Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 6*x^20 + 16*x^18 + 18*x^16 + 4*x^14 + 6*x^12 + 25*x^10 + 3*x^8 - 19*x^6 - 4*x^4 + 4*x^2 + 1)
gp: K = bnfinit(y^22 + 6*y^20 + 16*y^18 + 18*y^16 + 4*y^14 + 6*y^12 + 25*y^10 + 3*y^8 - 19*y^6 - 4*y^4 + 4*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + 6*x^20 + 16*x^18 + 18*x^16 + 4*x^14 + 6*x^12 + 25*x^10 + 3*x^8 - 19*x^6 - 4*x^4 + 4*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 6*x^20 + 16*x^18 + 18*x^16 + 4*x^14 + 6*x^12 + 25*x^10 + 3*x^8 - 19*x^6 - 4*x^4 + 4*x^2 + 1)
\( x^{22} + 6x^{20} + 16x^{18} + 18x^{16} + 4x^{14} + 6x^{12} + 25x^{10} + 3x^{8} - 19x^{6} - 4x^{4} + 4x^{2} + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-2613825179875044875466440704\)
\(\medspace = -\,2^{22}\cdot 971^{2}\cdot 25709231^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(17.63\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(971\), \(25709231\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{181121}a^{20}-\frac{52104}{181121}a^{18}-\frac{45455}{181121}a^{16}-\frac{40370}{181121}a^{14}-\frac{39711}{181121}a^{12}+\frac{32791}{181121}a^{10}-\frac{43471}{181121}a^{8}-\frac{6534}{181121}a^{6}-\frac{20759}{181121}a^{4}-\frac{84247}{181121}a^{2}-\frac{80745}{181121}$, $\frac{1}{181121}a^{21}-\frac{52104}{181121}a^{19}-\frac{45455}{181121}a^{17}-\frac{40370}{181121}a^{15}-\frac{39711}{181121}a^{13}+\frac{32791}{181121}a^{11}-\frac{43471}{181121}a^{9}-\frac{6534}{181121}a^{7}-\frac{20759}{181121}a^{5}-\frac{84247}{181121}a^{3}-\frac{80745}{181121}a$
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: | $10$ | 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: |
$\frac{25213}{181121}a^{20}+\frac{153582}{181121}a^{18}+\frac{439015}{181121}a^{16}+\frac{594573}{181121}a^{14}+\frac{365687}{181121}a^{12}+\frac{304360}{181121}a^{10}+\frac{653332}{181121}a^{8}+\frac{440610}{181121}a^{6}-\frac{138098}{181121}a^{4}-\frac{294765}{181121}a^{2}+\frac{157476}{181121}$, $\frac{251562}{181121}a^{20}+\frac{1427048}{181121}a^{18}+\frac{3583803}{181121}a^{16}+\frac{3518950}{181121}a^{14}+\frac{312415}{181121}a^{12}+\frac{1987049}{181121}a^{10}+\frac{5867908}{181121}a^{8}-\frac{938638}{181121}a^{6}-\frac{3717306}{181121}a^{4}+\frac{530001}{181121}a^{2}+\frac{527581}{181121}$, $a$, $\frac{249258}{181121}a^{20}+\frac{1391441}{181121}a^{18}+\frac{3443064}{181121}a^{16}+\frac{3254115}{181121}a^{14}+\frac{159333}{181121}a^{12}+\frac{1964042}{181121}a^{10}+\frac{5502937}{181121}a^{8}-\frac{1279587}{181121}a^{6}-\frac{3523393}{181121}a^{4}+\frac{292256}{181121}a^{2}+\frac{552794}{181121}$, $\frac{295083}{181121}a^{20}+\frac{1624905}{181121}a^{18}+\frac{3902552}{181121}a^{16}+\frac{3268759}{181121}a^{14}-\frac{780160}{181121}a^{12}+\frac{1669559}{181121}a^{10}+\frac{6318725}{181121}a^{8}-\frac{2393850}{181121}a^{6}-\frac{5006044}{181121}a^{4}+\frac{992080}{181121}a^{2}+\frac{896320}{181121}$, $\frac{383825}{181121}a^{21}+\frac{2193109}{181121}a^{19}+\frac{5510822}{181121}a^{17}+\frac{5319930}{181121}a^{15}-\frac{17941}{181121}a^{13}+\frac{2261858}{181121}a^{11}+\frac{8847116}{181121}a^{9}-\frac{1560152}{181121}a^{7}-\frac{7011862}{181121}a^{5}+\frac{332960}{181121}a^{3}+\frac{1294774}{181121}a$, $\frac{193767}{181121}a^{21}-\frac{170831}{181121}a^{20}+\frac{1097740}{181121}a^{19}-\frac{937605}{181121}a^{18}+\frac{2770939}{181121}a^{17}-\frac{2250980}{181121}a^{16}+\frac{2777894}{181121}a^{15}-\frac{1908057}{181121}a^{14}+\frac{425469}{181121}a^{13}+\frac{345028}{181121}a^{12}+\frac{1719106}{181121}a^{11}-\frac{1095759}{181121}a^{10}+\frac{4677115}{181121}a^{9}-\frac{3570140}{181121}a^{8}-\frac{400030}{181121}a^{7}+\frac{1409999}{181121}a^{6}-\frac{2609679}{181121}a^{5}+\frac{2467243}{181121}a^{4}+\frac{147281}{181121}a^{3}-\frac{599887}{181121}a^{2}+\frac{421170}{181121}a-\frac{426265}{181121}$, $\frac{212317}{181121}a^{21}+\frac{87823}{181121}a^{20}+\frac{1211317}{181121}a^{19}+\frac{454715}{181121}a^{18}+\frac{3061186}{181121}a^{17}+\frac{999101}{181121}a^{16}+\frac{3030850}{181121}a^{15}+\frac{572428}{181121}a^{14}+\frac{224405}{181121}a^{13}-\frac{597661}{181121}a^{12}+\frac{1425596}{181121}a^{11}+\frac{523456}{181121}a^{10}+\frac{4821878}{181121}a^{9}+\frac{1726015}{181121}a^{8}-\frac{798023}{181121}a^{7}-\frac{1312001}{181121}a^{6}-\frac{3531488}{181121}a^{5}-\frac{1221518}{181121}a^{4}+\frac{77419}{181121}a^{3}+\frac{693053}{181121}a^{2}+\frac{472090}{181121}a+\frac{343499}{181121}$, $\frac{42440}{181121}a^{21}+\frac{137358}{181121}a^{20}+\frac{193650}{181121}a^{19}+\frac{819567}{181121}a^{18}+\frac{371813}{181121}a^{17}+\frac{2168674}{181121}a^{16}+\frac{101860}{181121}a^{15}+\frac{2412649}{181121}a^{14}-\frac{185056}{181121}a^{13}+\frac{559861}{181121}a^{12}+\frac{640760}{181121}a^{11}+\frac{1075876}{181121}a^{10}+\frac{713750}{181121}a^{9}+\frac{3729930}{181121}a^{8}-\frac{731193}{181121}a^{7}+\frac{500746}{181121}a^{6}+\frac{322826}{181121}a^{5}-\frac{2381392}{181121}a^{4}+\frac{429223}{181121}a^{3}-\frac{178736}{181121}a^{2}-\frac{189601}{181121}a+\frac{516088}{181121}$, $\frac{153362}{181121}a^{21}+\frac{21341}{181121}a^{20}+\frac{828235}{181121}a^{19}+\frac{131476}{181121}a^{18}+\frac{1907669}{181121}a^{17}+\frac{391163}{181121}a^{16}+\frac{1303050}{181121}a^{15}+\frac{599790}{181121}a^{14}-\frac{1051483}{181121}a^{13}+\frac{536071}{181121}a^{12}+\frac{431019}{181121}a^{11}+\frac{484550}{181121}a^{10}+\frac{3162444}{181121}a^{9}+\frac{530514}{181121}a^{8}-\frac{1736025}{181121}a^{7}+\frac{383318}{181121}a^{6}-\frac{3156998}{181121}a^{5}+\frac{185268}{181121}a^{4}+\frac{883726}{181121}a^{3}-\frac{470423}{181121}a^{2}+\frac{1114806}{181121}a-\frac{356093}{181121}$
| 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: | \( 24986.318925 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 24986.318925 \cdot 1}{2\cdot\sqrt{2613825179875044875466440704}}\cr\approx \mathstrut & 0.14723551054 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 + 6*x^20 + 16*x^18 + 18*x^16 + 4*x^14 + 6*x^12 + 25*x^10 + 3*x^8 - 19*x^6 - 4*x^4 + 4*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^22 + 6*x^20 + 16*x^18 + 18*x^16 + 4*x^14 + 6*x^12 + 25*x^10 + 3*x^8 - 19*x^6 - 4*x^4 + 4*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^22 + 6*x^20 + 16*x^18 + 18*x^16 + 4*x^14 + 6*x^12 + 25*x^10 + 3*x^8 - 19*x^6 - 4*x^4 + 4*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^22 + 6*x^20 + 16*x^18 + 18*x^16 + 4*x^14 + 6*x^12 + 25*x^10 + 3*x^8 - 19*x^6 - 4*x^4 + 4*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
$C_2^{10}.(C_2\times S_{11})$ (as 22T53):
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 81749606400 |
| The 752 conjugacy class representatives for $C_2^{10}.(C_2\times S_{11})$ are not computed |
| Character table for $C_2^{10}.(C_2\times S_{11})$ is not computed |
Intermediate fields
| 11.3.24963663301.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
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 |
| Degree 44 siblings: | 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 | $22$ | ${\href{/padicField/5.11.0.1}{11} }^{2}$ | $16{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | $22$ | $16{,}\,{\href{/padicField/13.6.0.1}{6} }$ | $18{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.4.0.1}{4} }$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.7.0.1}{7} }^{2}{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.7.0.1}{7} }^{2}{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.9.0.1}{9} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | $18{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.10.0.1}{10} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.10.10.1 | $x^{10} + 4 x^{9} + 6 x^{8} + 80 x^{7} + 616 x^{6} + 2352 x^{5} + 6000 x^{4} + 11136 x^{3} + 13776 x^{2} + 9472 x + 2784$ | $2$ | $5$ | $10$ | $C_2^4 : C_5$ | $[2, 2, 2, 2]^{5}$ |
| 2.12.12.7 | $x^{12} + 2 x^{11} + 8 x^{10} + 108 x^{9} + 452 x^{8} + 1328 x^{7} + 5952 x^{6} + 25760 x^{5} + 72688 x^{4} + 129184 x^{3} + 143232 x^{2} + 92096 x + 26560$ | $2$ | $6$ | $12$ | 12T134 | $[2, 2, 2, 2, 2, 2]^{6}$ | |
|
\(971\)
| $\Q_{971}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{971}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
|
\(25709231\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |