LMFDB - Number field 20.2.400000000000000000000000000.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^18 + 15*x^14 - 120*x^12 + 371*x^10 - 505*x^8 + 420*x^6 - 140*x^4 - 80*x^2 - 16)

gp: K = bnfinit(y^20 - 5*y^18 + 15*y^14 - 120*y^12 + 371*y^10 - 505*y^8 + 420*y^6 - 140*y^4 - 80*y^2 - 16, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 5*x^18 + 15*x^14 - 120*x^12 + 371*x^10 - 505*x^8 + 420*x^6 - 140*x^4 - 80*x^2 - 16);

oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 5*x^18 + 15*x^14 - 120*x^12 + 371*x^10 - 505*x^8 + 420*x^6 - 140*x^4 - 80*x^2 - 16)

$ x^{20} - 5x^{18} + 15x^{14} - 120x^{12} + 371x^{10} - 505x^{8} + 420x^{6} - 140x^{4} - 80x^{2} - 16 $ x^20 - 5*x^18 + 15*x^14 - 120*x^12 + 371*x^10 - 505*x^8 + 420*x^6 - 140*x^4 - 80*x^2 - 16

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:	$[2, 9]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$-400000000000000000000000000$ -400000000000000000000000000 $\medspace = -\,2^{28}\cdot 5^{26}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$21.38$	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:	$2^{3/2}5^{13/10}\approx 22.91954538992328$
Ramified primes:	$2$, $5$ 2, 5	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{-1}) $
$\card{ \Aut(K/\Q) }$:	$2$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{8}+\frac{1}{4}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{9}+\frac{1}{4}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{8}a^{14}-\frac{1}{8}a^{13}-\frac{1}{8}a^{12}-\frac{1}{4}a^{11}+\frac{1}{8}a^{10}-\frac{1}{8}a^{9}+\frac{1}{8}a^{8}+\frac{1}{8}a^{6}+\frac{1}{8}a^{5}+\frac{3}{8}a^{4}+\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{8}a^{15}-\frac{1}{8}a^{12}-\frac{1}{8}a^{11}-\frac{1}{4}a^{9}-\frac{1}{8}a^{8}+\frac{1}{8}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{8}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{8}a^{16}-\frac{1}{8}a^{13}-\frac{1}{8}a^{12}-\frac{1}{4}a^{10}-\frac{1}{8}a^{9}+\frac{1}{8}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{1}{8}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{17}-\frac{1}{8}a^{12}-\frac{1}{4}a^{9}-\frac{1}{8}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{3}{8}a^{5}+\frac{1}{8}a^{4}+\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{5371420912}a^{18}-\frac{282229565}{5371420912}a^{16}+\frac{12883785}{671427614}a^{14}-\frac{45904041}{413186224}a^{12}-\frac{1}{4}a^{11}-\frac{5177735}{1342855228}a^{10}-\frac{90887453}{413186224}a^{8}-\frac{1}{4}a^{7}+\frac{356154411}{5371420912}a^{6}-\frac{3427641}{51648278}a^{4}-\frac{1}{4}a^{3}-\frac{147852691}{335713807}a^{2}+\frac{150363771}{671427614}$, $\frac{1}{5371420912}a^{19}-\frac{282229565}{5371420912}a^{17}+\frac{12883785}{671427614}a^{15}-\frac{45904041}{413186224}a^{13}-\frac{5177735}{1342855228}a^{11}-\frac{90887453}{413186224}a^{9}+\frac{356154411}{5371420912}a^{7}-\frac{3427641}{51648278}a^{5}-\frac{147852691}{335713807}a^{3}+\frac{150363771}{671427614}a$ 1, a, a^2, a^3, a^4, a^5, 1/2*a^6 - 1/2*a^4 - 1/2*a^2, 1/2*a^7 - 1/2*a^5 - 1/2*a^3, 1/2*a^8 - 1/2*a^2, 1/2*a^9 - 1/2*a^3, 1/2*a^10 - 1/2*a^4, 1/2*a^11 - 1/2*a^5, 1/4*a^12 - 1/4*a^8 + 1/4*a^4 - 1/2*a^2, 1/4*a^13 - 1/4*a^9 + 1/4*a^5 - 1/2*a^3, 1/8*a^14 - 1/8*a^13 - 1/8*a^12 - 1/4*a^11 + 1/8*a^10 - 1/8*a^9 + 1/8*a^8 + 1/8*a^6 + 1/8*a^5 + 3/8*a^4 + 1/4*a^2 - 1/2, 1/8*a^15 - 1/8*a^12 - 1/8*a^11 - 1/4*a^9 - 1/8*a^8 + 1/8*a^7 - 1/4*a^6 - 1/4*a^5 + 1/8*a^4 - 1/4*a^3 + 1/4*a^2 - 1/2*a - 1/2, 1/8*a^16 - 1/8*a^13 - 1/8*a^12 - 1/4*a^10 - 1/8*a^9 + 1/8*a^8 - 1/4*a^7 - 1/4*a^6 + 1/8*a^5 - 1/4*a^4 + 1/4*a^3 - 1/2*a^2 - 1/2*a, 1/8*a^17 - 1/8*a^12 - 1/4*a^9 - 1/8*a^8 - 1/4*a^7 - 1/4*a^6 - 3/8*a^5 + 1/8*a^4 + 1/4*a^2 - 1/2, 1/5371420912*a^18 - 282229565/5371420912*a^16 + 12883785/671427614*a^14 - 45904041/413186224*a^12 - 1/4*a^11 - 5177735/1342855228*a^10 - 90887453/413186224*a^8 - 1/4*a^7 + 356154411/5371420912*a^6 - 3427641/51648278*a^4 - 1/4*a^3 - 147852691/335713807*a^2 + 150363771/671427614, 1/5371420912*a^19 - 282229565/5371420912*a^17 + 12883785/671427614*a^15 - 45904041/413186224*a^13 - 5177735/1342855228*a^11 - 90887453/413186224*a^9 + 356154411/5371420912*a^7 - 3427641/51648278*a^5 - 147852691/335713807*a^3 + 150363771/671427614*a

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

Trivial group, which has order $1$ (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:	$10$	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:	$\frac{96375637}{5371420912}a^{19}-\frac{436139441}{5371420912}a^{17}-\frac{36128607}{1342855228}a^{15}+\frac{88272527}{413186224}a^{13}-\frac{1420915601}{671427614}a^{11}+\frac{2388498427}{413186224}a^{9}-\frac{40395385653}{5371420912}a^{7}+\frac{665800941}{103296556}a^{5}-\frac{799846319}{671427614}a^{3}-\frac{1195424523}{671427614}a$, $\frac{38680469}{2685710456}a^{18}-\frac{149597355}{2685710456}a^{16}-\frac{36153391}{671427614}a^{14}+\frac{25513269}{206593112}a^{12}-\frac{1103743339}{671427614}a^{10}+\frac{732530269}{206593112}a^{8}-\frac{11273182721}{2685710456}a^{6}+\frac{170975229}{51648278}a^{4}+\frac{6830127}{335713807}a^{2}-\frac{144958108}{335713807}$, $\frac{20459485}{2685710456}a^{18}-\frac{114122551}{2685710456}a^{16}+\frac{13654737}{671427614}a^{14}+\frac{23597285}{206593112}a^{12}-\frac{642622853}{671427614}a^{10}+\frac{703334429}{206593112}a^{8}-\frac{14265565573}{2685710456}a^{6}+\frac{292069779}{51648278}a^{4}-\frac{2487516601}{671427614}a^{2}-\frac{188766650}{335713807}$, $\frac{94383325}{2685710456}a^{19}+\frac{96375637}{5371420912}a^{18}-\frac{246188055}{1342855228}a^{17}-\frac{436139441}{5371420912}a^{16}+\frac{114122551}{2685710456}a^{15}-\frac{36128607}{1342855228}a^{14}+\frac{104702379}{206593112}a^{13}+\frac{88272527}{413186224}a^{12}-\frac{11632763705}{2685710456}a^{11}-\frac{1420915601}{671427614}a^{10}+\frac{2891284999}{206593112}a^{9}+\frac{2388498427}{413186224}a^{8}-\frac{28403463351}{1342855228}a^{7}-\frac{40395385653}{5371420912}a^{6}+\frac{4146658621}{206593112}a^{5}+\frac{665800941}{103296556}a^{4}-\frac{3550161751}{335713807}a^{3}-\frac{799846319}{671427614}a^{2}+\frac{599850101}{671427614}a-\frac{1195424523}{671427614}$, $\frac{37898295}{5371420912}a^{19}+\frac{5221825}{5371420912}a^{18}-\frac{164181579}{5371420912}a^{17}+\frac{231245}{5371420912}a^{16}-\frac{26380845}{2685710456}a^{15}-\frac{16916543}{1342855228}a^{14}+\frac{19681807}{413186224}a^{13}-\frac{8503589}{413186224}a^{12}-\frac{2195116387}{2685710456}a^{11}-\frac{178905823}{1342855228}a^{10}+\frac{915684887}{413186224}a^{9}-\frac{75195493}{413186224}a^{8}-\frac{18062107389}{5371420912}a^{7}+\frac{744898527}{5371420912}a^{6}+\frac{940740983}{206593112}a^{5}-\frac{20147453}{206593112}a^{4}-\frac{4325212127}{1342855228}a^{3}+\frac{1562134787}{1342855228}a^{2}+\frac{847971469}{671427614}a+\frac{69777878}{335713807}$, $\frac{37898295}{5371420912}a^{19}+\frac{72139113}{5371420912}a^{18}-\frac{164181579}{5371420912}a^{17}-\frac{299425955}{5371420912}a^{16}-\frac{26380845}{2685710456}a^{15}-\frac{55390239}{1342855228}a^{14}+\frac{19681807}{413186224}a^{13}+\frac{59530127}{413186224}a^{12}-\frac{2195116387}{2685710456}a^{11}-\frac{2028580855}{1342855228}a^{10}+\frac{915684887}{413186224}a^{9}+\frac{1540256031}{413186224}a^{8}-\frac{18062107389}{5371420912}a^{7}-\frac{23291263969}{5371420912}a^{6}+\frac{940740983}{206593112}a^{5}+\frac{704048369}{206593112}a^{4}-\frac{4325212127}{1342855228}a^{3}-\frac{1534814279}{1342855228}a^{2}+\frac{847971469}{671427614}a-\frac{214735986}{335713807}$, $\frac{53401093}{5371420912}a^{19}-\frac{76270465}{5371420912}a^{18}-\frac{124787659}{5371420912}a^{17}+\frac{197044873}{5371420912}a^{16}-\frac{264661479}{2685710456}a^{15}+\frac{339184405}{2685710456}a^{14}+\frac{11222445}{413186224}a^{13}-\frac{18828277}{413186224}a^{12}-\frac{2612214047}{2685710456}a^{11}+\frac{3788434491}{2685710456}a^{10}+\frac{325857361}{413186224}a^{9}-\frac{644100093}{413186224}a^{8}+\frac{6358820881}{5371420912}a^{7}-\frac{3430761577}{5371420912}a^{6}-\frac{86269343}{51648278}a^{5}+\frac{208657783}{206593112}a^{4}+\frac{1281807675}{335713807}a^{3}-\frac{4895562547}{1342855228}a^{2}-\frac{11992912}{335713807}a-\frac{204051810}{335713807}$, $\frac{96375637}{5371420912}a^{19}-\frac{412459}{122077748}a^{18}-\frac{436139441}{5371420912}a^{17}+\frac{1240631}{61038874}a^{16}-\frac{36128607}{1342855228}a^{15}-\frac{1213311}{122077748}a^{14}+\frac{88272527}{413186224}a^{13}-\frac{792089}{9390596}a^{12}-\frac{1420915601}{671427614}a^{11}+\frac{53704597}{122077748}a^{10}+\frac{2388498427}{413186224}a^{9}-\frac{14551629}{9390596}a^{8}-\frac{40395385653}{5371420912}a^{7}+\frac{66625659}{30519437}a^{6}+\frac{665800941}{103296556}a^{5}-\frac{6175171}{9390596}a^{4}-\frac{799846319}{671427614}a^{3}-\frac{11801224}{30519437}a^{2}-\frac{1195424523}{671427614}a+\frac{16556673}{30519437}$, $\frac{30744921}{2685710456}a^{18}-\frac{67304607}{2685710456}a^{16}-\frac{139766049}{1342855228}a^{14}-\frac{9022923}{206593112}a^{12}-\frac{1568202771}{1342855228}a^{10}+\frac{203575069}{206593112}a^{8}+\frac{258451529}{2685710456}a^{6}+\frac{52362920}{25824139}a^{4}+\frac{618509043}{671427614}a^{2}+\frac{124185309}{335713807}$, $\frac{32080525}{5371420912}a^{19}-\frac{155033513}{5371420912}a^{17}+\frac{2325361}{1342855228}a^{15}+\frac{26732271}{413186224}a^{13}-\frac{246706209}{335713807}a^{11}+\frac{900085363}{413186224}a^{9}-\frac{18907269701}{5371420912}a^{7}+\frac{395213491}{103296556}a^{5}-\frac{1691367907}{671427614}a^{3}+\frac{564557751}{671427614}a$ 96375637/5371420912a^19 - 436139441/5371420912a^17 - 36128607/1342855228a^15 + 88272527/413186224a^13 - 1420915601/671427614a^11 + 2388498427/413186224a^9 - 40395385653/5371420912a^7 + 665800941/103296556a^5 - 799846319/671427614a^3 - 1195424523/671427614a, 38680469/2685710456a^18 - 149597355/2685710456a^16 - 36153391/671427614a^14 + 25513269/206593112a^12 - 1103743339/671427614a^10 + 732530269/206593112a^8 - 11273182721/2685710456a^6 + 170975229/51648278a^4 + 6830127/335713807a^2 - 144958108/335713807, 20459485/2685710456a^18 - 114122551/2685710456a^16 + 13654737/671427614a^14 + 23597285/206593112a^12 - 642622853/671427614a^10 + 703334429/206593112a^8 - 14265565573/2685710456a^6 + 292069779/51648278a^4 - 2487516601/671427614a^2 - 188766650/335713807, 94383325/2685710456a^19 + 96375637/5371420912a^18 - 246188055/1342855228a^17 - 436139441/5371420912a^16 + 114122551/2685710456a^15 - 36128607/1342855228a^14 + 104702379/206593112a^13 + 88272527/413186224a^12 - 11632763705/2685710456a^11 - 1420915601/671427614a^10 + 2891284999/206593112a^9 + 2388498427/413186224a^8 - 28403463351/1342855228a^7 - 40395385653/5371420912a^6 + 4146658621/206593112a^5 + 665800941/103296556a^4 - 3550161751/335713807a^3 - 799846319/671427614a^2 + 599850101/671427614a - 1195424523/671427614, 37898295/5371420912a^19 + 5221825/5371420912a^18 - 164181579/5371420912a^17 + 231245/5371420912a^16 - 26380845/2685710456a^15 - 16916543/1342855228a^14 + 19681807/413186224a^13 - 8503589/413186224a^12 - 2195116387/2685710456a^11 - 178905823/1342855228a^10 + 915684887/413186224a^9 - 75195493/413186224a^8 - 18062107389/5371420912a^7 + 744898527/5371420912a^6 + 940740983/206593112a^5 - 20147453/206593112a^4 - 4325212127/1342855228a^3 + 1562134787/1342855228a^2 + 847971469/671427614a + 69777878/335713807, 37898295/5371420912a^19 + 72139113/5371420912a^18 - 164181579/5371420912a^17 - 299425955/5371420912a^16 - 26380845/2685710456a^15 - 55390239/1342855228a^14 + 19681807/413186224a^13 + 59530127/413186224a^12 - 2195116387/2685710456a^11 - 2028580855/1342855228a^10 + 915684887/413186224a^9 + 1540256031/413186224a^8 - 18062107389/5371420912a^7 - 23291263969/5371420912a^6 + 940740983/206593112a^5 + 704048369/206593112a^4 - 4325212127/1342855228a^3 - 1534814279/1342855228a^2 + 847971469/671427614a - 214735986/335713807, 53401093/5371420912a^19 - 76270465/5371420912a^18 - 124787659/5371420912a^17 + 197044873/5371420912a^16 - 264661479/2685710456a^15 + 339184405/2685710456a^14 + 11222445/413186224a^13 - 18828277/413186224a^12 - 2612214047/2685710456a^11 + 3788434491/2685710456a^10 + 325857361/413186224a^9 - 644100093/413186224a^8 + 6358820881/5371420912a^7 - 3430761577/5371420912a^6 - 86269343/51648278a^5 + 208657783/206593112a^4 + 1281807675/335713807a^3 - 4895562547/1342855228a^2 - 11992912/335713807a - 204051810/335713807, 96375637/5371420912a^19 - 412459/122077748a^18 - 436139441/5371420912a^17 + 1240631/61038874a^16 - 36128607/1342855228a^15 - 1213311/122077748a^14 + 88272527/413186224a^13 - 792089/9390596a^12 - 1420915601/671427614a^11 + 53704597/122077748a^10 + 2388498427/413186224a^9 - 14551629/9390596a^8 - 40395385653/5371420912a^7 + 66625659/30519437a^6 + 665800941/103296556a^5 - 6175171/9390596a^4 - 799846319/671427614a^3 - 11801224/30519437a^2 - 1195424523/671427614a + 16556673/30519437, 30744921/2685710456a^18 - 67304607/2685710456a^16 - 139766049/1342855228a^14 - 9022923/206593112a^12 - 1568202771/1342855228a^10 + 203575069/206593112a^8 + 258451529/2685710456a^6 + 52362920/25824139a^4 + 618509043/671427614a^2 + 124185309/335713807, 32080525/5371420912a^19 - 155033513/5371420912a^17 + 2325361/1342855228a^15 + 26732271/413186224a^13 - 246706209/335713807a^11 + 900085363/413186224a^9 - 18907269701/5371420912a^7 + 395213491/103296556a^5 - 1691367907/671427614a^3 + 564557751/671427614a (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:	$ 320036.323885 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{9}\cdot 320036.323885 \cdot 1}{2\cdot\sqrt{400000000000000000000000000}}\cr\approx \mathstrut & 0.488447721933 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^18 + 15*x^14 - 120*x^12 + 371*x^10 - 505*x^8 + 420*x^6 - 140*x^4 - 80*x^2 - 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))))

# self-contained Pari/GP code snippet to compute the analytic class number formula

K = bnfinit(x^20 - 5*x^18 + 15*x^14 - 120*x^12 + 371*x^10 - 505*x^8 + 420*x^6 - 140*x^4 - 80*x^2 - 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)))]

/* self-contained Magma code snippet to compute the analytic class number formula */

Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^20 - 5*x^18 + 15*x^14 - 120*x^12 + 371*x^10 - 505*x^8 + 420*x^6 - 140*x^4 - 80*x^2 - 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)));

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

Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 5*x^18 + 15*x^14 - 120*x^12 + 371*x^10 - 505*x^8 + 420*x^6 - 140*x^4 - 80*x^2 - 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

$D_{20}$ (as 20T10):

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 13 conjugacy class representatives for $D_{20}$

Character table for $D_{20}$

Intermediate fields

$\Q(\sqrt{5}) $, 4.2.400.1, 5.1.250000.1, 10.2.312500000000.1

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:	20.0.320000000000000000000000000.1
Minimal sibling:	20.0.320000000000000000000000000.1

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	$20$	R	$20$	${\href{/padicField/11.2.0.1}{2} }^{9}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$	${\href{/padicField/13.2.0.1}{2} }^{10}$	${\href{/padicField/17.2.0.1}{2} }^{10}$	${\href{/padicField/19.2.0.1}{2} }^{9}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$	$20$	${\href{/padicField/29.5.0.1}{5} }^{4}$	${\href{/padicField/31.2.0.1}{2} }^{9}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$	${\href{/padicField/37.2.0.1}{2} }^{10}$	${\href{/padicField/41.10.0.1}{10} }^{2}$	$20$	$20$	${\href{/padicField/53.2.0.1}{2} }^{10}$	${\href{/padicField/59.2.0.1}{2} }^{9}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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	2.4.4.3	$x^{4} + 2 x^{3} + 4 x^{2} + 12 x + 12$	$2$	$2$	$4$	$D_{4}$	$[2, 2]^{2}$
	2.8.12.13	$x^{8} + 8 x^{7} + 28 x^{6} + 58 x^{5} + 87 x^{4} + 98 x^{3} + 58 x^{2} - 2 x + 1$	$4$	$2$	$12$	$D_4$	$[2, 2]^{2}$
	2.8.12.13	$x^{8} + 8 x^{7} + 28 x^{6} + 58 x^{5} + 87 x^{4} + 98 x^{3} + 58 x^{2} - 2 x + 1$	$4$	$2$	$12$	$D_4$	$[2, 2]^{2}$
$5$ 5		Deg $20$	$10$	$2$	$26$

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