LMFDB - Number field 20.0.15233950180173557331984384.1

Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

comment:Define the number field

sage:x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 6*x^18 - 8*x^17 + 16*x^16 - 14*x^15 + 32*x^14 - 12*x^13 + 32*x^12 + 8*x^11 + 48*x^10 - 16*x^9 + 48*x^8 + 8*x^7 + 24*x^6 + 8*x^5 + 32*x^4 + 16*x^3 + 32*x^2 + 16*x + 16)

gp:K = bnfinit(y^20 - 2*y^19 + 6*y^18 - 8*y^17 + 16*y^16 - 14*y^15 + 32*y^14 - 12*y^13 + 32*y^12 + 8*y^11 + 48*y^10 - 16*y^9 + 48*y^8 + 8*y^7 + 24*y^6 + 8*y^5 + 32*y^4 + 16*y^3 + 32*y^2 + 16*y + 16, 1)

magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 2*x^19 + 6*x^18 - 8*x^17 + 16*x^16 - 14*x^15 + 32*x^14 - 12*x^13 + 32*x^12 + 8*x^11 + 48*x^10 - 16*x^9 + 48*x^8 + 8*x^7 + 24*x^6 + 8*x^5 + 32*x^4 + 16*x^3 + 32*x^2 + 16*x + 16);

oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 2*x^19 + 6*x^18 - 8*x^17 + 16*x^16 - 14*x^15 + 32*x^14 - 12*x^13 + 32*x^12 + 8*x^11 + 48*x^10 - 16*x^9 + 48*x^8 + 8*x^7 + 24*x^6 + 8*x^5 + 32*x^4 + 16*x^3 + 32*x^2 + 16*x + 16)

$ x^{20} - 2 x^{19} + 6 x^{18} - 8 x^{17} + 16 x^{16} - 14 x^{15} + 32 x^{14} - 12 x^{13} + 32 x^{12} + \cdots + 16 $ x^20 - 2*x^19 + 6*x^18 - 8*x^17 + 16*x^16 - 14*x^15 + 32*x^14 - 12*x^13 + 32*x^12 + 8*x^11 + 48*x^10 - 16*x^9 + 48*x^8 + 8*x^7 + 24*x^6 + 8*x^5 + 32*x^4 + 16*x^3 + 32*x^2 + 16*x + 16

comment:Defining polynomial

sage:K.defining_polynomial()

gp:K.pol

magma:DefiningPolynomial(K);

oscar:defining_polynomial(K)

Invariants

Degree:	$20$	comment:Degree over Q sage:K.degree() gp:poldegree(K.pol) magma:Degree(K); oscar:degree(K)
Signature:	$(0, 10)$	comment:Signature sage:K.signature() gp:K.sign magma:Signature(K); oscar:signature(K)
Discriminant:	$15233950180173557331984384$ 15233950180173557331984384 $\medspace = 2^{16}\cdot 3^{10}\cdot 89^{8}$	comment:Discriminant sage:K.disc() gp:K.disc magma:OK := Integers(K); Discriminant(OK); oscar:OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$18.16$	comment:Root discriminant sage:(K.disc().abs())^(1./K.degree()) gp:abs(K.disc)^(1/poldegree(K.pol)) magma:Abs(Discriminant(OK))^(1/Degree(K)); oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
Galois root discriminant:	$2^{4/5}3^{1/2}89^{1/2}\approx 28.44982682753188$
Ramified primes:	$2$, $3$, $89$ 2, 3, 89	comment:Ramified primes sage:K.disc().support() gp:factor(abs(K.disc))[,1]~ magma:PrimeDivisors(Discriminant(OK)); oscar:prime_divisors(discriminant(OK))
Discriminant root field:	$\Q$
$\Aut(K/\Q)$:	$C_2$	comment:Automorphisms sage:K.automorphisms() magma:Automorphisms(K); oscar:automorphism_group(K)
This field is not Galois over $\Q$.
This is not a CM field.
Maximal CM subfield:	$\Q(\sqrt{-3}) $

Integral basis (with respect to field generator $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{2}a^{9}$, $\frac{1}{4}a^{10}$, $\frac{1}{4}a^{11}$, $\frac{1}{4}a^{12}$, $\frac{1}{4}a^{13}$, $\frac{1}{12}a^{14}+\frac{1}{12}a^{11}+\frac{1}{12}a^{10}-\frac{1}{6}a^{9}-\frac{1}{6}a^{7}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{24}a^{15}-\frac{1}{12}a^{12}-\frac{1}{12}a^{11}-\frac{1}{12}a^{10}+\frac{1}{6}a^{8}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{24}a^{16}-\frac{1}{12}a^{13}-\frac{1}{12}a^{12}-\frac{1}{12}a^{11}+\frac{1}{6}a^{9}-\frac{1}{6}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{24}a^{17}-\frac{1}{12}a^{13}-\frac{1}{12}a^{12}+\frac{1}{12}a^{11}-\frac{1}{6}a^{9}-\frac{1}{6}a^{7}-\frac{1}{6}a^{6}-\frac{1}{6}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{2136}a^{18}-\frac{13}{712}a^{17}-\frac{7}{534}a^{16}-\frac{5}{534}a^{15}-\frac{37}{1068}a^{14}-\frac{11}{178}a^{13}+\frac{29}{534}a^{12}-\frac{13}{178}a^{11}+\frac{1}{178}a^{10}+\frac{1}{6}a^{9}+\frac{20}{89}a^{8}+\frac{95}{534}a^{7}-\frac{73}{534}a^{6}-\frac{3}{89}a^{5}+\frac{1}{267}a^{4}-\frac{4}{267}a^{3}+\frac{10}{267}a^{2}+\frac{37}{89}a+\frac{28}{89}$, $\frac{1}{98256}a^{19}-\frac{1}{24564}a^{18}+\frac{475}{24564}a^{17}+\frac{53}{12282}a^{16}-\frac{209}{49128}a^{15}-\frac{13}{24564}a^{14}+\frac{663}{8188}a^{13}-\frac{1381}{12282}a^{12}+\frac{863}{24564}a^{11}-\frac{793}{12282}a^{10}-\frac{860}{6141}a^{9}+\frac{1391}{12282}a^{8}-\frac{77}{6141}a^{7}+\frac{1873}{12282}a^{6}-\frac{937}{12282}a^{5}+\frac{5371}{12282}a^{4}-\frac{332}{6141}a^{3}+\frac{329}{2047}a^{2}-\frac{2777}{6141}a-\frac{2357}{6141}$ 1, a, a^2, a^3, a^4, 1/2*a^5, 1/2*a^6, 1/2*a^7, 1/2*a^8, 1/2*a^9, 1/4*a^10, 1/4*a^11, 1/4*a^12, 1/4*a^13, 1/12*a^14 + 1/12*a^11 + 1/12*a^10 - 1/6*a^9 - 1/6*a^7 - 1/3*a^3 - 1/3*a^2 - 1/3*a + 1/3, 1/24*a^15 - 1/12*a^12 - 1/12*a^11 - 1/12*a^10 + 1/6*a^8 + 1/3*a^4 + 1/3*a^3 + 1/3*a^2 - 1/3*a, 1/24*a^16 - 1/12*a^13 - 1/12*a^12 - 1/12*a^11 + 1/6*a^9 - 1/6*a^5 + 1/3*a^4 + 1/3*a^3 - 1/3*a^2, 1/24*a^17 - 1/12*a^13 - 1/12*a^12 + 1/12*a^11 - 1/6*a^9 - 1/6*a^7 - 1/6*a^6 - 1/6*a^5 + 1/3*a^4 + 1/3*a^3 - 1/3*a^2 - 1/3*a + 1/3, 1/2136*a^18 - 13/712*a^17 - 7/534*a^16 - 5/534*a^15 - 37/1068*a^14 - 11/178*a^13 + 29/534*a^12 - 13/178*a^11 + 1/178*a^10 + 1/6*a^9 + 20/89*a^8 + 95/534*a^7 - 73/534*a^6 - 3/89*a^5 + 1/267*a^4 - 4/267*a^3 + 10/267*a^2 + 37/89*a + 28/89, 1/98256*a^19 - 1/24564*a^18 + 475/24564*a^17 + 53/12282*a^16 - 209/49128*a^15 - 13/24564*a^14 + 663/8188*a^13 - 1381/12282*a^12 + 863/24564*a^11 - 793/12282*a^10 - 860/6141*a^9 + 1391/12282*a^8 - 77/6141*a^7 + 1873/12282*a^6 - 937/12282*a^5 + 5371/12282*a^4 - 332/6141*a^3 + 329/2047*a^2 - 2777/6141*a - 2357/6141

comment:Integral basis

sage:K.integral_basis()

gp:K.zk

magma:IntegralBasis(K);

oscar:basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$2$

Class group and class number

Ideal class group:		$C_{2}$, which has order $2$	comment:Class group sage:K.class_group().invariants() gp:K.clgp magma:ClassGroup(K); oscar:class_group(K)
Narrow class group:		$C_{2}$, which has order $2$	comment:Narrow class group sage:K.narrow_class_group().invariants() gp:bnfnarrow(K) magma:NarrowClassGroup(K);

Unit group

comment:Unit group

sage:UK = K.unit_group()

magma:UK, fUK := UnitGroup(K);

oscar:UK, fUK = unit_group(OK)

Rank:	$9$	comment:Unit rank sage:UK.rank() gp:K.fu magma:UnitRank(K); oscar:rank(UK)
Torsion generator:	$ -\frac{2533}{98256} a^{19} + \frac{1153}{24564} a^{18} - \frac{3035}{24564} a^{17} + \frac{875}{6141} a^{16} - \frac{13127}{49128} a^{15} + \frac{1347}{8188} a^{14} - \frac{12775}{24564} a^{13} + \frac{1133}{24564} a^{12} - \frac{9673}{24564} a^{11} - \frac{1577}{12282} a^{10} - \frac{7816}{6141} a^{9} + \frac{2030}{2047} a^{8} - \frac{13361}{12282} a^{7} - \frac{268}{6141} a^{6} - \frac{1797}{2047} a^{5} - \frac{1807}{12282} a^{4} - \frac{1899}{2047} a^{3} + \frac{374}{6141} a^{2} - \frac{2283}{2047} a + \frac{2747}{6141} $ -(2533)/(98256)a^(19) + (1153)/(24564)a^(18) - (3035)/(24564)a^(17) + (875)/(6141)a^(16) - (13127)/(49128)a^(15) + (1347)/(8188)a^(14) - (12775)/(24564)a^(13) + (1133)/(24564)a^(12) - (9673)/(24564)a^(11) - (1577)/(12282)a^(10) - (7816)/(6141)a^(9) + (2030)/(2047)a^(8) - (13361)/(12282)a^(7) - (268)/(6141)a^(6) - (1797)/(2047)a^(5) - (1807)/(12282)a^(4) - (1899)/(2047)a^(3) + (374)/(6141)a^(2) - (2283)/(2047)*a + (2747)/(6141) (order $6$)	comment:Generator for roots of unity 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:	$\frac{4039}{98256}a^{19}-\frac{549}{8188}a^{18}+\frac{8861}{49128}a^{17}-\frac{2821}{16376}a^{16}+\frac{8059}{24564}a^{15}-\frac{251}{24564}a^{14}+\frac{906}{2047}a^{13}+\frac{17935}{24564}a^{12}-\frac{4585}{12282}a^{11}+\frac{14763}{8188}a^{10}+\frac{2266}{6141}a^{9}+\frac{6218}{6141}a^{8}-\frac{1883}{6141}a^{7}+\frac{12352}{6141}a^{6}-\frac{1825}{12282}a^{5}+\frac{7071}{4094}a^{4}-\frac{292}{2047}a^{3}+\frac{2628}{2047}a^{2}+\frac{6202}{6141}a+\frac{3365}{6141}$, $\frac{2905}{98256}a^{19}-\frac{1653}{16376}a^{18}+\frac{12239}{49128}a^{17}-\frac{1839}{4094}a^{16}+\frac{4627}{6141}a^{15}-\frac{12109}{12282}a^{14}+\frac{11771}{8188}a^{13}-\frac{38033}{24564}a^{12}+\frac{15659}{12282}a^{11}-\frac{1412}{2047}a^{10}+\frac{7228}{6141}a^{9}-\frac{11335}{6141}a^{8}+\frac{18311}{12282}a^{7}-\frac{287}{6141}a^{6}+\frac{1583}{12282}a^{5}-\frac{3353}{4094}a^{4}+\frac{1487}{2047}a^{3}+\frac{1268}{2047}a^{2}-\frac{1706}{6141}a-\frac{31}{6141}$, $\frac{88}{2047}a^{19}-\frac{4607}{49128}a^{18}+\frac{2099}{8188}a^{17}-\frac{9289}{24564}a^{16}+\frac{17603}{24564}a^{15}-\frac{4735}{6141}a^{14}+\frac{9454}{6141}a^{13}-\frac{6436}{6141}a^{12}+\frac{18881}{12282}a^{11}-\frac{4411}{8188}a^{10}+\frac{13067}{6141}a^{9}-\frac{6477}{4094}a^{8}+\frac{86}{69}a^{7}+\frac{4037}{12282}a^{6}+\frac{4829}{6141}a^{5}-\frac{3121}{2047}a^{4}+\frac{11074}{6141}a^{3}-\frac{3181}{6141}a^{2}+\frac{628}{2047}a-\frac{1412}{2047}$, $\frac{569}{98256}a^{19}+\frac{395}{16376}a^{18}+\frac{415}{12282}a^{17}-\frac{433}{12282}a^{16}+\frac{1059}{4094}a^{15}-\frac{1229}{4094}a^{14}+\frac{11237}{12282}a^{13}-\frac{10085}{24564}a^{12}+\frac{47837}{24564}a^{11}-\frac{22615}{24564}a^{10}+\frac{18209}{12282}a^{9}+\frac{3923}{6141}a^{8}+\frac{2730}{2047}a^{7}-\frac{18851}{12282}a^{6}+\frac{22757}{12282}a^{5}+\frac{10715}{12282}a^{4}+\frac{2849}{2047}a^{3}+\frac{3485}{6141}a^{2}+\frac{4196}{6141}a+\frac{2366}{6141}$, $\frac{999}{32752}a^{19}-\frac{189}{4094}a^{18}+\frac{1979}{16376}a^{17}-\frac{1077}{16376}a^{16}+\frac{1177}{6141}a^{15}+\frac{211}{4094}a^{14}+\frac{844}{2047}a^{13}+\frac{9715}{24564}a^{12}+\frac{7237}{24564}a^{11}+\frac{24295}{24564}a^{10}+\frac{7341}{4094}a^{9}-\frac{2066}{6141}a^{8}+\frac{495}{2047}a^{7}+\frac{4903}{2047}a^{6}+\frac{3671}{4094}a^{5}-\frac{13739}{12282}a^{4}+\frac{7451}{6141}a^{3}+\frac{14867}{6141}a^{2}+\frac{8713}{6141}a+\frac{1385}{2047}$, $\frac{3323}{98256}a^{19}-\frac{3725}{49128}a^{18}+\frac{1205}{8188}a^{17}-\frac{649}{4094}a^{16}+\frac{1085}{8188}a^{15}+\frac{358}{2047}a^{14}-\frac{2275}{8188}a^{13}+\frac{13375}{12282}a^{12}-\frac{49847}{24564}a^{11}+\frac{63307}{24564}a^{10}-\frac{4150}{2047}a^{9}+\frac{4993}{4094}a^{8}-\frac{25453}{12282}a^{7}+\frac{18800}{6141}a^{6}-\frac{34319}{12282}a^{5}+\frac{28399}{12282}a^{4}-\frac{7493}{6141}a^{3}+\frac{353}{2047}a^{2}-\frac{1128}{2047}a+\frac{3329}{6141}$, $\frac{4199}{98256}a^{19}-\frac{95}{2047}a^{18}+\frac{6485}{49128}a^{17}-\frac{3707}{49128}a^{16}+\frac{2305}{12282}a^{15}+\frac{895}{6141}a^{14}+\frac{9757}{24564}a^{13}+\frac{1650}{2047}a^{12}-\frac{950}{6141}a^{11}+\frac{5414}{6141}a^{10}+\frac{13865}{12282}a^{9}+\frac{5}{12282}a^{8}-\frac{1979}{12282}a^{7}+\frac{1670}{2047}a^{6}+\frac{5260}{6141}a^{5}+\frac{3031}{12282}a^{4}+\frac{1375}{2047}a^{3}+\frac{1055}{6141}a^{2}-\frac{1417}{6141}a-\frac{1730}{6141}$, $\frac{275}{4272}a^{19}-\frac{185}{2136}a^{18}+\frac{287}{1068}a^{17}-\frac{257}{1068}a^{16}+\frac{421}{712}a^{15}-\frac{259}{1068}a^{14}+\frac{239}{178}a^{13}+\frac{80}{267}a^{12}+\frac{1307}{1068}a^{11}+\frac{149}{178}a^{10}+\frac{1637}{534}a^{9}-\frac{42}{89}a^{8}+\frac{335}{534}a^{7}+\frac{148}{89}a^{6}+\frac{443}{267}a^{5}-\frac{179}{534}a^{4}+\frac{512}{267}a^{3}+\frac{421}{267}a^{2}+\frac{196}{89}a+\frac{56}{267}$, $\frac{333}{32752}a^{19}-\frac{20}{2047}a^{18}+\frac{725}{24564}a^{17}+\frac{715}{24564}a^{16}-\frac{1873}{24564}a^{15}+\frac{97}{276}a^{14}-\frac{1791}{4094}a^{13}+\frac{28799}{24564}a^{12}-\frac{10011}{8188}a^{11}+\frac{27053}{12282}a^{10}-\frac{9635}{6141}a^{9}+\frac{13142}{6141}a^{8}-\frac{5259}{4094}a^{7}+\frac{32645}{12282}a^{6}-\frac{3969}{2047}a^{5}+\frac{30089}{12282}a^{4}-\frac{2032}{6141}a^{3}+\frac{10445}{6141}a^{2}-\frac{3988}{6141}a+\frac{684}{2047}$ 4039/98256a^19 - 549/8188a^18 + 8861/49128a^17 - 2821/16376a^16 + 8059/24564a^15 - 251/24564a^14 + 906/2047a^13 + 17935/24564a^12 - 4585/12282a^11 + 14763/8188a^10 + 2266/6141a^9 + 6218/6141a^8 - 1883/6141a^7 + 12352/6141a^6 - 1825/12282a^5 + 7071/4094a^4 - 292/2047a^3 + 2628/2047a^2 + 6202/6141a + 3365/6141, 2905/98256a^19 - 1653/16376a^18 + 12239/49128a^17 - 1839/4094a^16 + 4627/6141a^15 - 12109/12282a^14 + 11771/8188a^13 - 38033/24564a^12 + 15659/12282a^11 - 1412/2047a^10 + 7228/6141a^9 - 11335/6141a^8 + 18311/12282a^7 - 287/6141a^6 + 1583/12282a^5 - 3353/4094a^4 + 1487/2047a^3 + 1268/2047a^2 - 1706/6141a - 31/6141, 88/2047a^19 - 4607/49128a^18 + 2099/8188a^17 - 9289/24564a^16 + 17603/24564a^15 - 4735/6141a^14 + 9454/6141a^13 - 6436/6141a^12 + 18881/12282a^11 - 4411/8188a^10 + 13067/6141a^9 - 6477/4094a^8 + 86/69a^7 + 4037/12282a^6 + 4829/6141a^5 - 3121/2047a^4 + 11074/6141a^3 - 3181/6141a^2 + 628/2047a - 1412/2047, 569/98256a^19 + 395/16376a^18 + 415/12282a^17 - 433/12282a^16 + 1059/4094a^15 - 1229/4094a^14 + 11237/12282a^13 - 10085/24564a^12 + 47837/24564a^11 - 22615/24564a^10 + 18209/12282a^9 + 3923/6141a^8 + 2730/2047a^7 - 18851/12282a^6 + 22757/12282a^5 + 10715/12282a^4 + 2849/2047a^3 + 3485/6141a^2 + 4196/6141a + 2366/6141, 999/32752a^19 - 189/4094a^18 + 1979/16376a^17 - 1077/16376a^16 + 1177/6141a^15 + 211/4094a^14 + 844/2047a^13 + 9715/24564a^12 + 7237/24564a^11 + 24295/24564a^10 + 7341/4094a^9 - 2066/6141a^8 + 495/2047a^7 + 4903/2047a^6 + 3671/4094a^5 - 13739/12282a^4 + 7451/6141a^3 + 14867/6141a^2 + 8713/6141a + 1385/2047, 3323/98256a^19 - 3725/49128a^18 + 1205/8188a^17 - 649/4094a^16 + 1085/8188a^15 + 358/2047a^14 - 2275/8188a^13 + 13375/12282a^12 - 49847/24564a^11 + 63307/24564a^10 - 4150/2047a^9 + 4993/4094a^8 - 25453/12282a^7 + 18800/6141a^6 - 34319/12282a^5 + 28399/12282a^4 - 7493/6141a^3 + 353/2047a^2 - 1128/2047a + 3329/6141, 4199/98256a^19 - 95/2047a^18 + 6485/49128a^17 - 3707/49128a^16 + 2305/12282a^15 + 895/6141a^14 + 9757/24564a^13 + 1650/2047a^12 - 950/6141a^11 + 5414/6141a^10 + 13865/12282a^9 + 5/12282a^8 - 1979/12282a^7 + 1670/2047a^6 + 5260/6141a^5 + 3031/12282a^4 + 1375/2047a^3 + 1055/6141a^2 - 1417/6141a - 1730/6141, 275/4272a^19 - 185/2136a^18 + 287/1068a^17 - 257/1068a^16 + 421/712a^15 - 259/1068a^14 + 239/178a^13 + 80/267a^12 + 1307/1068a^11 + 149/178a^10 + 1637/534a^9 - 42/89a^8 + 335/534a^7 + 148/89a^6 + 443/267a^5 - 179/534a^4 + 512/267a^3 + 421/267a^2 + 196/89a + 56/267, 333/32752a^19 - 20/2047a^18 + 725/24564a^17 + 715/24564a^16 - 1873/24564a^15 + 97/276a^14 - 1791/4094a^13 + 28799/24564a^12 - 10011/8188a^11 + 27053/12282a^10 - 9635/6141a^9 + 13142/6141a^8 - 5259/4094a^7 + 32645/12282a^6 - 3969/2047a^5 + 30089/12282a^4 - 2032/6141a^3 + 10445/6141a^2 - 3988/6141*a + 684/2047	comment:Fundamental units 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:	$ 64033.5089322 $	comment:Regulator sage:K.regulator() gp:K.reg magma:Regulator(K); oscar:regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 64033.5089322 \cdot 2}{6\cdot\sqrt{15233950180173557331984384}}\cr\approx \mathstrut & 0.524419056076 \end{aligned}\]

comment:Analytic class number formula

sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 6*x^18 - 8*x^17 + 16*x^16 - 14*x^15 + 32*x^14 - 12*x^13 + 32*x^12 + 8*x^11 + 48*x^10 - 16*x^9 + 48*x^8 + 8*x^7 + 24*x^6 + 8*x^5 + 32*x^4 + 16*x^3 + 32*x^2 + 16*x + 16) 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))))

gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^20 - 2*x^19 + 6*x^18 - 8*x^17 + 16*x^16 - 14*x^15 + 32*x^14 - 12*x^13 + 32*x^12 + 8*x^11 + 48*x^10 - 16*x^9 + 48*x^8 + 8*x^7 + 24*x^6 + 8*x^5 + 32*x^4 + 16*x^3 + 32*x^2 + 16*x + 16, 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)))]

magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 2*x^19 + 6*x^18 - 8*x^17 + 16*x^16 - 14*x^15 + 32*x^14 - 12*x^13 + 32*x^12 + 8*x^11 + 48*x^10 - 16*x^9 + 48*x^8 + 8*x^7 + 24*x^6 + 8*x^5 + 32*x^4 + 16*x^3 + 32*x^2 + 16*x + 16); 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)));

oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 2*x^19 + 6*x^18 - 8*x^17 + 16*x^16 - 14*x^15 + 32*x^14 - 12*x^13 + 32*x^12 + 8*x^11 + 48*x^10 - 16*x^9 + 48*x^8 + 8*x^7 + 24*x^6 + 8*x^5 + 32*x^4 + 16*x^3 + 32*x^2 + 16*x + 16); 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

$S_5$ (as 20T32):

comment:Galois group

sage:K.galois_group()

gp:polgalois(K.pol)

magma:G = GaloisGroup(K);

oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)

A non-solvable group of order 120

The 7 conjugacy class representatives for $S_5$

Character table for $S_5$

Intermediate fields

$\Q(\sqrt{-3}) $, 10.4.433674369792.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

comment:Intermediate fields

sage:K.subfields()[1:-1]

gp:L = nfsubfields(K); L[2..length(L)]

magma:L := Subfields(K); L[2..#L];

oscar:subfields(K)[2:end-1]

Sibling fields

Degree 5 sibling:	5.3.380208.1
Degree 6 sibling:	6.0.3421872.4
Degree 10 siblings:	10.0.3903069328128.1, 10.4.433674369792.1
Degree 12 sibling:	12.0.11709207984384.1
Degree 15 sibling:	15.3.1483978183108890624.1
Degree 20 siblings:	deg 20, deg 20
Degree 24 sibling:	data not computed
Degree 30 siblings:	data not computed
Degree 40 sibling:	data not computed
Minimal sibling:	5.3.380208.1

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	R	${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$	${\href{/padicField/7.5.0.1}{5} }^{4}$	${\href{/padicField/11.6.0.1}{6} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }$	${\href{/padicField/13.3.0.1}{3} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$	${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$	${\href{/padicField/19.5.0.1}{5} }^{4}$	${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$	${\href{/padicField/29.2.0.1}{2} }^{10}$	${\href{/padicField/31.3.0.1}{3} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$	${\href{/padicField/37.5.0.1}{5} }^{4}$	${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$	${\href{/padicField/43.5.0.1}{5} }^{4}$	${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }$	${\href{/padicField/53.6.0.1}{6} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }$	${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

comment:Frobenius cycle types

sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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	2.2.5.8a1.1	$x^{10} + 5 x^{9} + 15 x^{8} + 30 x^{7} + 45 x^{6} + 51 x^{5} + 45 x^{4} + 30 x^{3} + 15 x^{2} + 5 x + 3$	$5$	$2$	$8$	$F_5$	$$[\ ]_{5}^{4}$$
$2$ 2	2.2.5.8a1.1	$x^{10} + 5 x^{9} + 15 x^{8} + 30 x^{7} + 45 x^{6} + 51 x^{5} + 45 x^{4} + 30 x^{3} + 15 x^{2} + 5 x + 3$	$5$	$2$	$8$	$F_5$	$$[\ ]_{5}^{4}$$
$3$ 3	3.1.2.1a1.1	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
	3.3.2.3a1.2	$x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 4 x + 4$	$2$	$3$	$3$	$C_6$	$$[\ ]_{2}^{3}$$
	3.3.2.3a1.2	$x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 4 x + 4$	$2$	$3$	$3$	$C_6$	$$[\ ]_{2}^{3}$$
	3.3.2.3a1.2	$x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 4 x + 4$	$2$	$3$	$3$	$C_6$	$$[\ ]_{2}^{3}$$
$89$ 89	89.2.1.0a1.1	$x^{2} + 82 x + 3$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
	89.2.1.0a1.1	$x^{2} + 82 x + 3$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
	89.2.2.2a1.1	$x^{4} + 164 x^{3} + 6730 x^{2} + 581 x + 9$	$2$	$2$	$2$	$C_4$	$$[\ ]_{2}^{2}$$
	89.2.2.2a1.1	$x^{4} + 164 x^{3} + 6730 x^{2} + 581 x + 9$	$2$	$2$	$2$	$C_4$	$$[\ ]_{2}^{2}$$
	89.2.2.2a1.1	$x^{4} + 164 x^{3} + 6730 x^{2} + 581 x + 9$	$2$	$2$	$2$	$C_4$	$$[\ ]_{2}^{2}$$
	89.2.2.2a1.1	$x^{4} + 164 x^{3} + 6730 x^{2} + 581 x + 9$	$2$	$2$	$2$	$C_4$	$$[\ ]_{2}^{2}$$

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