LMFDB - Number field 20.0.623866452951915562152862548828125.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 20*x^18 + 170*x^16 - 800*x^14 + 2275*x^12 - 3480*x^10 - 950*x^8 + 15700*x^6 - 25375*x^4 + 13000*x^2 + 28880)

gp: K = bnfinit(y^20 - 20*y^18 + 170*y^16 - 800*y^14 + 2275*y^12 - 3480*y^10 - 950*y^8 + 15700*y^6 - 25375*y^4 + 13000*y^2 + 28880, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 20*x^18 + 170*x^16 - 800*x^14 + 2275*x^12 - 3480*x^10 - 950*x^8 + 15700*x^6 - 25375*x^4 + 13000*x^2 + 28880);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 20*x^18 + 170*x^16 - 800*x^14 + 2275*x^12 - 3480*x^10 - 950*x^8 + 15700*x^6 - 25375*x^4 + 13000*x^2 + 28880)

$ x^{20} - 20 x^{18} + 170 x^{16} - 800 x^{14} + 2275 x^{12} - 3480 x^{10} - 950 x^{8} + 15700 x^{6} + \cdots + 28880 $ x^20 - 20*x^18 + 170*x^16 - 800*x^14 + 2275*x^12 - 3480*x^10 - 950*x^8 + 15700*x^6 - 25375*x^4 + 13000*x^2 + 28880

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:	$623866452951915562152862548828125$ 623866452951915562152862548828125 $\medspace = 5^{35}\cdot 11^{8}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$43.63$	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:	$5^{7/4}11^{1/2}\approx 55.449016844865795$
Ramified primes:	$5$, $11$ 5, 11	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{5}) $
$\card{ \Aut(K/\Q) }$:	$4$	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 $a$)

$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{4}a^{5}-\frac{1}{4}a^{3}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{7}-\frac{1}{2}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{8}a^{8}-\frac{1}{8}a^{7}-\frac{1}{8}a^{5}-\frac{1}{8}a^{3}+\frac{3}{8}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{7}-\frac{1}{8}a^{6}-\frac{1}{8}a^{5}-\frac{1}{8}a^{4}-\frac{1}{4}a^{3}+\frac{3}{8}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{176}a^{10}-\frac{5}{88}a^{8}-\frac{9}{176}a^{6}-\frac{3}{88}a^{4}-\frac{19}{176}a^{2}-\frac{1}{2}a-\frac{1}{44}$, $\frac{1}{352}a^{11}-\frac{5}{176}a^{9}+\frac{35}{352}a^{7}-\frac{1}{8}a^{6}+\frac{19}{176}a^{5}-\frac{1}{8}a^{4}-\frac{63}{352}a^{3}+\frac{3}{8}a^{2}-\frac{23}{88}a$, $\frac{1}{352}a^{12}-\frac{21}{352}a^{8}+\frac{9}{88}a^{6}-\frac{35}{352}a^{4}+\frac{57}{176}a^{2}-\frac{1}{2}a+\frac{17}{44}$, $\frac{1}{704}a^{13}-\frac{1}{704}a^{12}-\frac{1}{704}a^{11}+\frac{3}{64}a^{9}+\frac{21}{704}a^{8}-\frac{43}{704}a^{7}-\frac{9}{176}a^{6}-\frac{29}{704}a^{5}-\frac{141}{704}a^{4}-\frac{175}{704}a^{3}-\frac{57}{352}a^{2}-\frac{9}{176}a-\frac{17}{88}$, $\frac{1}{704}a^{14}-\frac{1}{704}a^{11}+\frac{1}{704}a^{10}-\frac{17}{352}a^{9}-\frac{1}{88}a^{8}-\frac{79}{704}a^{7}+\frac{31}{704}a^{6}+\frac{3}{352}a^{5}-\frac{53}{352}a^{4}-\frac{25}{704}a^{3}+\frac{167}{352}a^{2}-\frac{87}{176}a-\frac{1}{8}$, $\frac{1}{13376}a^{15}+\frac{1}{3344}a^{13}-\frac{1}{704}a^{12}-\frac{5}{13376}a^{11}-\frac{1}{352}a^{10}-\frac{35}{836}a^{9}+\frac{41}{704}a^{8}-\frac{1355}{13376}a^{7}-\frac{9}{352}a^{6}-\frac{545}{6688}a^{5}+\frac{47}{704}a^{4}-\frac{41}{1672}a^{3}+\frac{69}{176}a^{2}+\frac{305}{836}a+\frac{7}{22}$, $\frac{1}{26752}a^{16}-\frac{15}{26752}a^{14}+\frac{7}{13376}a^{12}-\frac{1}{704}a^{11}+\frac{29}{26752}a^{10}-\frac{17}{352}a^{9}-\frac{497}{13376}a^{8}+\frac{9}{704}a^{7}-\frac{3123}{26752}a^{6}-\frac{41}{352}a^{5}+\frac{4061}{26752}a^{4}-\frac{113}{704}a^{3}-\frac{205}{6688}a^{2}+\frac{45}{176}a-\frac{2}{11}$, $\frac{1}{26752}a^{17}-\frac{1}{26752}a^{15}-\frac{3}{13376}a^{13}+\frac{35}{26752}a^{11}-\frac{655}{13376}a^{9}-\frac{2105}{26752}a^{7}-\frac{1}{8}a^{6}+\frac{1037}{26752}a^{5}-\frac{1}{8}a^{4}-\frac{343}{1672}a^{3}+\frac{3}{8}a^{2}-\frac{461}{1672}a-\frac{1}{2}$, $\frac{1}{53504}a^{18}-\frac{1}{53504}a^{17}+\frac{1}{53504}a^{15}+\frac{17}{53504}a^{14}+\frac{3}{26752}a^{13}-\frac{27}{53504}a^{12}-\frac{35}{53504}a^{11}-\frac{27}{53504}a^{10}-\frac{1017}{26752}a^{9}-\frac{591}{53504}a^{8}+\frac{5449}{53504}a^{7}-\frac{303}{13376}a^{6}+\frac{2307}{53504}a^{5}+\frac{3285}{53504}a^{4}+\frac{761}{3344}a^{3}-\frac{65}{6688}a^{2}-\frac{375}{3344}a+\frac{5}{16}$, $\frac{1}{53504}a^{19}-\frac{1}{53504}a^{17}-\frac{1}{53504}a^{16}-\frac{1}{26752}a^{15}-\frac{23}{53504}a^{14}-\frac{25}{53504}a^{13}+\frac{31}{26752}a^{12}-\frac{1}{1408}a^{11}+\frac{85}{53504}a^{10}-\frac{2293}{53504}a^{9}+\frac{763}{26752}a^{8}+\frac{1317}{53504}a^{7}+\frac{3313}{53504}a^{6}-\frac{391}{13376}a^{5}-\frac{3605}{53504}a^{4}-\frac{2115}{13376}a^{3}+\frac{541}{1672}a^{2}+\frac{1087}{3344}a+\frac{59}{176}$ 1, a, a^2, 1/2*a^3 - 1/2*a^2 - 1/2*a, 1/2*a^4 - 1/2*a, 1/4*a^5 - 1/4*a^3 + 1/4*a - 1/2, 1/4*a^6 - 1/4*a^4 + 1/4*a^2 - 1/2*a, 1/4*a^7 - 1/2*a^2 + 1/4*a - 1/2, 1/8*a^8 - 1/8*a^7 - 1/8*a^5 - 1/8*a^3 + 3/8*a^2 - 1/2*a - 1/2, 1/8*a^9 - 1/8*a^7 - 1/8*a^6 - 1/8*a^5 - 1/8*a^4 - 1/4*a^3 + 3/8*a^2 - 1/2*a - 1/2, 1/176*a^10 - 5/88*a^8 - 9/176*a^6 - 3/88*a^4 - 19/176*a^2 - 1/2*a - 1/44, 1/352*a^11 - 5/176*a^9 + 35/352*a^7 - 1/8*a^6 + 19/176*a^5 - 1/8*a^4 - 63/352*a^3 + 3/8*a^2 - 23/88*a, 1/352*a^12 - 21/352*a^8 + 9/88*a^6 - 35/352*a^4 + 57/176*a^2 - 1/2*a + 17/44, 1/704*a^13 - 1/704*a^12 - 1/704*a^11 + 3/64*a^9 + 21/704*a^8 - 43/704*a^7 - 9/176*a^6 - 29/704*a^5 - 141/704*a^4 - 175/704*a^3 - 57/352*a^2 - 9/176*a - 17/88, 1/704*a^14 - 1/704*a^11 + 1/704*a^10 - 17/352*a^9 - 1/88*a^8 - 79/704*a^7 + 31/704*a^6 + 3/352*a^5 - 53/352*a^4 - 25/704*a^3 + 167/352*a^2 - 87/176*a - 1/8, 1/13376*a^15 + 1/3344*a^13 - 1/704*a^12 - 5/13376*a^11 - 1/352*a^10 - 35/836*a^9 + 41/704*a^8 - 1355/13376*a^7 - 9/352*a^6 - 545/6688*a^5 + 47/704*a^4 - 41/1672*a^3 + 69/176*a^2 + 305/836*a + 7/22, 1/26752*a^16 - 15/26752*a^14 + 7/13376*a^12 - 1/704*a^11 + 29/26752*a^10 - 17/352*a^9 - 497/13376*a^8 + 9/704*a^7 - 3123/26752*a^6 - 41/352*a^5 + 4061/26752*a^4 - 113/704*a^3 - 205/6688*a^2 + 45/176*a - 2/11, 1/26752*a^17 - 1/26752*a^15 - 3/13376*a^13 + 35/26752*a^11 - 655/13376*a^9 - 2105/26752*a^7 - 1/8*a^6 + 1037/26752*a^5 - 1/8*a^4 - 343/1672*a^3 + 3/8*a^2 - 461/1672*a - 1/2, 1/53504*a^18 - 1/53504*a^17 + 1/53504*a^15 + 17/53504*a^14 + 3/26752*a^13 - 27/53504*a^12 - 35/53504*a^11 - 27/53504*a^10 - 1017/26752*a^9 - 591/53504*a^8 + 5449/53504*a^7 - 303/13376*a^6 + 2307/53504*a^5 + 3285/53504*a^4 + 761/3344*a^3 - 65/6688*a^2 - 375/3344*a + 5/16, 1/53504*a^19 - 1/53504*a^17 - 1/53504*a^16 - 1/26752*a^15 - 23/53504*a^14 - 25/53504*a^13 + 31/26752*a^12 - 1/1408*a^11 + 85/53504*a^10 - 2293/53504*a^9 + 763/26752*a^8 + 1317/53504*a^7 + 3313/53504*a^6 - 391/13376*a^5 - 3605/53504*a^4 - 2115/13376*a^3 + 541/1672*a^2 + 1087/3344*a + 59/176

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$2$, $11$

Class group and class number

$C_{5}$, which has order $5$ (assuming GRH)

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{5}{13376} a^{15} - \frac{75}{13376} a^{13} + \frac{225}{6688} a^{11} - \frac{1}{352} a^{10} - \frac{125}{1216} a^{9} + \frac{5}{176} a^{8} + \frac{1125}{6688} a^{7} - \frac{35}{352} a^{6} + \frac{41}{13376} a^{5} + \frac{25}{176} a^{4} - \frac{8955}{13376} a^{3} - \frac{25}{352} a^{2} + \frac{2395}{3344} a - \frac{43}{88} $ (5)/(13376)a^(15) - (75)/(13376)a^(13) + (225)/(6688)a^(11) - (1)/(352)a^(10) - (125)/(1216)a^(9) + (5)/(176)a^(8) + (1125)/(6688)a^(7) - (35)/(352)a^(6) + (41)/(13376)a^(5) + (25)/(176)a^(4) - (8955)/(13376)a^(3) - (25)/(352)a^(2) + (2395)/(3344)*a - (43)/(88) (order $10$)	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:	$\frac{1}{13376}a^{15}-\frac{15}{13376}a^{13}+\frac{45}{6688}a^{11}-\frac{1}{352}a^{10}-\frac{25}{1216}a^{9}+\frac{5}{176}a^{8}+\frac{225}{6688}a^{7}-\frac{35}{352}a^{6}+\frac{677}{13376}a^{5}+\frac{25}{176}a^{4}-\frac{5135}{13376}a^{3}-\frac{25}{352}a^{2}+\frac{1315}{3344}a+\frac{1}{88}$, $\frac{3}{26752}a^{19}+\frac{17}{26752}a^{18}-\frac{5}{3344}a^{17}-\frac{127}{13376}a^{16}+\frac{211}{26752}a^{15}+\frac{1553}{26752}a^{14}-\frac{577}{26752}a^{13}-\frac{5079}{26752}a^{12}+\frac{921}{26752}a^{11}+\frac{9693}{26752}a^{10}+\frac{679}{26752}a^{9}-\frac{2443}{26752}a^{8}-\frac{2847}{13376}a^{7}-\frac{575}{418}a^{6}+\frac{5915}{26752}a^{5}+\frac{63803}{26752}a^{4}-\frac{141}{704}a^{3}-\frac{13315}{6688}a^{2}-\frac{1963}{3344}a-\frac{71}{22}$, $\frac{1}{4864}a^{19}-\frac{1}{2432}a^{18}-\frac{17}{4864}a^{17}+\frac{19}{2816}a^{16}+\frac{639}{26752}a^{15}-\frac{2407}{53504}a^{14}-\frac{4603}{53504}a^{13}+\frac{97}{608}a^{12}+\frac{251}{1408}a^{11}-\frac{17475}{53504}a^{10}-\frac{541}{4864}a^{9}+\frac{587}{3344}a^{8}-\frac{30733}{53504}a^{7}+\frac{57463}{53504}a^{6}+\frac{15307}{13376}a^{5}-\frac{111581}{53504}a^{4}-\frac{18019}{13376}a^{3}+\frac{490}{209}a^{2}-\frac{6293}{3344}a+\frac{523}{176}$, $\frac{3}{53504}a^{19}+\frac{1}{13376}a^{18}-\frac{67}{53504}a^{17}-\frac{89}{53504}a^{16}+\frac{299}{26752}a^{15}+\frac{833}{53504}a^{14}-\frac{2623}{53504}a^{13}-\frac{2007}{26752}a^{12}+\frac{135}{1408}a^{11}+\frac{9509}{53504}a^{10}+\frac{357}{53504}a^{9}-\frac{2283}{26752}a^{8}-\frac{18621}{53504}a^{7}-\frac{28811}{53504}a^{6}+\frac{1621}{3344}a^{5}+\frac{3669}{2816}a^{4}-\frac{131}{704}a^{3}-\frac{1245}{3344}a^{2}-\frac{1351}{3344}a-\frac{23}{16}$, $\frac{3}{53504}a^{19}-\frac{1}{13376}a^{18}-\frac{67}{53504}a^{17}+\frac{89}{53504}a^{16}+\frac{299}{26752}a^{15}-\frac{833}{53504}a^{14}-\frac{2623}{53504}a^{13}+\frac{2007}{26752}a^{12}+\frac{135}{1408}a^{11}-\frac{9509}{53504}a^{10}+\frac{357}{53504}a^{9}+\frac{2283}{26752}a^{8}-\frac{18621}{53504}a^{7}+\frac{28811}{53504}a^{6}+\frac{1621}{3344}a^{5}-\frac{3669}{2816}a^{4}-\frac{131}{704}a^{3}+\frac{1245}{3344}a^{2}-\frac{1351}{3344}a+\frac{23}{16}$, $\frac{13}{53504}a^{19}-\frac{3}{53504}a^{18}-\frac{65}{13376}a^{17}+\frac{49}{53504}a^{16}+\frac{2217}{53504}a^{15}-\frac{73}{13376}a^{14}-\frac{10543}{53504}a^{13}+\frac{387}{53504}a^{12}+\frac{31421}{53504}a^{11}+\frac{1031}{13376}a^{10}-\frac{59971}{53504}a^{9}-\frac{2491}{4864}a^{8}+\frac{3529}{3344}a^{7}+\frac{87235}{53504}a^{6}+\frac{16901}{53504}a^{5}-\frac{7135}{2432}a^{4}-\frac{3557}{6688}a^{3}+\frac{2873}{3344}a^{2}+\frac{2561}{3344}a+\frac{323}{88}$, $\frac{1}{4864}a^{19}+\frac{17}{53504}a^{18}-\frac{27}{6688}a^{17}-\frac{371}{53504}a^{16}+\frac{1675}{53504}a^{15}+\frac{35}{608}a^{14}-\frac{6909}{53504}a^{13}-\frac{13405}{53504}a^{12}+\frac{17095}{53504}a^{11}+\frac{4369}{6688}a^{10}-\frac{21281}{53504}a^{9}-\frac{49393}{53504}a^{8}-\frac{579}{1216}a^{7}-\frac{35373}{53504}a^{6}+\frac{97131}{53504}a^{5}+\frac{95977}{26752}a^{4}-\frac{10229}{3344}a^{3}-\frac{22175}{3344}a^{2}-\frac{479}{3344}a+\frac{109}{88}$, $\frac{7}{53504}a^{19}+\frac{3}{26752}a^{18}-\frac{129}{53504}a^{17}-\frac{85}{53504}a^{16}+\frac{259}{13376}a^{15}+\frac{427}{53504}a^{14}-\frac{5003}{53504}a^{13}-\frac{205}{13376}a^{12}+\frac{2197}{6688}a^{11}+\frac{527}{53504}a^{10}-\frac{47439}{53504}a^{9}-\frac{113}{13376}a^{8}+\frac{8423}{4864}a^{7}+\frac{3621}{53504}a^{6}-\frac{64455}{26752}a^{5}-\frac{47315}{53504}a^{4}+\frac{19559}{13376}a^{3}+\frac{137}{76}a^{2}+\frac{8919}{3344}a+\frac{301}{176}$, $\frac{5}{53504}a^{19}+\frac{1}{1408}a^{18}-\frac{113}{53504}a^{17}-\frac{751}{53504}a^{16}+\frac{543}{26752}a^{15}+\frac{6401}{53504}a^{14}-\frac{5785}{53504}a^{13}-\frac{971}{1672}a^{12}+\frac{9377}{26752}a^{11}+\frac{95413}{53504}a^{10}-\frac{36349}{53504}a^{9}-\frac{5501}{1672}a^{8}+\frac{22417}{53504}a^{7}+\frac{84411}{53504}a^{6}+\frac{21631}{13376}a^{5}+\frac{436043}{53504}a^{4}-\frac{56237}{13376}a^{3}-\frac{69797}{3344}a^{2}+\frac{16653}{3344}a+\frac{349}{16}$ 1/13376a^15 - 15/13376a^13 + 45/6688a^11 - 1/352a^10 - 25/1216a^9 + 5/176a^8 + 225/6688a^7 - 35/352a^6 + 677/13376a^5 + 25/176a^4 - 5135/13376a^3 - 25/352a^2 + 1315/3344a + 1/88, 3/26752a^19 + 17/26752a^18 - 5/3344a^17 - 127/13376a^16 + 211/26752a^15 + 1553/26752a^14 - 577/26752a^13 - 5079/26752a^12 + 921/26752a^11 + 9693/26752a^10 + 679/26752a^9 - 2443/26752a^8 - 2847/13376a^7 - 575/418a^6 + 5915/26752a^5 + 63803/26752a^4 - 141/704a^3 - 13315/6688a^2 - 1963/3344a - 71/22, 1/4864a^19 - 1/2432a^18 - 17/4864a^17 + 19/2816a^16 + 639/26752a^15 - 2407/53504a^14 - 4603/53504a^13 + 97/608a^12 + 251/1408a^11 - 17475/53504a^10 - 541/4864a^9 + 587/3344a^8 - 30733/53504a^7 + 57463/53504a^6 + 15307/13376a^5 - 111581/53504a^4 - 18019/13376a^3 + 490/209a^2 - 6293/3344a + 523/176, 3/53504a^19 + 1/13376a^18 - 67/53504a^17 - 89/53504a^16 + 299/26752a^15 + 833/53504a^14 - 2623/53504a^13 - 2007/26752a^12 + 135/1408a^11 + 9509/53504a^10 + 357/53504a^9 - 2283/26752a^8 - 18621/53504a^7 - 28811/53504a^6 + 1621/3344a^5 + 3669/2816a^4 - 131/704a^3 - 1245/3344a^2 - 1351/3344a - 23/16, 3/53504a^19 - 1/13376a^18 - 67/53504a^17 + 89/53504a^16 + 299/26752a^15 - 833/53504a^14 - 2623/53504a^13 + 2007/26752a^12 + 135/1408a^11 - 9509/53504a^10 + 357/53504a^9 + 2283/26752a^8 - 18621/53504a^7 + 28811/53504a^6 + 1621/3344a^5 - 3669/2816a^4 - 131/704a^3 + 1245/3344a^2 - 1351/3344a + 23/16, 13/53504a^19 - 3/53504a^18 - 65/13376a^17 + 49/53504a^16 + 2217/53504a^15 - 73/13376a^14 - 10543/53504a^13 + 387/53504a^12 + 31421/53504a^11 + 1031/13376a^10 - 59971/53504a^9 - 2491/4864a^8 + 3529/3344a^7 + 87235/53504a^6 + 16901/53504a^5 - 7135/2432a^4 - 3557/6688a^3 + 2873/3344a^2 + 2561/3344a + 323/88, 1/4864a^19 + 17/53504a^18 - 27/6688a^17 - 371/53504a^16 + 1675/53504a^15 + 35/608a^14 - 6909/53504a^13 - 13405/53504a^12 + 17095/53504a^11 + 4369/6688a^10 - 21281/53504a^9 - 49393/53504a^8 - 579/1216a^7 - 35373/53504a^6 + 97131/53504a^5 + 95977/26752a^4 - 10229/3344a^3 - 22175/3344a^2 - 479/3344a + 109/88, 7/53504a^19 + 3/26752a^18 - 129/53504a^17 - 85/53504a^16 + 259/13376a^15 + 427/53504a^14 - 5003/53504a^13 - 205/13376a^12 + 2197/6688a^11 + 527/53504a^10 - 47439/53504a^9 - 113/13376a^8 + 8423/4864a^7 + 3621/53504a^6 - 64455/26752a^5 - 47315/53504a^4 + 19559/13376a^3 + 137/76a^2 + 8919/3344a + 301/176, 5/53504a^19 + 1/1408a^18 - 113/53504a^17 - 751/53504a^16 + 543/26752a^15 + 6401/53504a^14 - 5785/53504a^13 - 971/1672a^12 + 9377/26752a^11 + 95413/53504a^10 - 36349/53504a^9 - 5501/1672a^8 + 22417/53504a^7 + 84411/53504a^6 + 21631/13376a^5 + 436043/53504a^4 - 56237/13376a^3 - 69797/3344a^2 + 16653/3344*a + 349/16 (assuming GRH)	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:	$ 403949827.0826757 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 403949827.0826757 \cdot 5}{10\cdot\sqrt{623866452951915562152862548828125}}\cr\approx \mathstrut & 0.775443746442569 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^20 - 20*x^18 + 170*x^16 - 800*x^14 + 2275*x^12 - 3480*x^10 - 950*x^8 + 15700*x^6 - 25375*x^4 + 13000*x^2 + 28880)

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 - 20*x^18 + 170*x^16 - 800*x^14 + 2275*x^12 - 3480*x^10 - 950*x^8 + 15700*x^6 - 25375*x^4 + 13000*x^2 + 28880, 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 - 20*x^18 + 170*x^16 - 800*x^14 + 2275*x^12 - 3480*x^10 - 950*x^8 + 15700*x^6 - 25375*x^4 + 13000*x^2 + 28880);

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 - 20*x^18 + 170*x^16 - 800*x^14 + 2275*x^12 - 3480*x^10 - 950*x^8 + 15700*x^6 - 25375*x^4 + 13000*x^2 + 28880);

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

$C_4\times D_5$ (as 20T6):

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 solvable group of order 40

The 16 conjugacy class representatives for $C_4\times D_5$

Character table for $C_4\times D_5$

Intermediate fields

$\Q(\sqrt{5}) $, $\Q(\zeta_{5})$, 5.1.47265625.1, 10.2.11170196533203125.1

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

Galois closure:	deg 40
Degree 20 sibling:	20.4.75487840807181783020496368408203125.1
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	${\href{/padicField/2.4.0.1}{4} }^{5}$	$20$	R	${\href{/padicField/7.4.0.1}{4} }^{5}$	R	${\href{/padicField/13.4.0.1}{4} }^{5}$	${\href{/padicField/17.4.0.1}{4} }^{5}$	${\href{/padicField/19.2.0.1}{2} }^{10}$	$20$	${\href{/padicField/29.2.0.1}{2} }^{10}$	${\href{/padicField/31.5.0.1}{5} }^{4}$	${\href{/padicField/37.4.0.1}{4} }^{5}$	${\href{/padicField/41.2.0.1}{2} }^{8}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$	${\href{/padicField/43.4.0.1}{4} }^{5}$	${\href{/padicField/47.4.0.1}{4} }^{5}$	$20$	${\href{/padicField/59.10.0.1}{10} }^{2}$

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
$5$ 5		Deg $20$	$20$	$1$	$35$
$11$ 11	$\Q_{11}$	$x + 9$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{11}$	$x + 9$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{11}$	$x + 9$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{11}$	$x + 9$	$1$	$1$	$0$	Trivial	$[\ ]$
	11.2.1.2	$x^{2} + 11$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	11.2.1.2	$x^{2} + 11$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	11.2.1.2	$x^{2} + 11$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	11.2.1.2	$x^{2} + 11$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	11.2.1.2	$x^{2} + 11$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	11.2.1.2	$x^{2} + 11$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	11.2.1.2	$x^{2} + 11$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	11.2.1.2	$x^{2} + 11$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	$\Q$	$C_1$	$1$	$1$
*	1.5.2t1.a.a	$1$	$ 5 $	$\Q(\sqrt{5}) $	$C_2$ (as 2T1)	$1$	$1$
	1.55.2t1.a.a	$1$	$ 5 \cdot 11 $	$\Q(\sqrt{-55}) $	$C_2$ (as 2T1)	$1$	$-1$
	1.11.2t1.a.a	$1$	$ 11 $	$\Q(\sqrt{-11}) $	$C_2$ (as 2T1)	$1$	$-1$
*	1.5.4t1.a.b	$1$	$ 5 $	$\Q(\zeta_{5})$	$C_4$ (as 4T1)	$0$	$-1$
	1.55.4t1.a.b	$1$	$ 5 \cdot 11 $	4.4.15125.1	$C_4$ (as 4T1)	$0$	$1$
	1.55.4t1.a.a	$1$	$ 5 \cdot 11 $	4.4.15125.1	$C_4$ (as 4T1)	$0$	$1$
*	1.5.4t1.a.a	$1$	$ 5 $	$\Q(\zeta_{5})$	$C_4$ (as 4T1)	$0$	$-1$
*	2.6875.10t3.a.b	$2$	$ 5^{4} \cdot 11 $	10.0.122872161865234375.5	$D_{10}$ (as 10T3)	$1$	$0$
*	2.6875.5t2.a.a	$2$	$ 5^{4} \cdot 11 $	5.1.47265625.1	$D_{5}$ (as 5T2)	$1$	$0$
*	2.6875.10t3.a.a	$2$	$ 5^{4} \cdot 11 $	10.0.122872161865234375.5	$D_{10}$ (as 10T3)	$1$	$0$
*	2.6875.5t2.a.b	$2$	$ 5^{4} \cdot 11 $	5.1.47265625.1	$D_{5}$ (as 5T2)	$1$	$0$
*	2.6875.20t6.a.b	$2$	$ 5^{4} \cdot 11 $	20.0.623866452951915562152862548828125.1	$C_4\times D_5$ (as 20T6)	$0$	$0$
*	2.6875.20t6.a.d	$2$	$ 5^{4} \cdot 11 $	20.0.623866452951915562152862548828125.1	$C_4\times D_5$ (as 20T6)	$0$	$0$
*	2.6875.20t6.a.a	$2$	$ 5^{4} \cdot 11 $	20.0.623866452951915562152862548828125.1	$C_4\times D_5$ (as 20T6)	$0$	$0$
*	2.6875.20t6.a.c	$2$	$ 5^{4} \cdot 11 $	20.0.623866452951915562152862548828125.1	$C_4\times D_5$ (as 20T6)	$0$	$0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.

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