LMFDB - Number field 20.0.48544842802805942483246326446533203125.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 100*x^18 + 4250*x^16 - 151*x^15 + 100000*x^14 - 11325*x^13 + 1421875*x^12 - 339750*x^11 + 12538426*x^10 - 5190625*x^9 + 68327550*x^8 - 42468750*x^7 + 228935250*x^6 - 182755451*x^5 + 484303125*x^4 - 439980025*x^3 + 600406250*x^2 - 725290750*x + 924979351)

gp: K = bnfinit(y^20 + 100*y^18 + 4250*y^16 - 151*y^15 + 100000*y^14 - 11325*y^13 + 1421875*y^12 - 339750*y^11 + 12538426*y^10 - 5190625*y^9 + 68327550*y^8 - 42468750*y^7 + 228935250*y^6 - 182755451*y^5 + 484303125*y^4 - 439980025*y^3 + 600406250*y^2 - 725290750*y + 924979351, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + 100*x^18 + 4250*x^16 - 151*x^15 + 100000*x^14 - 11325*x^13 + 1421875*x^12 - 339750*x^11 + 12538426*x^10 - 5190625*x^9 + 68327550*x^8 - 42468750*x^7 + 228935250*x^6 - 182755451*x^5 + 484303125*x^4 - 439980025*x^3 + 600406250*x^2 - 725290750*x + 924979351);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 + 100*x^18 + 4250*x^16 - 151*x^15 + 100000*x^14 - 11325*x^13 + 1421875*x^12 - 339750*x^11 + 12538426*x^10 - 5190625*x^9 + 68327550*x^8 - 42468750*x^7 + 228935250*x^6 - 182755451*x^5 + 484303125*x^4 - 439980025*x^3 + 600406250*x^2 - 725290750*x + 924979351)

$ x^{20} + 100 x^{18} + 4250 x^{16} - 151 x^{15} + 100000 x^{14} - 11325 x^{13} + 1421875 x^{12} + \cdots + 924979351 $ x^20 + 100*x^18 + 4250*x^16 - 151*x^15 + 100000*x^14 - 11325*x^13 + 1421875*x^12 - 339750*x^11 + 12538426*x^10 - 5190625*x^9 + 68327550*x^8 - 42468750*x^7 + 228935250*x^6 - 182755451*x^5 + 484303125*x^4 - 439980025*x^3 + 600406250*x^2 - 725290750*x + 924979351

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:	$48544842802805942483246326446533203125$ 48544842802805942483246326446533203125 $\medspace = 3^{10}\cdot 5^{35}\cdot 7^{10}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$76.61$	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:	$3^{1/2}5^{7/4}7^{1/2}\approx 76.61382669555769$
Ramified primes:	$3$, $5$, $7$ 3, 5, 7	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}) $
$\card{ \Gal(K/\Q) }$:	$20$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:	$525=3\cdot 5^{2}\cdot 7$
Dirichlet character group:	$\lbrace$$\chi_{525}(64,·)$, $\chi_{525}(1,·)$, $\chi_{525}(398,·)$, $\chi_{525}(272,·)$, $\chi_{525}(274,·)$, $\chi_{525}(211,·)$, $\chi_{525}(421,·)$, $\chi_{525}(188,·)$, $\chi_{525}(482,·)$, $\chi_{525}(484,·)$, $\chi_{525}(293,·)$, $\chi_{525}(167,·)$, $\chi_{525}(169,·)$, $\chi_{525}(106,·)$, $\chi_{525}(83,·)$, $\chi_{525}(503,·)$, $\chi_{525}(377,·)$, $\chi_{525}(379,·)$, $\chi_{525}(316,·)$, $\chi_{525}(62,·)$$\rbrace$
This is a CM field.
Reflex fields:	unavailable$^{512}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{41}a^{10}+\frac{9}{41}a^{8}+\frac{14}{41}a^{6}-\frac{14}{41}a^{5}+\frac{18}{41}a^{4}+\frac{19}{41}a^{3}+\frac{4}{41}a^{2}+\frac{13}{41}a+\frac{9}{41}$, $\frac{1}{41}a^{11}+\frac{9}{41}a^{9}+\frac{14}{41}a^{7}-\frac{14}{41}a^{6}+\frac{18}{41}a^{5}+\frac{19}{41}a^{4}+\frac{4}{41}a^{3}+\frac{13}{41}a^{2}+\frac{9}{41}a$, $\frac{1}{41}a^{12}+\frac{15}{41}a^{8}-\frac{14}{41}a^{7}+\frac{15}{41}a^{6}-\frac{19}{41}a^{5}+\frac{6}{41}a^{4}+\frac{6}{41}a^{3}+\frac{14}{41}a^{2}+\frac{6}{41}a+\frac{1}{41}$, $\frac{1}{30479834641}a^{13}-\frac{34265820}{30479834641}a^{12}+\frac{65}{30479834641}a^{11}+\frac{174282603}{30479834641}a^{10}+\frac{1625}{30479834641}a^{9}-\frac{1654220940}{30479834641}a^{8}-\frac{5203854707}{30479834641}a^{7}+\frac{14646569665}{30479834641}a^{6}+\frac{113750}{30479834641}a^{5}-\frac{620780076}{30479834641}a^{4}+\frac{11151443390}{30479834641}a^{3}-\frac{1064194416}{30479834641}a^{2}-\frac{14124598294}{30479834641}a-\frac{14420410979}{30479834641}$, $\frac{1}{30479834641}a^{14}+\frac{70}{30479834641}a^{12}+\frac{171329100}{30479834641}a^{11}+\frac{1925}{30479834641}a^{10}+\frac{3475815692}{30479834641}a^{9}+\frac{26250}{30479834641}a^{8}+\frac{12273697563}{30479834641}a^{7}-\frac{4460279856}{30479834641}a^{6}+\frac{12052444098}{30479834641}a^{5}+\frac{10408360914}{30479834641}a^{4}-\frac{14729180642}{30479834641}a^{3}-\frac{8920161587}{30479834641}a^{2}-\frac{8781895834}{30479834641}a+\frac{156250}{30479834641}$, $\frac{1}{30479834641}a^{15}+\frac{339704697}{30479834641}a^{12}-\frac{2625}{30479834641}a^{11}+\frac{196960694}{30479834641}a^{10}-\frac{87500}{30479834641}a^{9}-\frac{7974976620}{30479834641}a^{8}-\frac{5205055457}{30479834641}a^{7}-\frac{7372889299}{30479834641}a^{6}-\frac{11158509015}{30479834641}a^{5}-\frac{6958284170}{30479834641}a^{4}+\frac{6671554784}{30479834641}a^{3}+\frac{9212507610}{30479834641}a^{2}-\frac{5961347308}{30479834641}a-\frac{9787165441}{30479834641}$, $\frac{1}{30479834641}a^{16}-\frac{3000}{30479834641}a^{12}-\frac{324937182}{30479834641}a^{11}-\frac{110000}{30479834641}a^{10}-\frac{206926692}{30479834641}a^{9}-\frac{1687500}{30479834641}a^{8}+\frac{10759239136}{30479834641}a^{7}-\frac{13393990818}{30479834641}a^{6}-\frac{12575223349}{30479834641}a^{5}+\frac{699660601}{30479834641}a^{4}-\frac{4458135037}{30479834641}a^{3}+\frac{3660803005}{30479834641}a^{2}-\frac{10418579948}{30479834641}a-\frac{11718750}{30479834641}$, $\frac{1}{30479834641}a^{17}+\frac{211676357}{30479834641}a^{12}+\frac{85000}{30479834641}a^{11}+\frac{23229805}{30479834641}a^{10}+\frac{3187500}{30479834641}a^{9}+\frac{2184664200}{30479834641}a^{8}-\frac{2927742404}{30479834641}a^{7}-\frac{2527038}{301780541}a^{6}-\frac{10110248414}{30479834641}a^{5}+\frac{14030457493}{30479834641}a^{4}+\frac{8230981010}{30479834641}a^{3}-\frac{154542528}{743410601}a^{2}+\frac{6544941058}{30479834641}a-\frac{8117349618}{30479834641}$, $\frac{1}{30479834641}a^{18}+\frac{102000}{30479834641}a^{12}-\frac{8642502}{743410601}a^{11}+\frac{4207500}{30479834641}a^{10}-\frac{7998056076}{30479834641}a^{9}+\frac{68850000}{30479834641}a^{8}+\frac{9135802227}{30479834641}a^{7}+\frac{13173480217}{30479834641}a^{6}-\frac{3671049892}{30479834641}a^{5}+\frac{13063659015}{30479834641}a^{4}+\frac{4050667063}{30479834641}a^{3}-\frac{2693717957}{30479834641}a^{2}-\frac{8238435206}{30479834641}a+\frac{531250000}{30479834641}$, $\frac{1}{30479834641}a^{19}-\frac{13937883}{30479834641}a^{12}-\frac{2422500}{30479834641}a^{11}-\frac{211754353}{30479834641}a^{10}-\frac{96900000}{30479834641}a^{9}-\frac{11561102369}{30479834641}a^{8}+\frac{6724016611}{30479834641}a^{7}+\frac{13831227521}{30479834641}a^{6}-\frac{8203178798}{30479834641}a^{5}+\frac{11797995499}{30479834641}a^{4}+\frac{6213972694}{30479834641}a^{3}-\frac{7276676418}{30479834641}a^{2}+\frac{627996828}{30479834641}a+\frac{8362070652}{30479834641}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, 1/41*a^10 + 9/41*a^8 + 14/41*a^6 - 14/41*a^5 + 18/41*a^4 + 19/41*a^3 + 4/41*a^2 + 13/41*a + 9/41, 1/41*a^11 + 9/41*a^9 + 14/41*a^7 - 14/41*a^6 + 18/41*a^5 + 19/41*a^4 + 4/41*a^3 + 13/41*a^2 + 9/41*a, 1/41*a^12 + 15/41*a^8 - 14/41*a^7 + 15/41*a^6 - 19/41*a^5 + 6/41*a^4 + 6/41*a^3 + 14/41*a^2 + 6/41*a + 1/41, 1/30479834641*a^13 - 34265820/30479834641*a^12 + 65/30479834641*a^11 + 174282603/30479834641*a^10 + 1625/30479834641*a^9 - 1654220940/30479834641*a^8 - 5203854707/30479834641*a^7 + 14646569665/30479834641*a^6 + 113750/30479834641*a^5 - 620780076/30479834641*a^4 + 11151443390/30479834641*a^3 - 1064194416/30479834641*a^2 - 14124598294/30479834641*a - 14420410979/30479834641, 1/30479834641*a^14 + 70/30479834641*a^12 + 171329100/30479834641*a^11 + 1925/30479834641*a^10 + 3475815692/30479834641*a^9 + 26250/30479834641*a^8 + 12273697563/30479834641*a^7 - 4460279856/30479834641*a^6 + 12052444098/30479834641*a^5 + 10408360914/30479834641*a^4 - 14729180642/30479834641*a^3 - 8920161587/30479834641*a^2 - 8781895834/30479834641*a + 156250/30479834641, 1/30479834641*a^15 + 339704697/30479834641*a^12 - 2625/30479834641*a^11 + 196960694/30479834641*a^10 - 87500/30479834641*a^9 - 7974976620/30479834641*a^8 - 5205055457/30479834641*a^7 - 7372889299/30479834641*a^6 - 11158509015/30479834641*a^5 - 6958284170/30479834641*a^4 + 6671554784/30479834641*a^3 + 9212507610/30479834641*a^2 - 5961347308/30479834641*a - 9787165441/30479834641, 1/30479834641*a^16 - 3000/30479834641*a^12 - 324937182/30479834641*a^11 - 110000/30479834641*a^10 - 206926692/30479834641*a^9 - 1687500/30479834641*a^8 + 10759239136/30479834641*a^7 - 13393990818/30479834641*a^6 - 12575223349/30479834641*a^5 + 699660601/30479834641*a^4 - 4458135037/30479834641*a^3 + 3660803005/30479834641*a^2 - 10418579948/30479834641*a - 11718750/30479834641, 1/30479834641*a^17 + 211676357/30479834641*a^12 + 85000/30479834641*a^11 + 23229805/30479834641*a^10 + 3187500/30479834641*a^9 + 2184664200/30479834641*a^8 - 2927742404/30479834641*a^7 - 2527038/301780541*a^6 - 10110248414/30479834641*a^5 + 14030457493/30479834641*a^4 + 8230981010/30479834641*a^3 - 154542528/743410601*a^2 + 6544941058/30479834641*a - 8117349618/30479834641, 1/30479834641*a^18 + 102000/30479834641*a^12 - 8642502/743410601*a^11 + 4207500/30479834641*a^10 - 7998056076/30479834641*a^9 + 68850000/30479834641*a^8 + 9135802227/30479834641*a^7 + 13173480217/30479834641*a^6 - 3671049892/30479834641*a^5 + 13063659015/30479834641*a^4 + 4050667063/30479834641*a^3 - 2693717957/30479834641*a^2 - 8238435206/30479834641*a + 531250000/30479834641, 1/30479834641*a^19 - 13937883/30479834641*a^12 - 2422500/30479834641*a^11 - 211754353/30479834641*a^10 - 96900000/30479834641*a^9 - 11561102369/30479834641*a^8 + 6724016611/30479834641*a^7 + 13831227521/30479834641*a^6 - 8203178798/30479834641*a^5 + 11797995499/30479834641*a^4 + 6213972694/30479834641*a^3 - 7276676418/30479834641*a^2 + 627996828/30479834641*a + 8362070652/30479834641

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

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{122}\times C_{122}$, which has order $238144$ (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:	$ -1 $ -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:	$\frac{151}{743410601}a^{15}+\frac{11325}{743410601}a^{13}+\frac{339750}{743410601}a^{11}-\frac{25926}{743410601}a^{10}+\frac{5190625}{743410601}a^{9}-\frac{1296300}{743410601}a^{8}+\frac{42468750}{743410601}a^{7}-\frac{22685250}{743410601}a^{6}+\frac{178368750}{743410601}a^{5}-\frac{162037500}{743410601}a^{4}+\frac{330312500}{743410601}a^{3}-\frac{405093750}{743410601}a^{2}+\frac{176953125}{743410601}a-\frac{905448101}{743410601}$, $\frac{151}{30479834641}a^{19}+\frac{14345}{30479834641}a^{17}-\frac{625}{30479834641}a^{16}+\frac{573800}{30479834641}a^{15}-\frac{75926}{30479834641}a^{14}+\frac{12551875}{30479834641}a^{13}-\frac{3439820}{30479834641}a^{12}+\frac{163268750}{30479834641}a^{11}-\frac{77407550}{30479834641}a^{10}+\frac{1291661701}{30479834641}a^{9}-\frac{938370000}{30479834641}a^{8}+\frac{6218526545}{30479834641}a^{7}-\frac{6092606250}{30479834641}a^{6}+\frac{18662663800}{30479834641}a^{5}-\frac{20390448101}{30479834641}a^{4}+\frac{36504816250}{30479834641}a^{3}-\frac{41488649520}{30479834641}a^{2}+\frac{25345332764}{30479834641}a-\frac{1116114300}{743410601}$, $\frac{48576}{30479834641}a^{13}-\frac{136681}{30479834641}a^{12}+\frac{3157440}{30479834641}a^{11}-\frac{8200860}{30479834641}a^{10}+\frac{78936000}{30479834641}a^{9}-\frac{184519350}{30479834641}a^{8}+\frac{947232000}{30479834641}a^{7}-\frac{1913534000}{30479834641}a^{6}+\frac{5525520000}{30479834641}a^{5}-\frac{8969690625}{30479834641}a^{4}+\frac{13813800000}{30479834641}a^{3}-\frac{15376612500}{30479834641}a^{2}+\frac{9867000000}{30479834641}a-\frac{4271281250}{30479834641}$, $\frac{151}{30479834641}a^{18}+\frac{125}{30479834641}a^{17}+\frac{13590}{30479834641}a^{16}+\frac{10625}{30479834641}a^{15}+\frac{509625}{30479834641}a^{14}+\frac{345949}{30479834641}a^{13}+\frac{10286875}{30479834641}a^{12}+\frac{5221060}{30479834641}a^{11}+\frac{120328125}{30479834641}a^{10}+\frac{30917125}{30479834641}a^{9}+\frac{819786701}{30479834641}a^{8}-\frac{64035000}{30479834641}a^{7}+\frac{3181311790}{30479834641}a^{6}-\frac{1441125000}{30479834641}a^{5}+\frac{7324991125}{30479834641}a^{4}-\frac{4989666851}{30479834641}a^{3}+\frac{6967235269}{30479834641}a^{2}-\frac{12751643390}{30479834641}a-\frac{39121259810}{30479834641}$, $\frac{96}{30479834641}a^{19}+\frac{9120}{30479834641}a^{17}+\frac{364800}{30479834641}a^{15}+\frac{7980000}{30479834641}a^{13}+\frac{103740000}{30479834641}a^{11}+\frac{815100000}{30479834641}a^{9}+\frac{3762000000}{30479834641}a^{7}-\frac{64457461}{30479834641}a^{6}+\frac{9405000000}{30479834641}a^{5}-\frac{1933723830}{30479834641}a^{4}+\frac{10687500000}{30479834641}a^{3}-\frac{14502928725}{30479834641}a^{2}+\frac{3562500000}{30479834641}a-\frac{16114365250}{30479834641}$, $\frac{150}{30479834641}a^{18}-\frac{906}{30479834641}a^{17}+\frac{13500}{30479834641}a^{16}-\frac{77010}{30479834641}a^{15}+\frac{506250}{30479834641}a^{14}-\frac{2718000}{30479834641}a^{13}+\frac{10393056}{30479834641}a^{12}-\frac{51528750}{30479834641}a^{11}+\frac{129989610}{30479834641}a^{10}-\frac{566250000}{30479834641}a^{9}+\frac{1049202000}{30479834641}a^{8}-\frac{3644657706}{30479834641}a^{7}+\frac{5581777500}{30479834641}a^{6}-\frac{13605207210}{30479834641}a^{5}+\frac{18480937500}{30479834641}a^{4}-\frac{29349130249}{30479834641}a^{3}+\frac{34484407356}{30479834641}a^{2}-\frac{22496208735}{30479834641}a+\frac{1339337160}{743410601}$, $\frac{55}{30479834641}a^{19}+\frac{5225}{30479834641}a^{17}+\frac{1661}{30479834641}a^{16}+\frac{209000}{30479834641}a^{15}+\frac{124575}{30479834641}a^{14}+\frac{4571875}{30479834641}a^{13}+\frac{3737250}{30479834641}a^{12}+\frac{59149189}{30479834641}a^{11}+\frac{57096875}{30479834641}a^{10}+\frac{11042075}{743410601}a^{9}+\frac{467156250}{30479834641}a^{8}+\frac{1905774750}{30479834641}a^{7}+\frac{2010309961}{30479834641}a^{6}+\frac{3605868750}{30479834641}a^{5}+\frac{4337638229}{30479834641}a^{4}+\frac{1667015625}{30479834641}a^{3}-\frac{2064642670}{30479834641}a^{2}-\frac{7918913486}{30479834641}a-\frac{25107102300}{30479834641}$, $\frac{17621}{30479834641}a^{14}+\frac{1233470}{30479834641}a^{12}-\frac{379561}{30479834641}a^{11}+\frac{33920425}{30479834641}a^{10}-\frac{20875855}{30479834641}a^{9}+\frac{462551250}{30479834641}a^{8}-\frac{417517100}{30479834641}a^{7}+\frac{3237858750}{30479834641}a^{6}-\frac{3653274625}{30479834641}a^{5}+\frac{10792862500}{30479834641}a^{4}-\frac{13047409375}{30479834641}a^{3}+\frac{13491078125}{30479834641}a^{2}-\frac{13047409375}{30479834641}a+\frac{2753281250}{30479834641}$, $\frac{781}{30479834641}a^{17}+\frac{66385}{30479834641}a^{15}+\frac{2323475}{30479834641}a^{13}+\frac{43150250}{30479834641}a^{11}+\frac{456396875}{30479834641}a^{9}-\frac{8275601}{30479834641}a^{8}+\frac{2738381250}{30479834641}a^{7}-\frac{331024040}{30479834641}a^{6}+\frac{8713031250}{30479834641}a^{5}-\frac{4137800500}{30479834641}a^{4}+\frac{12447187500}{30479834641}a^{3}-\frac{16551202000}{30479834641}a^{2}+\frac{5186328125}{30479834641}a-\frac{10344501250}{30479834641}$ 151/743410601a^15 + 11325/743410601a^13 + 339750/743410601a^11 - 25926/743410601a^10 + 5190625/743410601a^9 - 1296300/743410601a^8 + 42468750/743410601a^7 - 22685250/743410601a^6 + 178368750/743410601a^5 - 162037500/743410601a^4 + 330312500/743410601a^3 - 405093750/743410601a^2 + 176953125/743410601a - 905448101/743410601, 151/30479834641a^19 + 14345/30479834641a^17 - 625/30479834641a^16 + 573800/30479834641a^15 - 75926/30479834641a^14 + 12551875/30479834641a^13 - 3439820/30479834641a^12 + 163268750/30479834641a^11 - 77407550/30479834641a^10 + 1291661701/30479834641a^9 - 938370000/30479834641a^8 + 6218526545/30479834641a^7 - 6092606250/30479834641a^6 + 18662663800/30479834641a^5 - 20390448101/30479834641a^4 + 36504816250/30479834641a^3 - 41488649520/30479834641a^2 + 25345332764/30479834641a - 1116114300/743410601, 48576/30479834641a^13 - 136681/30479834641a^12 + 3157440/30479834641a^11 - 8200860/30479834641a^10 + 78936000/30479834641a^9 - 184519350/30479834641a^8 + 947232000/30479834641a^7 - 1913534000/30479834641a^6 + 5525520000/30479834641a^5 - 8969690625/30479834641a^4 + 13813800000/30479834641a^3 - 15376612500/30479834641a^2 + 9867000000/30479834641a - 4271281250/30479834641, 151/30479834641a^18 + 125/30479834641a^17 + 13590/30479834641a^16 + 10625/30479834641a^15 + 509625/30479834641a^14 + 345949/30479834641a^13 + 10286875/30479834641a^12 + 5221060/30479834641a^11 + 120328125/30479834641a^10 + 30917125/30479834641a^9 + 819786701/30479834641a^8 - 64035000/30479834641a^7 + 3181311790/30479834641a^6 - 1441125000/30479834641a^5 + 7324991125/30479834641a^4 - 4989666851/30479834641a^3 + 6967235269/30479834641a^2 - 12751643390/30479834641a - 39121259810/30479834641, 96/30479834641a^19 + 9120/30479834641a^17 + 364800/30479834641a^15 + 7980000/30479834641a^13 + 103740000/30479834641a^11 + 815100000/30479834641a^9 + 3762000000/30479834641a^7 - 64457461/30479834641a^6 + 9405000000/30479834641a^5 - 1933723830/30479834641a^4 + 10687500000/30479834641a^3 - 14502928725/30479834641a^2 + 3562500000/30479834641a - 16114365250/30479834641, 150/30479834641a^18 - 906/30479834641a^17 + 13500/30479834641a^16 - 77010/30479834641a^15 + 506250/30479834641a^14 - 2718000/30479834641a^13 + 10393056/30479834641a^12 - 51528750/30479834641a^11 + 129989610/30479834641a^10 - 566250000/30479834641a^9 + 1049202000/30479834641a^8 - 3644657706/30479834641a^7 + 5581777500/30479834641a^6 - 13605207210/30479834641a^5 + 18480937500/30479834641a^4 - 29349130249/30479834641a^3 + 34484407356/30479834641a^2 - 22496208735/30479834641a + 1339337160/743410601, 55/30479834641a^19 + 5225/30479834641a^17 + 1661/30479834641a^16 + 209000/30479834641a^15 + 124575/30479834641a^14 + 4571875/30479834641a^13 + 3737250/30479834641a^12 + 59149189/30479834641a^11 + 57096875/30479834641a^10 + 11042075/743410601a^9 + 467156250/30479834641a^8 + 1905774750/30479834641a^7 + 2010309961/30479834641a^6 + 3605868750/30479834641a^5 + 4337638229/30479834641a^4 + 1667015625/30479834641a^3 - 2064642670/30479834641a^2 - 7918913486/30479834641a - 25107102300/30479834641, 17621/30479834641a^14 + 1233470/30479834641a^12 - 379561/30479834641a^11 + 33920425/30479834641a^10 - 20875855/30479834641a^9 + 462551250/30479834641a^8 - 417517100/30479834641a^7 + 3237858750/30479834641a^6 - 3653274625/30479834641a^5 + 10792862500/30479834641a^4 - 13047409375/30479834641a^3 + 13491078125/30479834641a^2 - 13047409375/30479834641a + 2753281250/30479834641, 781/30479834641a^17 + 66385/30479834641a^15 + 2323475/30479834641a^13 + 43150250/30479834641a^11 + 456396875/30479834641a^9 - 8275601/30479834641a^8 + 2738381250/30479834641a^7 - 331024040/30479834641a^6 + 8713031250/30479834641a^5 - 4137800500/30479834641a^4 + 12447187500/30479834641a^3 - 16551202000/30479834641a^2 + 5186328125/30479834641*a - 10344501250/30479834641 (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:	$ 161406.837641 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 161406.837641 \cdot 238144}{2\cdot\sqrt{48544842802805942483246326446533203125}}\cr\approx \mathstrut & 0.264520120010 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^20 + 100*x^18 + 4250*x^16 - 151*x^15 + 100000*x^14 - 11325*x^13 + 1421875*x^12 - 339750*x^11 + 12538426*x^10 - 5190625*x^9 + 68327550*x^8 - 42468750*x^7 + 228935250*x^6 - 182755451*x^5 + 484303125*x^4 - 439980025*x^3 + 600406250*x^2 - 725290750*x + 924979351)

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 + 100*x^18 + 4250*x^16 - 151*x^15 + 100000*x^14 - 11325*x^13 + 1421875*x^12 - 339750*x^11 + 12538426*x^10 - 5190625*x^9 + 68327550*x^8 - 42468750*x^7 + 228935250*x^6 - 182755451*x^5 + 484303125*x^4 - 439980025*x^3 + 600406250*x^2 - 725290750*x + 924979351, 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 + 100*x^18 + 4250*x^16 - 151*x^15 + 100000*x^14 - 11325*x^13 + 1421875*x^12 - 339750*x^11 + 12538426*x^10 - 5190625*x^9 + 68327550*x^8 - 42468750*x^7 + 228935250*x^6 - 182755451*x^5 + 484303125*x^4 - 439980025*x^3 + 600406250*x^2 - 725290750*x + 924979351);

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 + 100*x^18 + 4250*x^16 - 151*x^15 + 100000*x^14 - 11325*x^13 + 1421875*x^12 - 339750*x^11 + 12538426*x^10 - 5190625*x^9 + 68327550*x^8 - 42468750*x^7 + 228935250*x^6 - 182755451*x^5 + 484303125*x^4 - 439980025*x^3 + 600406250*x^2 - 725290750*x + 924979351);

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

$C_{20}$ (as 20T1):

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 cyclic group of order 20

The 20 conjugacy class representatives for $C_{20}$

Character table for $C_{20}$

Intermediate fields

$\Q(\sqrt{5}) $, 4.0.55125.1, 5.5.390625.1, $\Q(\zeta_{25})^+$

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]

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	$20$	R	R	R	${\href{/padicField/11.10.0.1}{10} }^{2}$	$20$	$20$	${\href{/padicField/19.5.0.1}{5} }^{4}$	$20$	${\href{/padicField/29.5.0.1}{5} }^{4}$	${\href{/padicField/31.10.0.1}{10} }^{2}$	$20$	${\href{/padicField/41.5.0.1}{5} }^{4}$	${\href{/padicField/43.4.0.1}{4} }^{5}$	$20$	$20$	${\href{/padicField/59.10.0.1}{10} }^{2}$

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
$3$ 3	3.20.10.2	$x^{20} + 162 x^{12} - 486 x^{10} + 1458 x^{8} - 19683 x^{2} + 118098$	$2$	$10$	$10$	20T1	$[\ ]_{2}^{10}$
$5$ 5		Deg $20$	$20$	$1$	$35$
$7$ 7	7.4.2.2	$x^{4} - 42 x^{2} + 147$	$2$	$2$	$2$	$C_4$	$[\ ]_{2}^{2}$
	7.4.2.2	$x^{4} - 42 x^{2} + 147$	$2$	$2$	$2$	$C_4$	$[\ ]_{2}^{2}$
	7.4.2.2	$x^{4} - 42 x^{2} + 147$	$2$	$2$	$2$	$C_4$	$[\ ]_{2}^{2}$
	7.4.2.2	$x^{4} - 42 x^{2} + 147$	$2$	$2$	$2$	$C_4$	$[\ ]_{2}^{2}$
	7.4.2.2	$x^{4} - 42 x^{2} + 147$	$2$	$2$	$2$	$C_4$	$[\ ]_{2}^{2}$

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