Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 6*x^18 + 18*x^16 + 39*x^14 + 65*x^12 + 73*x^10 + 45*x^8 + 4*x^6 - 11*x^4 - 2*x^2 + 1)
gp: K = bnfinit(y^20 + 6*y^18 + 18*y^16 + 39*y^14 + 65*y^12 + 73*y^10 + 45*y^8 + 4*y^6 - 11*y^4 - 2*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + 6*x^18 + 18*x^16 + 39*x^14 + 65*x^12 + 73*x^10 + 45*x^8 + 4*x^6 - 11*x^4 - 2*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 + 6*x^18 + 18*x^16 + 39*x^14 + 65*x^12 + 73*x^10 + 45*x^8 + 4*x^6 - 11*x^4 - 2*x^2 + 1)
\( x^{20} + 6x^{18} + 18x^{16} + 39x^{14} + 65x^{12} + 73x^{10} + 45x^{8} + 4x^{6} - 11x^{4} - 2x^{2} + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(2948433986992424882176\)
\(\medspace = 2^{12}\cdot 41^{2}\cdot 4549^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.84\) | 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: | $2^{7/4}41^{1/2}4549^{1/2}\approx 1452.6212370873382$ | ||
| Ramified primes: |
\(2\), \(41\), \(4549\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{3358}a^{18}-\frac{17}{73}a^{16}-\frac{1}{2}a^{15}+\frac{41}{3358}a^{14}-\frac{8}{73}a^{12}-\frac{209}{1679}a^{10}+\frac{373}{3358}a^{8}+\frac{1625}{3358}a^{6}-\frac{1}{2}a^{5}+\frac{581}{3358}a^{4}-\frac{1}{2}a^{3}-\frac{1151}{3358}a^{2}+\frac{163}{1679}$, $\frac{1}{3358}a^{19}-\frac{17}{73}a^{17}+\frac{41}{3358}a^{15}+\frac{57}{146}a^{13}-\frac{1}{2}a^{12}-\frac{209}{1679}a^{11}+\frac{373}{3358}a^{9}+\frac{1625}{3358}a^{7}+\frac{581}{3358}a^{5}+\frac{264}{1679}a^{3}-\frac{1}{2}a^{2}-\frac{1353}{3358}a-\frac{1}{2}$
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 $1$
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: | $9$ | 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: |
$\frac{1157}{1679}a^{19}+\frac{301}{73}a^{17}+\frac{20573}{1679}a^{15}+\frac{1928}{73}a^{13}+\frac{73802}{1679}a^{11}+\frac{82329}{1679}a^{9}+\frac{50015}{1679}a^{7}+\frac{5654}{1679}a^{5}-\frac{10334}{1679}a^{3}-\frac{2272}{1679}a$, $\frac{1632}{1679}a^{19}+\frac{430}{73}a^{17}+\frac{29974}{1679}a^{15}+\frac{2869}{73}a^{13}+\frac{111991}{1679}a^{11}+\frac{130221}{1679}a^{9}+\frac{89846}{1679}a^{7}+\frac{23063}{1679}a^{5}-\frac{9705}{1679}a^{3}-\frac{1890}{1679}a$, $\frac{1224}{1679}a^{18}+\frac{286}{73}a^{16}+\frac{18283}{1679}a^{14}+\frac{1659}{73}a^{12}+\frac{60907}{1679}a^{10}+\frac{61987}{1679}a^{8}+\frac{32965}{1679}a^{6}+\frac{927}{1679}a^{4}-\frac{6859}{1679}a^{2}-\frac{578}{1679}$, $a^{19}+6a^{17}+18a^{15}+39a^{13}+65a^{11}+73a^{9}+45a^{7}+4a^{5}-11a^{3}-2a$, $\frac{578}{1679}a^{19}+\frac{204}{73}a^{17}+\frac{16982}{1679}a^{15}+\frac{1775}{73}a^{13}+\frac{75727}{1679}a^{11}+\frac{103101}{1679}a^{9}+\frac{87997}{1679}a^{7}+\frac{35277}{1679}a^{5}-\frac{5431}{1679}a^{3}-\frac{6336}{1679}a$, $\frac{641}{1679}a^{19}+\frac{853}{1679}a^{18}+\frac{179}{73}a^{17}+\frac{469}{146}a^{16}+\frac{12849}{1679}a^{15}+\frac{16504}{1679}a^{14}+\frac{2483}{146}a^{13}+\frac{3145}{146}a^{12}+\frac{49393}{1679}a^{11}+\frac{61517}{1679}a^{10}+\frac{59440}{1679}a^{9}+\frac{71356}{1679}a^{8}+\frac{42620}{1679}a^{7}+\frac{94245}{3358}a^{6}+\frac{13115}{1679}a^{5}+\frac{19045}{3358}a^{4}-\frac{3099}{3358}a^{3}-\frac{14287}{3358}a^{2}-\frac{139}{3358}a-\frac{2951}{3358}$, $\frac{577}{1679}a^{19}-\frac{187}{3358}a^{18}+\frac{257}{146}a^{17}-\frac{33}{73}a^{16}+\frac{15413}{3358}a^{15}-\frac{2994}{1679}a^{14}+\frac{1319}{146}a^{13}-\frac{329}{73}a^{12}+\frac{22417}{1679}a^{11}-\frac{14645}{1679}a^{10}+\frac{18778}{1679}a^{9}-\frac{42887}{3358}a^{8}+\frac{6523}{3358}a^{7}-\frac{41951}{3358}a^{6}-\frac{7279}{1679}a^{5}-\frac{11509}{1679}a^{4}-\frac{2601}{1679}a^{3}-\frac{2356}{1679}a^{2}+\frac{5145}{3358}a+\frac{1420}{1679}$, $\frac{1147}{1679}a^{19}+\frac{633}{3358}a^{18}+\frac{625}{146}a^{17}+\frac{159}{146}a^{16}+\frac{45363}{3358}a^{15}+\frac{5421}{1679}a^{14}+\frac{2234}{73}a^{13}+\frac{1041}{146}a^{12}+\frac{89735}{1679}a^{11}+\frac{20492}{1679}a^{10}+\frac{110500}{1679}a^{9}+\frac{48061}{3358}a^{8}+\frac{169949}{3358}a^{7}+\frac{18168}{1679}a^{6}+\frac{30066}{1679}a^{5}+\frac{15183}{3358}a^{4}-\frac{9401}{3358}a^{3}+\frac{103}{3358}a^{2}-\frac{5532}{1679}a-\frac{3517}{3358}$, $\frac{1222}{1679}a^{19}-\frac{187}{3358}a^{18}+\frac{635}{146}a^{17}-\frac{33}{73}a^{16}+\frac{44797}{3358}a^{15}-\frac{2994}{1679}a^{14}+\frac{4331}{146}a^{13}-\frac{329}{73}a^{12}+\frac{85249}{1679}a^{11}-\frac{14645}{1679}a^{10}+\frac{101537}{1679}a^{9}-\frac{42887}{3358}a^{8}+\frac{148417}{3358}a^{7}-\frac{41951}{3358}a^{6}+\frac{23271}{1679}a^{5}-\frac{11509}{1679}a^{4}-\frac{4557}{1679}a^{3}-\frac{2356}{1679}a^{2}-\frac{7497}{3358}a+\frac{1420}{1679}$
| 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: | \( 198.57668435 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 198.57668435 \cdot 1}{2\cdot\sqrt{2948433986992424882176}}\cr\approx \mathstrut & 0.17534817030 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 + 6*x^18 + 18*x^16 + 39*x^14 + 65*x^12 + 73*x^10 + 45*x^8 + 4*x^6 - 11*x^4 - 2*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^20 + 6*x^18 + 18*x^16 + 39*x^14 + 65*x^12 + 73*x^10 + 45*x^8 + 4*x^6 - 11*x^4 - 2*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^20 + 6*x^18 + 18*x^16 + 39*x^14 + 65*x^12 + 73*x^10 + 45*x^8 + 4*x^6 - 11*x^4 - 2*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^20 + 6*x^18 + 18*x^16 + 39*x^14 + 65*x^12 + 73*x^10 + 45*x^8 + 4*x^6 - 11*x^4 - 2*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
$C_2^{10}.S_5$ (as 20T799):
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 122880 |
| The 252 conjugacy class representatives for $C_2^{10}.S_5$ |
| Character table for $C_2^{10}.S_5$ |
Intermediate fields
| 5.1.4549.1, 10.0.1324377664.1, 10.0.54299484224.2, 10.2.848429441.1 |
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 20 siblings: | data not computed |
| Degree 40 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 | ${\href{/padicField/3.10.0.1}{10} }^{2}$ | ${\href{/padicField/5.5.0.1}{5} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 2.12.12.23 | $x^{12} + 56 x^{10} - 152 x^{9} - 764 x^{8} + 2976 x^{7} - 960 x^{6} - 11008 x^{5} + 20336 x^{4} - 14976 x^{3} + 80768 x^{2} + 6016 x + 38848$ | $2$ | $6$ | $12$ | $C_2^2 \times A_4$ | $[2, 2, 2]^{6}$ | |
|
\(41\)
| 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.1.2 | $x^{2} + 123$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 41.2.1.2 | $x^{2} + 123$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 41.6.0.1 | $x^{6} + 4 x^{4} + 33 x^{3} + 39 x^{2} + 6 x + 6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 41.6.0.1 | $x^{6} + 4 x^{4} + 33 x^{3} + 39 x^{2} + 6 x + 6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(4549\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |