LMFDB - Number field 20.0.161053520503936294265094144.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 12*x^18 - 23*x^17 + 66*x^16 - 117*x^15 + 222*x^14 - 297*x^13 + 411*x^12 - 454*x^11 + 456*x^10 - 312*x^9 + 147*x^8 + 69*x^7 - 150*x^6 + 127*x^5 - 78*x^4 + 21*x^3 + 16*x^2 - 15*x + 9)

gp: K = bnfinit(y^20 - 3*y^19 + 12*y^18 - 23*y^17 + 66*y^16 - 117*y^15 + 222*y^14 - 297*y^13 + 411*y^12 - 454*y^11 + 456*y^10 - 312*y^9 + 147*y^8 + 69*y^7 - 150*y^6 + 127*y^5 - 78*y^4 + 21*y^3 + 16*y^2 - 15*y + 9, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 3*x^19 + 12*x^18 - 23*x^17 + 66*x^16 - 117*x^15 + 222*x^14 - 297*x^13 + 411*x^12 - 454*x^11 + 456*x^10 - 312*x^9 + 147*x^8 + 69*x^7 - 150*x^6 + 127*x^5 - 78*x^4 + 21*x^3 + 16*x^2 - 15*x + 9);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 3*x^19 + 12*x^18 - 23*x^17 + 66*x^16 - 117*x^15 + 222*x^14 - 297*x^13 + 411*x^12 - 454*x^11 + 456*x^10 - 312*x^9 + 147*x^8 + 69*x^7 - 150*x^6 + 127*x^5 - 78*x^4 + 21*x^3 + 16*x^2 - 15*x + 9)

$ x^{20} - 3 x^{19} + 12 x^{18} - 23 x^{17} + 66 x^{16} - 117 x^{15} + 222 x^{14} - 297 x^{13} + 411 x^{12} + \cdots + 9 $ x^20 - 3*x^19 + 12*x^18 - 23*x^17 + 66*x^16 - 117*x^15 + 222*x^14 - 297*x^13 + 411*x^12 - 454*x^11 + 456*x^10 - 312*x^9 + 147*x^8 + 69*x^7 - 150*x^6 + 127*x^5 - 78*x^4 + 21*x^3 + 16*x^2 - 15*x + 9

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:	$161053520503936294265094144$ 161053520503936294265094144 $\medspace = 2^{16}\cdot 3^{22}\cdot 23^{8}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$20.43$	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:	$2^{4/5}3^{7/6}23^{1/2}\approx 30.08357041040144$
Ramified primes:	$2$, $3$, $23$ 2, 3, 23	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Aut(K/\Q) }$:	$2$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{7}+\frac{1}{3}a^{4}-\frac{1}{3}a$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{8}+\frac{1}{3}a^{5}-\frac{1}{3}a^{2}$, $\frac{1}{9}a^{12}+\frac{1}{9}a^{11}-\frac{1}{9}a^{10}+\frac{1}{9}a^{9}-\frac{2}{9}a^{8}+\frac{4}{9}a^{6}+\frac{4}{9}a^{5}-\frac{1}{9}a^{4}-\frac{2}{9}a^{3}+\frac{4}{9}a^{2}+\frac{1}{3}a$, $\frac{1}{9}a^{13}+\frac{1}{9}a^{11}-\frac{1}{9}a^{10}+\frac{2}{9}a^{8}+\frac{1}{9}a^{7}-\frac{2}{9}a^{5}-\frac{4}{9}a^{4}-\frac{1}{9}a^{2}+\frac{1}{3}a$, $\frac{1}{9}a^{14}+\frac{1}{9}a^{11}+\frac{1}{9}a^{10}+\frac{1}{9}a^{9}+\frac{1}{3}a^{6}+\frac{4}{9}a^{5}+\frac{1}{9}a^{4}+\frac{1}{9}a^{3}-\frac{4}{9}a^{2}-\frac{1}{3}a$, $\frac{1}{18}a^{15}-\frac{1}{18}a^{12}+\frac{1}{9}a^{11}-\frac{1}{9}a^{9}-\frac{4}{9}a^{8}+\frac{1}{3}a^{7}-\frac{2}{9}a^{6}-\frac{2}{9}a^{5}-\frac{1}{2}a^{3}+\frac{2}{9}a^{2}+\frac{1}{3}a-\frac{1}{2}$, $\frac{1}{18}a^{16}-\frac{1}{18}a^{13}-\frac{1}{9}a^{11}+\frac{1}{9}a^{9}+\frac{2}{9}a^{8}+\frac{4}{9}a^{7}+\frac{1}{3}a^{6}-\frac{4}{9}a^{5}-\frac{7}{18}a^{4}+\frac{1}{9}a^{3}-\frac{4}{9}a^{2}-\frac{1}{6}a$, $\frac{1}{18}a^{17}-\frac{1}{18}a^{14}+\frac{1}{9}a^{11}-\frac{1}{9}a^{8}+\frac{1}{18}a^{5}-\frac{1}{18}a^{2}$, $\frac{1}{162}a^{18}+\frac{1}{81}a^{17}-\frac{1}{54}a^{16}+\frac{1}{81}a^{15}-\frac{1}{81}a^{14}+\frac{1}{54}a^{13}-\frac{1}{162}a^{12}-\frac{4}{81}a^{11}-\frac{1}{9}a^{10}+\frac{8}{81}a^{9}-\frac{2}{81}a^{8}+\frac{1}{27}a^{7}+\frac{79}{162}a^{6}-\frac{23}{81}a^{5}-\frac{5}{54}a^{4}+\frac{7}{81}a^{3}-\frac{19}{81}a^{2}-\frac{5}{54}a-\frac{5}{18}$, $\frac{1}{160749696474}a^{19}+\frac{20075465}{26791616079}a^{18}-\frac{1824513287}{80374848237}a^{17}-\frac{2406521503}{160749696474}a^{16}-\frac{219269486}{8930538693}a^{15}-\frac{2660712952}{80374848237}a^{14}+\frac{2118915334}{80374848237}a^{13}+\frac{86657285}{8930538693}a^{12}-\frac{1115018149}{80374848237}a^{11}+\frac{6778201823}{80374848237}a^{10}-\frac{821106472}{26791616079}a^{9}-\frac{7782749927}{80374848237}a^{8}-\frac{44605974011}{160749696474}a^{7}+\frac{6294788179}{26791616079}a^{6}+\frac{22498882786}{80374848237}a^{5}+\frac{41567703659}{160749696474}a^{4}+\frac{7642784165}{26791616079}a^{3}-\frac{11592075811}{80374848237}a^{2}-\frac{2680035904}{26791616079}a-\frac{1691606518}{8930538693}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, 1/3*a^9 + 1/3*a^8 + 1/3*a^7 + 1/3*a^3 + 1/3*a^2 + 1/3*a, 1/3*a^10 - 1/3*a^7 + 1/3*a^4 - 1/3*a, 1/3*a^11 - 1/3*a^8 + 1/3*a^5 - 1/3*a^2, 1/9*a^12 + 1/9*a^11 - 1/9*a^10 + 1/9*a^9 - 2/9*a^8 + 4/9*a^6 + 4/9*a^5 - 1/9*a^4 - 2/9*a^3 + 4/9*a^2 + 1/3*a, 1/9*a^13 + 1/9*a^11 - 1/9*a^10 + 2/9*a^8 + 1/9*a^7 - 2/9*a^5 - 4/9*a^4 - 1/9*a^2 + 1/3*a, 1/9*a^14 + 1/9*a^11 + 1/9*a^10 + 1/9*a^9 + 1/3*a^6 + 4/9*a^5 + 1/9*a^4 + 1/9*a^3 - 4/9*a^2 - 1/3*a, 1/18*a^15 - 1/18*a^12 + 1/9*a^11 - 1/9*a^9 - 4/9*a^8 + 1/3*a^7 - 2/9*a^6 - 2/9*a^5 - 1/2*a^3 + 2/9*a^2 + 1/3*a - 1/2, 1/18*a^16 - 1/18*a^13 - 1/9*a^11 + 1/9*a^9 + 2/9*a^8 + 4/9*a^7 + 1/3*a^6 - 4/9*a^5 - 7/18*a^4 + 1/9*a^3 - 4/9*a^2 - 1/6*a, 1/18*a^17 - 1/18*a^14 + 1/9*a^11 - 1/9*a^8 + 1/18*a^5 - 1/18*a^2, 1/162*a^18 + 1/81*a^17 - 1/54*a^16 + 1/81*a^15 - 1/81*a^14 + 1/54*a^13 - 1/162*a^12 - 4/81*a^11 - 1/9*a^10 + 8/81*a^9 - 2/81*a^8 + 1/27*a^7 + 79/162*a^6 - 23/81*a^5 - 5/54*a^4 + 7/81*a^3 - 19/81*a^2 - 5/54*a - 5/18, 1/160749696474*a^19 + 20075465/26791616079*a^18 - 1824513287/80374848237*a^17 - 2406521503/160749696474*a^16 - 219269486/8930538693*a^15 - 2660712952/80374848237*a^14 + 2118915334/80374848237*a^13 + 86657285/8930538693*a^12 - 1115018149/80374848237*a^11 + 6778201823/80374848237*a^10 - 821106472/26791616079*a^9 - 7782749927/80374848237*a^8 - 44605974011/160749696474*a^7 + 6294788179/26791616079*a^6 + 22498882786/80374848237*a^5 + 41567703659/160749696474*a^4 + 7642784165/26791616079*a^3 - 11592075811/80374848237*a^2 - 2680035904/26791616079*a - 1691606518/8930538693

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

$C_{2}$, which has order $2$

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:	$ -\frac{232294184}{2976846231} a^{19} + \frac{4375258585}{17861077386} a^{18} - \frac{1864489903}{1984564154} a^{17} + \frac{16716790891}{8930538693} a^{16} - \frac{45999889924}{8930538693} a^{15} + \frac{171213886615}{17861077386} a^{14} - \frac{155791007006}{8930538693} a^{13} + \frac{436345370111}{17861077386} a^{12} - \frac{293119401211}{8930538693} a^{11} + \frac{346710509297}{8930538693} a^{10} - \frac{345828051655}{8930538693} a^{9} + \frac{270456812641}{8930538693} a^{8} - \frac{156481297835}{8930538693} a^{7} + \frac{60526036675}{17861077386} a^{6} + \frac{93045477361}{17861077386} a^{5} - \frac{27530735567}{8930538693} a^{4} + \frac{11474653574}{2976846231} a^{3} - \frac{36576409739}{17861077386} a^{2} + \frac{122165722}{2976846231} a + \frac{1047062401}{1984564154} $ -(232294184)/(2976846231)a^(19) + (4375258585)/(17861077386)a^(18) - (1864489903)/(1984564154)a^(17) + (16716790891)/(8930538693)a^(16) - (45999889924)/(8930538693)a^(15) + (171213886615)/(17861077386)a^(14) - (155791007006)/(8930538693)a^(13) + (436345370111)/(17861077386)a^(12) - (293119401211)/(8930538693)a^(11) + (346710509297)/(8930538693)a^(10) - (345828051655)/(8930538693)a^(9) + (270456812641)/(8930538693)a^(8) - (156481297835)/(8930538693)a^(7) + (60526036675)/(17861077386)a^(6) + (93045477361)/(17861077386)a^(5) - (27530735567)/(8930538693)a^(4) + (11474653574)/(2976846231)a^(3) - (36576409739)/(17861077386)a^(2) + (122165722)/(2976846231)*a + (1047062401)/(1984564154) (order $6$)	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:	$\frac{8790299405}{160749696474}a^{19}-\frac{631088647}{2976846231}a^{18}+\frac{65448212309}{80374848237}a^{17}-\frac{151559429458}{80374848237}a^{16}+\frac{261979653119}{53583232158}a^{15}-\frac{798878491865}{80374848237}a^{14}+\frac{3000320133685}{160749696474}a^{13}-\frac{1540073104855}{53583232158}a^{12}+\frac{3197858912038}{80374848237}a^{11}-\frac{3918080524121}{80374848237}a^{10}+\frac{463078193501}{8930538693}a^{9}-\frac{3618933903961}{80374848237}a^{8}+\frac{4693141301909}{160749696474}a^{7}-\frac{198472121813}{26791616079}a^{6}-\frac{714672914461}{80374848237}a^{5}+\frac{1030096993613}{80374848237}a^{4}-\frac{461453323589}{53583232158}a^{3}+\frac{202763110753}{80374848237}a^{2}+\frac{88159284689}{53583232158}a-\frac{18888148765}{17861077386}$, $\frac{232294184}{2976846231}a^{19}-\frac{4375258585}{17861077386}a^{18}+\frac{1864489903}{1984564154}a^{17}-\frac{16716790891}{8930538693}a^{16}+\frac{45999889924}{8930538693}a^{15}-\frac{171213886615}{17861077386}a^{14}+\frac{155791007006}{8930538693}a^{13}-\frac{436345370111}{17861077386}a^{12}+\frac{293119401211}{8930538693}a^{11}-\frac{346710509297}{8930538693}a^{10}+\frac{345828051655}{8930538693}a^{9}-\frac{270456812641}{8930538693}a^{8}+\frac{156481297835}{8930538693}a^{7}-\frac{60526036675}{17861077386}a^{6}-\frac{93045477361}{17861077386}a^{5}+\frac{27530735567}{8930538693}a^{4}-\frac{11474653574}{2976846231}a^{3}+\frac{36576409739}{17861077386}a^{2}+\frac{2854680509}{2976846231}a-\frac{1047062401}{1984564154}$, $\frac{131168344}{26791616079}a^{19}-\frac{171825118}{8930538693}a^{18}+\frac{2902494974}{26791616079}a^{17}-\frac{6475799446}{26791616079}a^{16}+\frac{14018849795}{17861077386}a^{15}-\frac{36609872390}{26791616079}a^{14}+\frac{93696736895}{26791616079}a^{13}-\frac{88327518775}{17861077386}a^{12}+\frac{237647651914}{26791616079}a^{11}-\frac{262165459223}{26791616079}a^{10}+\frac{120297569102}{8930538693}a^{9}-\frac{324291972871}{26791616079}a^{8}+\frac{281400943138}{26791616079}a^{7}-\frac{46805965162}{8930538693}a^{6}+\frac{9628744379}{26791616079}a^{5}+\frac{72425351309}{26791616079}a^{4}-\frac{57926242633}{17861077386}a^{3}+\frac{40151084938}{26791616079}a^{2}-\frac{12397553399}{8930538693}a-\frac{763318987}{5953692462}$, $\frac{59493167}{8930538693}a^{19}+\frac{351504113}{17861077386}a^{18}-\frac{65041897}{1984564154}a^{17}+\frac{833609363}{2976846231}a^{16}-\frac{924494126}{2976846231}a^{15}+\frac{8144417683}{5953692462}a^{14}-\frac{6161140609}{2976846231}a^{13}+\frac{77243002721}{17861077386}a^{12}-\frac{38518465243}{8930538693}a^{11}+\frac{46588253008}{8930538693}a^{10}-\frac{4098645002}{992282077}a^{9}+\frac{22448434121}{8930538693}a^{8}+\frac{26798727658}{8930538693}a^{7}-\frac{43092374417}{5953692462}a^{6}+\frac{176511914581}{17861077386}a^{5}-\frac{53174856041}{8930538693}a^{4}+\frac{445675446}{992282077}a^{3}+\frac{8096492287}{17861077386}a^{2}-\frac{3197770787}{2976846231}a+\frac{2280304913}{1984564154}$, $\frac{13768305367}{80374848237}a^{19}-\frac{89887163761}{160749696474}a^{18}+\frac{171165249709}{80374848237}a^{17}-\frac{690119104955}{160749696474}a^{16}+\frac{936310528136}{80374848237}a^{15}-\frac{1741568664886}{80374848237}a^{14}+\frac{6380092537231}{160749696474}a^{13}-\frac{8697616965551}{160749696474}a^{12}+\frac{5820872747054}{80374848237}a^{11}-\frac{6474876953948}{80374848237}a^{10}+\frac{6414630980293}{80374848237}a^{9}-\frac{4150929103853}{80374848237}a^{8}+\frac{1731998179840}{80374848237}a^{7}+\frac{2591516871791}{160749696474}a^{6}-\frac{2338776390116}{80374848237}a^{5}+\frac{3452271231337}{160749696474}a^{4}-\frac{720213238612}{80374848237}a^{3}+\frac{287725545524}{80374848237}a^{2}+\frac{228107558467}{53583232158}a-\frac{67092503603}{17861077386}$, $\frac{833215090}{26791616079}a^{19}-\frac{6267694247}{80374848237}a^{18}+\frac{59427857635}{160749696474}a^{17}-\frac{17546690195}{26791616079}a^{16}+\frac{176162056589}{80374848237}a^{15}-\frac{568630274911}{160749696474}a^{14}+\frac{69285315953}{8930538693}a^{13}-\frac{838426509883}{80374848237}a^{12}+\frac{1359261057901}{80374848237}a^{11}-\frac{511147921682}{26791616079}a^{10}+\frac{1831535247715}{80374848237}a^{9}-\frac{1598143062670}{80374848237}a^{8}+\frac{152211339449}{8930538693}a^{7}-\frac{598310908085}{80374848237}a^{6}+\frac{406351434997}{160749696474}a^{5}+\frac{20025887744}{8930538693}a^{4}-\frac{251957046289}{80374848237}a^{3}+\frac{220512020069}{160749696474}a^{2}-\frac{18422400368}{26791616079}a-\frac{214813235}{8930538693}$, $\frac{8586405989}{80374848237}a^{19}-\frac{12930673303}{80374848237}a^{18}+\frac{66580039598}{80374848237}a^{17}-\frac{51871620443}{80374848237}a^{16}+\frac{600043050269}{160749696474}a^{15}-\frac{221468657468}{80374848237}a^{14}+\frac{557346521641}{80374848237}a^{13}-\frac{25258171423}{160749696474}a^{12}+\frac{254186572393}{80374848237}a^{11}+\frac{652719194156}{80374848237}a^{10}-\frac{1007920008844}{80374848237}a^{9}+\frac{2237548306319}{80374848237}a^{8}-\frac{1871009001550}{80374848237}a^{7}+\frac{2171869843184}{80374848237}a^{6}-\frac{618963999358}{80374848237}a^{5}-\frac{225094201310}{80374848237}a^{4}+\frac{233913827273}{160749696474}a^{3}-\frac{405087883574}{80374848237}a^{2}+\frac{117811242685}{26791616079}a+\frac{27317387087}{17861077386}$, $\frac{4097738534}{80374848237}a^{19}-\frac{6687543715}{53583232158}a^{18}+\frac{41555475844}{80374848237}a^{17}-\frac{136308931201}{160749696474}a^{16}+\frac{145078093415}{53583232158}a^{15}-\frac{347809726951}{80374848237}a^{14}+\frac{1300350359873}{160749696474}a^{13}-\frac{267568406960}{26791616079}a^{12}+\frac{1118645131295}{80374848237}a^{11}-\frac{1222565262940}{80374848237}a^{10}+\frac{364867830463}{26791616079}a^{9}-\frac{654849914747}{80374848237}a^{8}+\frac{229356247673}{80374848237}a^{7}+\frac{61279628845}{17861077386}a^{6}-\frac{544104887936}{80374848237}a^{5}+\frac{740715936011}{160749696474}a^{4}-\frac{127947336209}{17861077386}a^{3}+\frac{117354736487}{80374848237}a^{2}+\frac{69663760075}{53583232158}a-\frac{20042193745}{8930538693}$, $\frac{12840854473}{160749696474}a^{19}-\frac{17432499437}{80374848237}a^{18}+\frac{23836368646}{26791616079}a^{17}-\frac{126506651543}{80374848237}a^{16}+\frac{768728988025}{160749696474}a^{15}-\frac{213891145987}{26791616079}a^{14}+\frac{2463277891571}{160749696474}a^{13}-\frac{3117561618563}{160749696474}a^{12}+\frac{27100199469}{992282077}a^{11}-\frac{2364919770511}{80374848237}a^{10}+\frac{2314562250607}{80374848237}a^{9}-\frac{491393050895}{26791616079}a^{8}+\frac{1238771932081}{160749696474}a^{7}+\frac{398126786395}{80374848237}a^{6}-\frac{265452731380}{26791616079}a^{5}+\frac{613400999245}{80374848237}a^{4}-\frac{999060851837}{160749696474}a^{3}+\frac{5169450455}{26791616079}a^{2}+\frac{17598317125}{17861077386}a-\frac{3863471939}{1984564154}$ 8790299405/160749696474a^19 - 631088647/2976846231a^18 + 65448212309/80374848237a^17 - 151559429458/80374848237a^16 + 261979653119/53583232158a^15 - 798878491865/80374848237a^14 + 3000320133685/160749696474a^13 - 1540073104855/53583232158a^12 + 3197858912038/80374848237a^11 - 3918080524121/80374848237a^10 + 463078193501/8930538693a^9 - 3618933903961/80374848237a^8 + 4693141301909/160749696474a^7 - 198472121813/26791616079a^6 - 714672914461/80374848237a^5 + 1030096993613/80374848237a^4 - 461453323589/53583232158a^3 + 202763110753/80374848237a^2 + 88159284689/53583232158a - 18888148765/17861077386, 232294184/2976846231a^19 - 4375258585/17861077386a^18 + 1864489903/1984564154a^17 - 16716790891/8930538693a^16 + 45999889924/8930538693a^15 - 171213886615/17861077386a^14 + 155791007006/8930538693a^13 - 436345370111/17861077386a^12 + 293119401211/8930538693a^11 - 346710509297/8930538693a^10 + 345828051655/8930538693a^9 - 270456812641/8930538693a^8 + 156481297835/8930538693a^7 - 60526036675/17861077386a^6 - 93045477361/17861077386a^5 + 27530735567/8930538693a^4 - 11474653574/2976846231a^3 + 36576409739/17861077386a^2 + 2854680509/2976846231a - 1047062401/1984564154, 131168344/26791616079a^19 - 171825118/8930538693a^18 + 2902494974/26791616079a^17 - 6475799446/26791616079a^16 + 14018849795/17861077386a^15 - 36609872390/26791616079a^14 + 93696736895/26791616079a^13 - 88327518775/17861077386a^12 + 237647651914/26791616079a^11 - 262165459223/26791616079a^10 + 120297569102/8930538693a^9 - 324291972871/26791616079a^8 + 281400943138/26791616079a^7 - 46805965162/8930538693a^6 + 9628744379/26791616079a^5 + 72425351309/26791616079a^4 - 57926242633/17861077386a^3 + 40151084938/26791616079a^2 - 12397553399/8930538693a - 763318987/5953692462, 59493167/8930538693a^19 + 351504113/17861077386a^18 - 65041897/1984564154a^17 + 833609363/2976846231a^16 - 924494126/2976846231a^15 + 8144417683/5953692462a^14 - 6161140609/2976846231a^13 + 77243002721/17861077386a^12 - 38518465243/8930538693a^11 + 46588253008/8930538693a^10 - 4098645002/992282077a^9 + 22448434121/8930538693a^8 + 26798727658/8930538693a^7 - 43092374417/5953692462a^6 + 176511914581/17861077386a^5 - 53174856041/8930538693a^4 + 445675446/992282077a^3 + 8096492287/17861077386a^2 - 3197770787/2976846231a + 2280304913/1984564154, 13768305367/80374848237a^19 - 89887163761/160749696474a^18 + 171165249709/80374848237a^17 - 690119104955/160749696474a^16 + 936310528136/80374848237a^15 - 1741568664886/80374848237a^14 + 6380092537231/160749696474a^13 - 8697616965551/160749696474a^12 + 5820872747054/80374848237a^11 - 6474876953948/80374848237a^10 + 6414630980293/80374848237a^9 - 4150929103853/80374848237a^8 + 1731998179840/80374848237a^7 + 2591516871791/160749696474a^6 - 2338776390116/80374848237a^5 + 3452271231337/160749696474a^4 - 720213238612/80374848237a^3 + 287725545524/80374848237a^2 + 228107558467/53583232158a - 67092503603/17861077386, 833215090/26791616079a^19 - 6267694247/80374848237a^18 + 59427857635/160749696474a^17 - 17546690195/26791616079a^16 + 176162056589/80374848237a^15 - 568630274911/160749696474a^14 + 69285315953/8930538693a^13 - 838426509883/80374848237a^12 + 1359261057901/80374848237a^11 - 511147921682/26791616079a^10 + 1831535247715/80374848237a^9 - 1598143062670/80374848237a^8 + 152211339449/8930538693a^7 - 598310908085/80374848237a^6 + 406351434997/160749696474a^5 + 20025887744/8930538693a^4 - 251957046289/80374848237a^3 + 220512020069/160749696474a^2 - 18422400368/26791616079a - 214813235/8930538693, 8586405989/80374848237a^19 - 12930673303/80374848237a^18 + 66580039598/80374848237a^17 - 51871620443/80374848237a^16 + 600043050269/160749696474a^15 - 221468657468/80374848237a^14 + 557346521641/80374848237a^13 - 25258171423/160749696474a^12 + 254186572393/80374848237a^11 + 652719194156/80374848237a^10 - 1007920008844/80374848237a^9 + 2237548306319/80374848237a^8 - 1871009001550/80374848237a^7 + 2171869843184/80374848237a^6 - 618963999358/80374848237a^5 - 225094201310/80374848237a^4 + 233913827273/160749696474a^3 - 405087883574/80374848237a^2 + 117811242685/26791616079a + 27317387087/17861077386, 4097738534/80374848237a^19 - 6687543715/53583232158a^18 + 41555475844/80374848237a^17 - 136308931201/160749696474a^16 + 145078093415/53583232158a^15 - 347809726951/80374848237a^14 + 1300350359873/160749696474a^13 - 267568406960/26791616079a^12 + 1118645131295/80374848237a^11 - 1222565262940/80374848237a^10 + 364867830463/26791616079a^9 - 654849914747/80374848237a^8 + 229356247673/80374848237a^7 + 61279628845/17861077386a^6 - 544104887936/80374848237a^5 + 740715936011/160749696474a^4 - 127947336209/17861077386a^3 + 117354736487/80374848237a^2 + 69663760075/53583232158a - 20042193745/8930538693, 12840854473/160749696474a^19 - 17432499437/80374848237a^18 + 23836368646/26791616079a^17 - 126506651543/80374848237a^16 + 768728988025/160749696474a^15 - 213891145987/26791616079a^14 + 2463277891571/160749696474a^13 - 3117561618563/160749696474a^12 + 27100199469/992282077a^11 - 2364919770511/80374848237a^10 + 2314562250607/80374848237a^9 - 491393050895/26791616079a^8 + 1238771932081/160749696474a^7 + 398126786395/80374848237a^6 - 265452731380/26791616079a^5 + 613400999245/80374848237a^4 - 999060851837/160749696474a^3 + 5169450455/26791616079a^2 + 17598317125/17861077386*a - 3863471939/1984564154	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:	$ 314083.885561 $	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 314083.885561 \cdot 2}{6\cdot\sqrt{161053520503936294265094144}}\cr\approx \mathstrut & 0.791112000694 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 12*x^18 - 23*x^17 + 66*x^16 - 117*x^15 + 222*x^14 - 297*x^13 + 411*x^12 - 454*x^11 + 456*x^10 - 312*x^9 + 147*x^8 + 69*x^7 - 150*x^6 + 127*x^5 - 78*x^4 + 21*x^3 + 16*x^2 - 15*x + 9)

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 - 3*x^19 + 12*x^18 - 23*x^17 + 66*x^16 - 117*x^15 + 222*x^14 - 297*x^13 + 411*x^12 - 454*x^11 + 456*x^10 - 312*x^9 + 147*x^8 + 69*x^7 - 150*x^6 + 127*x^5 - 78*x^4 + 21*x^3 + 16*x^2 - 15*x + 9, 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 - 3*x^19 + 12*x^18 - 23*x^17 + 66*x^16 - 117*x^15 + 222*x^14 - 297*x^13 + 411*x^12 - 454*x^11 + 456*x^10 - 312*x^9 + 147*x^8 + 69*x^7 - 150*x^6 + 127*x^5 - 78*x^4 + 21*x^3 + 16*x^2 - 15*x + 9);

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 - 3*x^19 + 12*x^18 - 23*x^17 + 66*x^16 - 117*x^15 + 222*x^14 - 297*x^13 + 411*x^12 - 454*x^11 + 456*x^10 - 312*x^9 + 147*x^8 + 69*x^7 - 150*x^6 + 127*x^5 - 78*x^4 + 21*x^3 + 16*x^2 - 15*x + 9);

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):

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 120

The 7 conjugacy class representatives for $S_5$

Character table for $S_5$

Intermediate fields

$\Q(\sqrt{-3}) $, 10.4.1410076263168.1

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 5 sibling:	5.3.228528.1
Degree 6 sibling:	6.0.18510768.2
Degree 10 siblings:	data not computed
Degree 12 sibling:	data not computed
Degree 15 sibling:	data not computed
Degree 20 siblings:	data not computed
Degree 24 sibling:	data not computed
Degree 30 siblings:	data not computed
Degree 40 sibling:	data not computed
Minimal sibling:	5.3.228528.1

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	${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.2.0.1}{2} }$	${\href{/padicField/7.5.0.1}{5} }^{4}$	${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$	${\href{/padicField/13.5.0.1}{5} }^{4}$	${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$	${\href{/padicField/19.3.0.1}{3} }^{6}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$	R	${\href{/padicField/29.6.0.1}{6} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }$	${\href{/padicField/31.3.0.1}{3} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$	${\href{/padicField/37.3.0.1}{3} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$	${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$	${\href{/padicField/43.5.0.1}{5} }^{4}$	${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$	${\href{/padicField/53.6.0.1}{6} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }$	${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }$

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

# 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	2.10.8.1	$x^{10} + 5 x^{9} + 15 x^{8} + 30 x^{7} + 45 x^{6} + 55 x^{5} + 55 x^{4} + 10 x^{3} - 25 x^{2} - 5 x + 7$	$5$	$2$	$8$	$F_5$	$[\ ]_{5}^{4}$
$2$ 2	2.10.8.1	$x^{10} + 5 x^{9} + 15 x^{8} + 30 x^{7} + 45 x^{6} + 55 x^{5} + 55 x^{4} + 10 x^{3} - 25 x^{2} - 5 x + 7$	$5$	$2$	$8$	$F_5$	$[\ ]_{5}^{4}$
$3$ 3	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.6.7.5	$x^{6} + 6 x^{2} + 3$	$6$	$1$	$7$	$D_{6}$	$[3/2]_{2}^{2}$
	3.12.14.11	$x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 792 x^{8} + 1728 x^{7} + 2918 x^{6} + 3684 x^{5} + 3156 x^{4} + 1376 x^{3} - 36 x^{2} - 168 x + 25$	$6$	$2$	$14$	$D_6$	$[3/2]_{2}^{2}$
$23$ 23	23.2.0.1	$x^{2} + 21 x + 5$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	23.2.0.1	$x^{2} + 21 x + 5$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	23.4.2.2	$x^{4} - 483 x^{2} + 2645$	$2$	$2$	$2$	$C_4$	$[\ ]_{2}^{2}$
	23.4.2.2	$x^{4} - 483 x^{2} + 2645$	$2$	$2$	$2$	$C_4$	$[\ ]_{2}^{2}$
	23.4.2.2	$x^{4} - 483 x^{2} + 2645$	$2$	$2$	$2$	$C_4$	$[\ ]_{2}^{2}$
	23.4.2.2	$x^{4} - 483 x^{2} + 2645$	$2$	$2$	$2$	$C_4$	$[\ ]_{2}^{2}$

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