdots\!29}{20}a-\frac{23\!\cdots\!49}{5}$, $\frac{64\!\cdots\!19}{1210}a^{10}+\frac{10\!\cdots\!43}{9680}a^{9}+\frac{22\!\cdots\!99}{19360}a^{8}-\frac{28\!\cdots\!57}{880}a^{7}-\frac{50\!\cdots\!87}{1760}a^{6}-\frac{29\!\cdots\!47}{440}a^{5}-\frac{95\!\cdots\!51}{80}a^{4}-\frac{28\!\cdots\!97}{40}a^{3}+\frac{30\!\cdots\!77}{80}a^{2}+\frac{11\!\cdots\!71}{10}a+\frac{15\!\cdots\!77}{5}$, $\frac{11\!\cdots\!19}{4840}a^{10}+\frac{20\!\cdots\!79}{19360}a^{9}+\frac{22\!\cdots\!19}{4840}a^{8}+\frac{35\!\cdots\!49}{1760}a^{7}-\frac{30\!\cdots\!63}{110}a^{6}-\frac{10\!\cdots\!91}{880}a^{5}-\frac{10\!\cdots\!91}{20}a^{4}-\frac{18\!\cdots\!91}{80}a^{3}+\frac{38\!\cdots\!71}{10}a^{2}+\frac{34\!\cdots\!43}{20}a+\frac{37\!\cdots\!13}{5}$, $\frac{17\!\cdots\!23}{38720}a^{10}-\frac{68\!\cdots\!51}{4840}a^{9}+\frac{26\!\cdots\!03}{2420}a^{8}+\frac{56\!\cdots\!23}{880}a^{7}-\frac{10\!\cdots\!43}{440}a^{6}+\frac{39\!\cdots\!71}{55}a^{5}-\frac{44\!\cdots\!27}{10}a^{4}-\frac{14\!\cdots\!37}{40}a^{3}+\frac{32\!\cdots\!77}{80}a^{2}-\frac{98\!\cdots\!09}{10}a+\frac{87\!\cdots\!87}{5}$
| 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: | \( 1430081294600 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{5}\cdot 1430081294600 \cdot 1}{2\cdot\sqrt{75125419892107285091969606352896}}\cr\approx \mathstrut & 1.61572190055419 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^11 - 484*x^7 + 58564*x^3 - 1362944)
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 - 484*x^7 + 58564*x^3 - 1362944, 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 - 484*x^7 + 58564*x^3 - 1362944);
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 - 484*x^7 + 58564*x^3 - 1362944);
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 | ${\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | R | R | ${\href{/padicField/17.11.0.1}{11} }$ | ${\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.11.0.1}{11} }$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.8.31.180 | $x^{8} + 16 x^{7} + 12 x^{4} + 8 x^{2} + 16 x + 18$ | $8$ | $1$ | $31$ | $D_{8}$ | $[2, 3, 4, 5]$ | |
|
\(11\)
| 11.11.20.20 | $x^{11} + 55 x^{10} + 11$ | $11$ | $1$ | $20$ | $F_{11}$ | $[2]^{10}$ |
|
\(13\)
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 13.9.0.1 | $x^{9} + 12 x^{4} + 8 x^{3} + 12 x^{2} + 12 x + 11$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |