Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 23*x^18 - 66*x^17 + 154*x^16 - 304*x^15 + 519*x^14 - 778*x^13 + 1036*x^12 - 1233*x^11 + 1317*x^10 - 1265*x^9 + 1093*x^8 - 845*x^7 + 580*x^6 - 349*x^5 + 180*x^4 - 76*x^3 + 25*x^2 - 6*x + 1)
gp: K = bnfinit(y^20 - 6*y^19 + 23*y^18 - 66*y^17 + 154*y^16 - 304*y^15 + 519*y^14 - 778*y^13 + 1036*y^12 - 1233*y^11 + 1317*y^10 - 1265*y^9 + 1093*y^8 - 845*y^7 + 580*y^6 - 349*y^5 + 180*y^4 - 76*y^3 + 25*y^2 - 6*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 6*x^19 + 23*x^18 - 66*x^17 + 154*x^16 - 304*x^15 + 519*x^14 - 778*x^13 + 1036*x^12 - 1233*x^11 + 1317*x^10 - 1265*x^9 + 1093*x^8 - 845*x^7 + 580*x^6 - 349*x^5 + 180*x^4 - 76*x^3 + 25*x^2 - 6*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 6*x^19 + 23*x^18 - 66*x^17 + 154*x^16 - 304*x^15 + 519*x^14 - 778*x^13 + 1036*x^12 - 1233*x^11 + 1317*x^10 - 1265*x^9 + 1093*x^8 - 845*x^7 + 580*x^6 - 349*x^5 + 180*x^4 - 76*x^3 + 25*x^2 - 6*x + 1)
\( x^{20} - 6 x^{19} + 23 x^{18} - 66 x^{17} + 154 x^{16} - 304 x^{15} + 519 x^{14} - 778 x^{13} + 1036 x^{12} + \cdots + 1 \)
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: |
\(405424975045226129161\)
\(\medspace = 7^{2}\cdot 53^{6}\cdot 139^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(10.72\) | 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: | $7^{1/2}53^{1/2}139^{1/2}\approx 227.08808863522543$ | ||
| Ramified primes: |
\(7\), \(53\), \(139\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{193}a^{19}-\frac{70}{193}a^{18}+\frac{64}{193}a^{17}+\frac{84}{193}a^{16}-\frac{11}{193}a^{15}+\frac{14}{193}a^{14}+\frac{9}{193}a^{13}-\frac{3}{193}a^{12}+\frac{70}{193}a^{11}+\frac{77}{193}a^{10}+\frac{56}{193}a^{9}-\frac{24}{193}a^{8}-\frac{73}{193}a^{7}-\frac{33}{193}a^{6}-\frac{10}{193}a^{5}-\frac{95}{193}a^{4}+\frac{84}{193}a^{3}-\frac{48}{193}a^{2}+\frac{9}{193}a-\frac{3}{193}$
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: |
$\frac{784}{193}a^{19}-\frac{4507}{193}a^{18}+\frac{16594}{193}a^{17}-\frac{46084}{193}a^{16}+\frac{103895}{193}a^{15}-\frac{198429}{193}a^{14}+\frac{327243}{193}a^{13}-\frac{473272}{193}a^{12}+\frac{607439}{193}a^{11}-\frac{695420}{193}a^{10}+\frac{712649}{193}a^{9}-\frac{655137}{193}a^{8}+\frac{539524}{193}a^{7}-\frac{394888}{193}a^{6}+\frac{253868}{193}a^{5}-\frac{141258}{193}a^{4}+\frac{65277}{193}a^{3}-\frac{23350}{193}a^{2}+\frac{5898}{193}a-\frac{1001}{193}$, $\frac{266}{193}a^{19}-\frac{1636}{193}a^{18}+\frac{6216}{193}a^{17}-\frac{17800}{193}a^{16}+\frac{41271}{193}a^{15}-\frac{81003}{193}a^{14}+\frac{137301}{193}a^{13}-\frac{204220}{193}a^{12}+\frac{269520}{193}a^{11}-\frac{317654}{193}a^{10}+\frac{335276}{193}a^{9}-\frac{317693}{193}a^{8}+\frac{270082}{193}a^{7}-\frac{204673}{193}a^{6}+\frac{136686}{193}a^{5}-\frac{79696}{193}a^{4}+\frac{39135}{193}a^{3}-\frac{15277}{193}a^{2}+\frac{4324}{193}a-\frac{991}{193}$, $\frac{1001}{193}a^{19}-\frac{5222}{193}a^{18}+\frac{18516}{193}a^{17}-\frac{49472}{193}a^{16}+\frac{108070}{193}a^{15}-\frac{200409}{193}a^{14}+\frac{321090}{193}a^{13}-\frac{451535}{193}a^{12}+\frac{563764}{193}a^{11}-\frac{626794}{193}a^{10}+\frac{622897}{193}a^{9}-\frac{553616}{193}a^{8}+\frac{438956}{193}a^{7}-\frac{306321}{193}a^{6}+\frac{185692}{193}a^{5}-\frac{95481}{193}a^{4}+\frac{38922}{193}a^{3}-\frac{10799}{193}a^{2}+\frac{1675}{193}a-\frac{108}{193}$, $\frac{156}{193}a^{19}-\frac{1270}{193}a^{18}+\frac{5159}{193}a^{17}-\frac{15653}{193}a^{16}+\frac{37463}{193}a^{15}-\frac{75209}{193}a^{14}+\frac{129363}{193}a^{13}-\frac{193854}{193}a^{12}+\frac{256416}{193}a^{11}-\frac{301806}{193}a^{10}+\frac{317343}{193}a^{9}-\frac{299034}{193}a^{8}+\frac{252250}{193}a^{7}-\frac{190042}{193}a^{6}+\frac{125820}{193}a^{5}-\frac{72527}{193}a^{4}+\frac{35106}{193}a^{3}-\frac{13664}{193}a^{2}+\frac{3913}{193}a-\frac{854}{193}$, $\frac{230}{193}a^{19}-\frac{1046}{193}a^{18}+\frac{3719}{193}a^{17}-\frac{9630}{193}a^{16}+\frac{20823}{193}a^{15}-\frac{38082}{193}a^{14}+\frac{60163}{193}a^{13}-\frac{83487}{193}a^{12}+\frac{102371}{193}a^{11}-\frac{111214}{193}a^{10}+\frac{107643}{193}a^{9}-\frac{91984}{193}a^{8}+\frac{69481}{193}a^{7}-\frac{45032}{193}a^{6}+\frac{24527}{193}a^{5}-\frac{10077}{193}a^{4}+\frac{2336}{193}a^{3}+\frac{540}{193}a^{2}-\frac{825}{193}a+\frac{275}{193}$, $\frac{26}{193}a^{19}-\frac{276}{193}a^{18}+\frac{1085}{193}a^{17}-\frac{3606}{193}a^{16}+\frac{8978}{193}a^{15}-\frac{19129}{193}a^{14}+\frac{34588}{193}a^{13}-\frac{54504}{193}a^{12}+\frac{75739}{193}a^{11}-\frac{93147}{193}a^{10}+\frac{102009}{193}a^{9}-\frac{100212}{193}a^{8}+\frac{87654}{193}a^{7}-\frac{68408}{193}a^{6}+\frac{46832}{193}a^{5}-\frac{28332}{193}a^{4}+\frac{13957}{193}a^{3}-\frac{5687}{193}a^{2}+\frac{1778}{193}a-\frac{464}{193}$, $\frac{564}{193}a^{19}-\frac{2617}{193}a^{18}+\frac{8883}{193}a^{17}-\frac{22490}{193}a^{16}+\frac{47064}{193}a^{15}-\frac{83586}{193}a^{14}+\frac{128403}{193}a^{13}-\frac{173269}{193}a^{12}+\frac{207969}{193}a^{11}-\frac{221947}{193}a^{10}+\frac{211846}{193}a^{9}-\frac{180481}{193}a^{8}+\frac{136967}{193}a^{7}-\frac{90215}{193}a^{6}+\frac{51488}{193}a^{5}-\frac{24051}{193}a^{4}+\frac{8390}{193}a^{3}-\frac{1403}{193}a^{2}+\frac{58}{193}a+\frac{238}{193}$, $\frac{970}{193}a^{19}-\frac{5561}{193}a^{18}+\frac{20392}{193}a^{17}-\frac{56515}{193}a^{16}+\frac{127132}{193}a^{15}-\frac{242338}{193}a^{14}+\frac{398976}{193}a^{13}-\frac{575927}{193}a^{12}+\frac{737996}{193}a^{11}-\frac{843218}{193}a^{10}+\frac{862218}{193}a^{9}-\frac{790648}{193}a^{8}+\frac{649080}{193}a^{7}-\frac{473015}{193}a^{6}+\frac{302381}{193}a^{5}-\frac{167034}{193}a^{4}+\frac{76269}{193}a^{3}-\frac{26681}{193}a^{2}+\frac{6800}{193}a-\frac{1173}{193}$, $\frac{690}{193}a^{19}-\frac{3717}{193}a^{18}+\frac{13280}{193}a^{17}-\frac{35838}{193}a^{16}+\frac{78681}{193}a^{15}-\frac{146670}{193}a^{14}+\frac{235880}{193}a^{13}-\frac{332872}{193}a^{12}+\frac{416930}{193}a^{11}-\frac{465268}{193}a^{10}+\frac{464591}{193}a^{9}-\frac{415491}{193}a^{8}+\frac{332156}{193}a^{7}-\frac{235070}{193}a^{6}+\frac{145184}{193}a^{5}-\frac{77130}{193}a^{4}+\frac{33256}{193}a^{3}-\frac{10732}{193}a^{2}+\frac{2350}{193}a-\frac{333}{193}$
| 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: | \( 62.9466310057 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 62.9466310057 \cdot 1}{2\cdot\sqrt{405424975045226129161}}\cr\approx \mathstrut & 0.149894580001 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 23*x^18 - 66*x^17 + 154*x^16 - 304*x^15 + 519*x^14 - 778*x^13 + 1036*x^12 - 1233*x^11 + 1317*x^10 - 1265*x^9 + 1093*x^8 - 845*x^7 + 580*x^6 - 349*x^5 + 180*x^4 - 76*x^3 + 25*x^2 - 6*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 - 6*x^19 + 23*x^18 - 66*x^17 + 154*x^16 - 304*x^15 + 519*x^14 - 778*x^13 + 1036*x^12 - 1233*x^11 + 1317*x^10 - 1265*x^9 + 1093*x^8 - 845*x^7 + 580*x^6 - 349*x^5 + 180*x^4 - 76*x^3 + 25*x^2 - 6*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 - 6*x^19 + 23*x^18 - 66*x^17 + 154*x^16 - 304*x^15 + 519*x^14 - 778*x^13 + 1036*x^12 - 1233*x^11 + 1317*x^10 - 1265*x^9 + 1093*x^8 - 845*x^7 + 580*x^6 - 349*x^5 + 180*x^4 - 76*x^3 + 25*x^2 - 6*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 - 6*x^19 + 23*x^18 - 66*x^17 + 154*x^16 - 304*x^15 + 519*x^14 - 778*x^13 + 1036*x^12 - 1233*x^11 + 1317*x^10 - 1265*x^9 + 1093*x^8 - 845*x^7 + 580*x^6 - 349*x^5 + 180*x^4 - 76*x^3 + 25*x^2 - 6*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
$C_2^{10}.S_5$ (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
| 5.3.7367.1, 10.2.2876452517.1, 10.0.379908823.1, 10.4.20135167619.1 |
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 | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | ${\href{/padicField/3.10.0.1}{10} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/11.5.0.1}{5} }^{4}$ | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.10.0.1}{10} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.6.0.1 | $x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 7.6.0.1 | $x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(53\)
| 53.4.0.1 | $x^{4} + 9 x^{2} + 38 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 53.4.2.1 | $x^{4} + 4974 x^{3} + 6304741 x^{2} + 297375564 x + 18587952$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 53.4.2.1 | $x^{4} + 4974 x^{3} + 6304741 x^{2} + 297375564 x + 18587952$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 53.4.2.1 | $x^{4} + 4974 x^{3} + 6304741 x^{2} + 297375564 x + 18587952$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 53.4.0.1 | $x^{4} + 9 x^{2} + 38 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(139\)
| $\Q_{139}$ | $x + 137$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{139}$ | $x + 137$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{139}$ | $x + 137$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{139}$ | $x + 137$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 139.4.0.1 | $x^{4} + 7 x^{2} + 96 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 139.4.2.1 | $x^{4} + 276 x^{3} + 19326 x^{2} + 38916 x + 2665885$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 139.4.2.1 | $x^{4} + 276 x^{3} + 19326 x^{2} + 38916 x + 2665885$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 139.4.0.1 | $x^{4} + 7 x^{2} + 96 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |