Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 + 43*x^20 - 45*x^19 - 147*x^18 + 411*x^17 + 19*x^16 - 1070*x^15 + 605*x^14 + 1553*x^13 - 1312*x^12 - 1553*x^11 + 1432*x^10 + 1195*x^9 - 928*x^8 - 712*x^7 + 352*x^6 + 291*x^5 - 71*x^4 - 67*x^3 + 7*x^2 + 7*x - 1)
gp: K = bnfinit(y^22 - 11*y^21 + 43*y^20 - 45*y^19 - 147*y^18 + 411*y^17 + 19*y^16 - 1070*y^15 + 605*y^14 + 1553*y^13 - 1312*y^12 - 1553*y^11 + 1432*y^10 + 1195*y^9 - 928*y^8 - 712*y^7 + 352*y^6 + 291*y^5 - 71*y^4 - 67*y^3 + 7*y^2 + 7*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 11*x^21 + 43*x^20 - 45*x^19 - 147*x^18 + 411*x^17 + 19*x^16 - 1070*x^15 + 605*x^14 + 1553*x^13 - 1312*x^12 - 1553*x^11 + 1432*x^10 + 1195*x^9 - 928*x^8 - 712*x^7 + 352*x^6 + 291*x^5 - 71*x^4 - 67*x^3 + 7*x^2 + 7*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 11*x^21 + 43*x^20 - 45*x^19 - 147*x^18 + 411*x^17 + 19*x^16 - 1070*x^15 + 605*x^14 + 1553*x^13 - 1312*x^12 - 1553*x^11 + 1432*x^10 + 1195*x^9 - 928*x^8 - 712*x^7 + 352*x^6 + 291*x^5 - 71*x^4 - 67*x^3 + 7*x^2 + 7*x - 1)
\( x^{22} - 11 x^{21} + 43 x^{20} - 45 x^{19} - 147 x^{18} + 411 x^{17} + 19 x^{16} - 1070 x^{15} + \cdots - 1 \)
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: | $[10, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(70442637605798451973179258449\)
\(\medspace = 71^{2}\cdot 1282529\cdot 3300850529^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(20.48\) | 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: | $71^{1/2}1282529^{1/2}3300850529^{1/2}\approx 548246289.0851672$ | ||
| Ramified primes: |
\(71\), \(1282529\), \(3300850529\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{1282529}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Yes | |
| 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: | $15$ | 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: |
$a^{2}-a-1$, $a^{20}-10a^{19}+34a^{18}-21a^{17}-133a^{16}+248a^{15}+160a^{14}-666a^{13}-12a^{12}+1008a^{11}-134a^{10}-1030a^{9}+88a^{8}+724a^{7}+49a^{6}-318a^{5}-79a^{4}+65a^{3}+28a^{2}-2a-2$, $a^{14}-7a^{13}+12a^{12}+19a^{11}-67a^{10}-6a^{9}+138a^{8}-21a^{7}-158a^{6}+13a^{5}+105a^{4}+12a^{3}-32a^{2}-9a+3$, $a^{12}-6a^{11}+8a^{10}+15a^{9}-36a^{8}-12a^{7}+55a^{6}+6a^{5}-41a^{4}-5a^{3}+13a^{2}+2a-1$, $a^{20}-10a^{19}+34a^{18}-21a^{17}-134a^{16}+256a^{15}+141a^{14}-673a^{13}+73a^{12}+953a^{11}-286a^{10}-886a^{9}+260a^{8}+569a^{7}-100a^{6}-239a^{5}+11a^{4}+55a^{3}+a^{2}-5a+1$, $a$, $a^{21}-10a^{20}+33a^{19}-12a^{18}-159a^{17}+252a^{16}+271a^{15}-799a^{14}-194a^{13}+1359a^{12}+46a^{11}-1501a^{10}-77a^{9}+1103a^{8}+211a^{7}-489a^{6}-192a^{5}+93a^{4}+63a^{3}+a^{2}-5a$, $a^{18}-9a^{17}+26a^{16}-4a^{15}-111a^{14}+133a^{13}+182a^{12}-351a^{11}-181a^{10}+476a^{9}+161a^{8}-393a^{7}-144a^{6}+187a^{5}+92a^{4}-39a^{3}-26a^{2}+1$, $a^{21}-11a^{20}+43a^{19}-46a^{18}-138a^{17}+386a^{16}+15a^{15}-940a^{14}+480a^{13}+1279a^{12}-893a^{11}-1207a^{10}+753a^{9}+870a^{8}-273a^{7}-458a^{6}-24a^{5}+145a^{4}+43a^{3}-23a^{2}-8a+3$, $a^{21}-10a^{20}+33a^{19}-12a^{18}-159a^{17}+251a^{16}+278a^{15}-811a^{14}-213a^{13}+1425a^{12}+58a^{11}-1647a^{10}-71a^{9}+1296a^{8}+213a^{7}-647a^{6}-219a^{5}+161a^{4}+88a^{3}-6a^{2}-10a-2$, $a^{21}-10a^{20}+33a^{19}-11a^{18}-168a^{17}+278a^{16}+267a^{15}-911a^{14}-53a^{13}+1522a^{12}-310a^{11}-1607a^{10}+349a^{9}+1154a^{8}-81a^{7}-538a^{6}-82a^{5}+126a^{4}+47a^{3}-3a^{2}-5a-1$, $a^{20}-10a^{19}+34a^{18}-20a^{17}-143a^{16}+282a^{15}+137a^{14}-785a^{13}+213a^{12}+1122a^{11}-651a^{10}-1003a^{9}+723a^{8}+613a^{7}-438a^{6}-265a^{5}+148a^{4}+75a^{3}-25a^{2}-9a+2$, $a^{21}-10a^{20}+34a^{19}-20a^{18}-143a^{17}+282a^{16}+136a^{15}-777a^{14}+194a^{13}+1116a^{12}-571a^{11}-1055a^{10}+589a^{9}+741a^{8}-303a^{7}-398a^{6}+57a^{5}+143a^{4}+12a^{3}-25a^{2}-5a+2$, $a^{21}-11a^{20}+43a^{19}-46a^{18}-138a^{17}+386a^{16}+15a^{15}-940a^{14}+480a^{13}+1279a^{12}-893a^{11}-1208a^{10}+758a^{9}+866a^{8}-288a^{7}-436a^{6}-9a^{5}+114a^{4}+36a^{3}-4a^{2}-7a-1$, $a^{21}-10a^{20}+33a^{19}-11a^{18}-167a^{17}+269a^{16}+292a^{15}-907a^{14}-185a^{13}+1662a^{12}-69a^{11}-2039a^{10}+134a^{9}+1756a^{8}+48a^{7}-1020a^{6}-168a^{5}+351a^{4}+100a^{3}-59a^{2}-19a+4$
| 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: | \( 1426414.38682 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{6}\cdot 1426414.38682 \cdot 1}{2\cdot\sqrt{70442637605798451973179258449}}\cr\approx \mathstrut & 0.169307843120 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 + 43*x^20 - 45*x^19 - 147*x^18 + 411*x^17 + 19*x^16 - 1070*x^15 + 605*x^14 + 1553*x^13 - 1312*x^12 - 1553*x^11 + 1432*x^10 + 1195*x^9 - 928*x^8 - 712*x^7 + 352*x^6 + 291*x^5 - 71*x^4 - 67*x^3 + 7*x^2 + 7*x - 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 - 11*x^21 + 43*x^20 - 45*x^19 - 147*x^18 + 411*x^17 + 19*x^16 - 1070*x^15 + 605*x^14 + 1553*x^13 - 1312*x^12 - 1553*x^11 + 1432*x^10 + 1195*x^9 - 928*x^8 - 712*x^7 + 352*x^6 + 291*x^5 - 71*x^4 - 67*x^3 + 7*x^2 + 7*x - 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 - 11*x^21 + 43*x^20 - 45*x^19 - 147*x^18 + 411*x^17 + 19*x^16 - 1070*x^15 + 605*x^14 + 1553*x^13 - 1312*x^12 - 1553*x^11 + 1432*x^10 + 1195*x^9 - 928*x^8 - 712*x^7 + 352*x^6 + 291*x^5 - 71*x^4 - 67*x^3 + 7*x^2 + 7*x - 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 - 11*x^21 + 43*x^20 - 45*x^19 - 147*x^18 + 411*x^17 + 19*x^16 - 1070*x^15 + 605*x^14 + 1553*x^13 - 1312*x^12 - 1553*x^11 + 1432*x^10 + 1195*x^9 - 928*x^8 - 712*x^7 + 352*x^6 + 291*x^5 - 71*x^4 - 67*x^3 + 7*x^2 + 7*x - 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 $C_2^{10}.(C_2\times S_{11})$ |
| Character table for $C_2^{10}.(C_2\times S_{11})$ |
Intermediate fields
| 11.5.234360387559.1 |
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 | ${\href{/padicField/2.11.0.1}{11} }^{2}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}{,}\,{\href{/padicField/3.6.0.1}{6} }$ | ${\href{/padicField/5.11.0.1}{11} }^{2}$ | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}{,}\,{\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.5.0.1}{5} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.7.0.1}{7} }^{2}{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.4.0.1}{4} }^{3}$ | ${\href{/padicField/23.9.0.1}{9} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/31.6.0.1}{6} }$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.7.0.1}{7} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(71\)
| $\Q_{71}$ | $x + 64$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{71}$ | $x + 64$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 71.2.1.1 | $x^{2} + 497$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.1.1 | $x^{2} + 497$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 71.8.0.1 | $x^{8} + 53 x^{3} + 22 x^{2} + 19 x + 7$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
|
\(1282529\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
|
\(3300850529\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ |