Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 10*x^18 - 17*x^17 + 26*x^16 - 29*x^15 + 24*x^14 - 5*x^13 - 7*x^12 + 20*x^11 - 14*x^10 + 8*x^9 + 12*x^8 + x^7 - 6*x^6 + 9*x^5 + 9*x^4 - 2*x^3 - x^2 + x + 1)
gp: K = bnfinit(y^20 - 4*y^19 + 10*y^18 - 17*y^17 + 26*y^16 - 29*y^15 + 24*y^14 - 5*y^13 - 7*y^12 + 20*y^11 - 14*y^10 + 8*y^9 + 12*y^8 + y^7 - 6*y^6 + 9*y^5 + 9*y^4 - 2*y^3 - y^2 + y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 4*x^19 + 10*x^18 - 17*x^17 + 26*x^16 - 29*x^15 + 24*x^14 - 5*x^13 - 7*x^12 + 20*x^11 - 14*x^10 + 8*x^9 + 12*x^8 + x^7 - 6*x^6 + 9*x^5 + 9*x^4 - 2*x^3 - x^2 + x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 4*x^19 + 10*x^18 - 17*x^17 + 26*x^16 - 29*x^15 + 24*x^14 - 5*x^13 - 7*x^12 + 20*x^11 - 14*x^10 + 8*x^9 + 12*x^8 + x^7 - 6*x^6 + 9*x^5 + 9*x^4 - 2*x^3 - x^2 + x + 1)
\( x^{20} - 4 x^{19} + 10 x^{18} - 17 x^{17} + 26 x^{16} - 29 x^{15} + 24 x^{14} - 5 x^{13} - 7 x^{12} + \cdots + 1 \)
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: |
\(1083064869622628748241\)
\(\medspace = 37^{4}\cdot 4903^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.27\) | 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: | $37^{1/2}4903^{1/2}\approx 425.92370208759223$ | ||
| Ramified primes: |
\(37\), \(4903\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{1722023}a^{19}+\frac{166001}{1722023}a^{18}-\frac{538054}{1722023}a^{17}-\frac{43300}{1722023}a^{16}-\frac{292472}{1722023}a^{15}+\frac{624096}{1722023}a^{14}-\frac{735268}{1722023}a^{13}+\frac{547918}{1722023}a^{12}-\frac{127277}{1722023}a^{11}+\frac{603845}{1722023}a^{10}+\frac{608358}{1722023}a^{9}+\frac{708940}{1722023}a^{8}-\frac{633177}{1722023}a^{7}+\frac{14013}{1722023}a^{6}-\frac{225014}{1722023}a^{5}+\frac{673855}{1722023}a^{4}+\frac{685204}{1722023}a^{3}+\frac{782776}{1722023}a^{2}-\frac{847724}{1722023}a+\frac{740987}{1722023}$
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: |
$\frac{207632}{1722023}a^{19}-\frac{892736}{1722023}a^{18}+\frac{2458043}{1722023}a^{17}-\frac{4949586}{1722023}a^{16}+\frac{9204906}{1722023}a^{15}-\frac{13706262}{1722023}a^{14}+\frac{18003919}{1722023}a^{13}-\frac{19081572}{1722023}a^{12}+\frac{20051170}{1722023}a^{11}-\frac{15003751}{1722023}a^{10}+\frac{11089298}{1722023}a^{9}-\frac{3340006}{1722023}a^{8}+\frac{3483117}{1722023}a^{7}+\frac{4494415}{1722023}a^{6}-\frac{1622858}{1722023}a^{5}+\frac{2936656}{1722023}a^{4}+\frac{1902737}{1722023}a^{3}+\frac{4815692}{1722023}a^{2}-\frac{1492669}{1722023}a+\frac{189872}{1722023}$, $\frac{403147}{1722023}a^{19}-\frac{1896725}{1722023}a^{18}+\frac{4937326}{1722023}a^{17}-\frac{8728064}{1722023}a^{16}+\frac{13203633}{1722023}a^{15}-\frac{16126602}{1722023}a^{14}+\frac{14838916}{1722023}a^{13}-\frac{7890471}{1722023}a^{12}+\frac{3222658}{1722023}a^{11}-\frac{2369272}{1722023}a^{10}+\frac{5464943}{1722023}a^{9}-\frac{9177291}{1722023}a^{8}+\frac{12725547}{1722023}a^{7}-\frac{4102598}{1722023}a^{6}-\frac{9601579}{1722023}a^{5}+\frac{10049389}{1722023}a^{4}+\frac{1339466}{1722023}a^{3}-\frac{4138908}{1722023}a^{2}-\frac{1258802}{1722023}a+\frac{2190210}{1722023}$, $\frac{600097}{1722023}a^{19}-\frac{2328453}{1722023}a^{18}+\frac{5053400}{1722023}a^{17}-\frac{7483145}{1722023}a^{16}+\frac{9068537}{1722023}a^{15}-\frac{6366981}{1722023}a^{14}-\frac{3333775}{1722023}a^{13}+\frac{16374633}{1722023}a^{12}-\frac{24046049}{1722023}a^{11}+\frac{22659374}{1722023}a^{10}-\frac{14007527}{1722023}a^{9}-\frac{1625085}{1722023}a^{8}+\frac{10532965}{1722023}a^{7}-\frac{4645117}{1722023}a^{6}-\frac{6402928}{1722023}a^{5}+\frac{2590937}{1722023}a^{4}+\frac{5934871}{1722023}a^{3}-\frac{3958829}{1722023}a^{2}-\frac{38614}{1722023}a-\frac{147367}{1722023}$, $\frac{1150634}{1722023}a^{19}-\frac{3840572}{1722023}a^{18}+\frac{8814862}{1722023}a^{17}-\frac{12936925}{1722023}a^{16}+\frac{18781803}{1722023}a^{15}-\frac{15398642}{1722023}a^{14}+\frac{7262011}{1722023}a^{13}+\frac{13571620}{1722023}a^{12}-\frac{17017813}{1722023}a^{11}+\frac{26856012}{1722023}a^{10}-\frac{8394574}{1722023}a^{9}+\frac{3006791}{1722023}a^{8}+\frac{20892921}{1722023}a^{7}+\frac{3976939}{1722023}a^{6}-\frac{878803}{1722023}a^{5}+\frac{7564159}{1722023}a^{4}+\frac{11453062}{1722023}a^{3}+\frac{3492087}{1722023}a^{2}-\frac{792942}{1722023}a+\frac{252044}{1722023}$, $\frac{147367}{1722023}a^{19}+\frac{10629}{1722023}a^{18}-\frac{854783}{1722023}a^{17}+\frac{2548161}{1722023}a^{16}-\frac{3651603}{1722023}a^{15}+\frac{4794894}{1722023}a^{14}-\frac{2830173}{1722023}a^{13}-\frac{4070610}{1722023}a^{12}+\frac{15343064}{1722023}a^{11}-\frac{21098709}{1722023}a^{10}+\frac{20596236}{1722023}a^{9}-\frac{12828591}{1722023}a^{8}+\frac{143319}{1722023}a^{7}+\frac{10680332}{1722023}a^{6}-\frac{5529319}{1722023}a^{5}-\frac{5076625}{1722023}a^{4}+\frac{3917240}{1722023}a^{3}+\frac{5640137}{1722023}a^{2}-\frac{4106196}{1722023}a+\frac{108753}{1722023}$, $\frac{317006}{1722023}a^{19}-\frac{1709874}{1722023}a^{18}+\frac{5197895}{1722023}a^{17}-\frac{10446605}{1722023}a^{16}+\frac{17281741}{1722023}a^{15}-\frac{23432193}{1722023}a^{14}+\frac{25885902}{1722023}a^{13}-\frac{19220663}{1722023}a^{12}+\frac{9758366}{1722023}a^{11}-\frac{1032656}{1722023}a^{10}-\frac{1385691}{1722023}a^{9}+\frac{2177979}{1722023}a^{8}+\frac{4980910}{1722023}a^{7}-\frac{4058308}{1722023}a^{6}-\frac{4595424}{1722023}a^{5}+\frac{11179141}{1722023}a^{4}-\frac{479973}{1722023}a^{3}-\frac{5713736}{1722023}a^{2}+\frac{148967}{1722023}a+\frac{3055584}{1722023}$, $\frac{1408178}{1722023}a^{19}-\frac{5666072}{1722023}a^{18}+\frac{14314388}{1722023}a^{17}-\frac{24825338}{1722023}a^{16}+\frac{38045354}{1722023}a^{15}-\frac{42675583}{1722023}a^{14}+\frac{34933805}{1722023}a^{13}-\frac{6996022}{1722023}a^{12}-\frac{14293650}{1722023}a^{11}+\frac{34503654}{1722023}a^{10}-\frac{26646730}{1722023}a^{9}+\frac{12205622}{1722023}a^{8}+\frac{18645641}{1722023}a^{7}-\frac{6751335}{1722023}a^{6}-\frac{2366423}{1722023}a^{5}+\frac{11120362}{1722023}a^{4}+\frac{6992975}{1722023}a^{3}-\frac{1366471}{1722023}a^{2}+\frac{1385280}{1722023}a+\frac{2419112}{1722023}$, $\frac{15421}{1722023}a^{19}-\frac{746780}{1722023}a^{18}+\frac{2820126}{1722023}a^{17}-\frac{6472468}{1722023}a^{16}+\frac{10099663}{1722023}a^{15}-\frac{13978315}{1722023}a^{14}+\frac{13007788}{1722023}a^{13}-\frac{5689452}{1722023}a^{12}-\frac{8242512}{1722023}a^{11}+\frac{14691568}{1722023}a^{10}-\frac{17312816}{1722023}a^{9}+\frac{8049828}{1722023}a^{8}+\frac{3091939}{1722023}a^{7}-\frac{14656609}{1722023}a^{6}+\frac{3379497}{1722023}a^{5}+\frac{5997242}{1722023}a^{4}-\frac{4968313}{1722023}a^{3}-\frac{5358603}{1722023}a^{2}+\frac{2568835}{1722023}a+\frac{1137922}{1722023}$, $\frac{448229}{1722023}a^{19}-\frac{2151601}{1722023}a^{18}+\frac{6524899}{1722023}a^{17}-\frac{13170651}{1722023}a^{16}+\frac{22121155}{1722023}a^{15}-\frac{28818688}{1722023}a^{14}+\frac{30205874}{1722023}a^{13}-\frac{17681245}{1722023}a^{12}-\frac{342466}{1722023}a^{11}+\frac{20817733}{1722023}a^{10}-\frac{27918459}{1722023}a^{9}+\frac{26671392}{1722023}a^{8}-\frac{8570995}{1722023}a^{7}-\frac{2628950}{1722023}a^{6}+\frac{2808927}{1722023}a^{5}+\frac{3684664}{1722023}a^{4}-\frac{1386426}{1722023}a^{3}-\frac{4448615}{1722023}a^{2}+\frac{1948315}{1722023}a-\frac{1602079}{1722023}$
| 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: | \( 109.689161601 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 109.689161601 \cdot 1}{2\cdot\sqrt{1083064869622628748241}}\cr\approx \mathstrut & 0.159810448487 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 10*x^18 - 17*x^17 + 26*x^16 - 29*x^15 + 24*x^14 - 5*x^13 - 7*x^12 + 20*x^11 - 14*x^10 + 8*x^9 + 12*x^8 + x^7 - 6*x^6 + 9*x^5 + 9*x^4 - 2*x^3 - x^2 + 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^20 - 4*x^19 + 10*x^18 - 17*x^17 + 26*x^16 - 29*x^15 + 24*x^14 - 5*x^13 - 7*x^12 + 20*x^11 - 14*x^10 + 8*x^9 + 12*x^8 + x^7 - 6*x^6 + 9*x^5 + 9*x^4 - 2*x^3 - x^2 + 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^20 - 4*x^19 + 10*x^18 - 17*x^17 + 26*x^16 - 29*x^15 + 24*x^14 - 5*x^13 - 7*x^12 + 20*x^11 - 14*x^10 + 8*x^9 + 12*x^8 + x^7 - 6*x^6 + 9*x^5 + 9*x^4 - 2*x^3 - x^2 + 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^20 - 4*x^19 + 10*x^18 - 17*x^17 + 26*x^16 - 29*x^15 + 24*x^14 - 5*x^13 - 7*x^12 + 20*x^11 - 14*x^10 + 8*x^9 + 12*x^8 + x^7 - 6*x^6 + 9*x^5 + 9*x^4 - 2*x^3 - x^2 + 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
$C_3^5.D_6$ (as 20T669):
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 61440 |
| The 126 conjugacy class representatives for $C_3^5.D_6$ |
| Character table for $C_3^5.D_6$ |
Intermediate fields
| 5.3.4903.1, 10.2.32909950921.1, 10.2.889458133.2, 10.2.889458133.1 |
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: | 20.4.15046165299292489048629012821532792588969889.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }^{4}$ | ${\href{/padicField/11.5.0.1}{5} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/41.8.0.1}{8} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.5.0.1}{5} }^{4}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(37\)
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 37.2.1.2 | $x^{2} + 74$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 37.2.1.2 | $x^{2} + 74$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 37.4.2.1 | $x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 37.4.0.1 | $x^{4} + 6 x^{2} + 24 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 37.4.0.1 | $x^{4} + 6 x^{2} + 24 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(4903\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |