Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 25*x^18 - 35*x^17 + 270*x^16 - 919*x^15 + 4850*x^14 + 24400*x^13 + 158005*x^12 + 323200*x^11 + 240476*x^10 - 2451535*x^9 - 9806600*x^8 - 22103305*x^7 - 18499285*x^6 + 36486691*x^5 + 184536915*x^4 + 378693200*x^3 + 498508960*x^2 + 402067640*x + 168988496)
gp: K = bnfinit(y^20 - 5*y^19 + 25*y^18 - 35*y^17 + 270*y^16 - 919*y^15 + 4850*y^14 + 24400*y^13 + 158005*y^12 + 323200*y^11 + 240476*y^10 - 2451535*y^9 - 9806600*y^8 - 22103305*y^7 - 18499285*y^6 + 36486691*y^5 + 184536915*y^4 + 378693200*y^3 + 498508960*y^2 + 402067640*y + 168988496, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 5*x^19 + 25*x^18 - 35*x^17 + 270*x^16 - 919*x^15 + 4850*x^14 + 24400*x^13 + 158005*x^12 + 323200*x^11 + 240476*x^10 - 2451535*x^9 - 9806600*x^8 - 22103305*x^7 - 18499285*x^6 + 36486691*x^5 + 184536915*x^4 + 378693200*x^3 + 498508960*x^2 + 402067640*x + 168988496);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 5*x^19 + 25*x^18 - 35*x^17 + 270*x^16 - 919*x^15 + 4850*x^14 + 24400*x^13 + 158005*x^12 + 323200*x^11 + 240476*x^10 - 2451535*x^9 - 9806600*x^8 - 22103305*x^7 - 18499285*x^6 + 36486691*x^5 + 184536915*x^4 + 378693200*x^3 + 498508960*x^2 + 402067640*x + 168988496)
\( x^{20} - 5 x^{19} + 25 x^{18} - 35 x^{17} + 270 x^{16} - 919 x^{15} + 4850 x^{14} + 24400 x^{13} + \cdots + 168988496 \)
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: |
\(9387703148876316845417022705078125\)
\(\medspace = 5^{31}\cdot 17^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(49.96\) | 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: | $5^{31/20}17^{1/2}\approx 49.96063704619908$ | ||
| Ramified primes: |
\(5\), \(17\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}+\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{17}-\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{2}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{76}a^{18}+\frac{7}{76}a^{16}-\frac{2}{19}a^{14}+\frac{5}{38}a^{13}-\frac{5}{38}a^{12}+\frac{1}{38}a^{11}+\frac{1}{76}a^{10}-\frac{1}{38}a^{9}-\frac{4}{19}a^{8}+\frac{5}{38}a^{7}-\frac{5}{19}a^{6}+\frac{2}{19}a^{5}-\frac{29}{76}a^{4}+\frac{9}{38}a^{3}-\frac{11}{76}a^{2}-\frac{9}{19}a+\frac{3}{19}$, $\frac{1}{18\!\cdots\!68}a^{19}+\frac{81\!\cdots\!87}{18\!\cdots\!68}a^{18}+\frac{10\!\cdots\!37}{18\!\cdots\!68}a^{17}-\frac{21\!\cdots\!83}{18\!\cdots\!68}a^{16}+\frac{16\!\cdots\!38}{23\!\cdots\!21}a^{15}-\frac{13\!\cdots\!51}{18\!\cdots\!68}a^{14}-\frac{84\!\cdots\!29}{92\!\cdots\!84}a^{13}-\frac{27\!\cdots\!53}{46\!\cdots\!42}a^{12}-\frac{37\!\cdots\!05}{18\!\cdots\!68}a^{11}+\frac{86\!\cdots\!95}{46\!\cdots\!42}a^{10}+\frac{95\!\cdots\!07}{92\!\cdots\!84}a^{9}-\frac{56\!\cdots\!03}{18\!\cdots\!68}a^{8}+\frac{61\!\cdots\!73}{92\!\cdots\!84}a^{7}-\frac{79\!\cdots\!17}{18\!\cdots\!68}a^{6}-\frac{14\!\cdots\!93}{18\!\cdots\!68}a^{5}+\frac{47\!\cdots\!93}{96\!\cdots\!72}a^{4}-\frac{50\!\cdots\!23}{18\!\cdots\!68}a^{3}+\frac{10\!\cdots\!68}{23\!\cdots\!21}a^{2}-\frac{31\!\cdots\!93}{23\!\cdots\!21}a+\frac{93\!\cdots\!24}{23\!\cdots\!21}$
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 $2$ (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: |
\( -1 \)
(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: |
$\frac{33\!\cdots\!83}{18\!\cdots\!68}a^{19}-\frac{29\!\cdots\!05}{18\!\cdots\!68}a^{18}+\frac{14\!\cdots\!19}{18\!\cdots\!68}a^{17}-\frac{34\!\cdots\!89}{18\!\cdots\!68}a^{16}+\frac{10\!\cdots\!21}{23\!\cdots\!21}a^{15}-\frac{49\!\cdots\!45}{18\!\cdots\!68}a^{14}+\frac{13\!\cdots\!27}{92\!\cdots\!84}a^{13}+\frac{83\!\cdots\!81}{48\!\cdots\!36}a^{12}+\frac{11\!\cdots\!09}{18\!\cdots\!68}a^{11}-\frac{21\!\cdots\!23}{46\!\cdots\!42}a^{10}-\frac{11\!\cdots\!97}{92\!\cdots\!84}a^{9}-\frac{43\!\cdots\!71}{18\!\cdots\!68}a^{8}+\frac{45\!\cdots\!15}{92\!\cdots\!84}a^{7}+\frac{40\!\cdots\!17}{18\!\cdots\!68}a^{6}+\frac{85\!\cdots\!05}{18\!\cdots\!68}a^{5}+\frac{20\!\cdots\!45}{18\!\cdots\!68}a^{4}-\frac{20\!\cdots\!85}{18\!\cdots\!68}a^{3}-\frac{28\!\cdots\!57}{92\!\cdots\!84}a^{2}-\frac{16\!\cdots\!07}{46\!\cdots\!42}a-\frac{45\!\cdots\!78}{23\!\cdots\!21}$, $\frac{65\!\cdots\!45}{43\!\cdots\!72}a^{19}+\frac{39\!\cdots\!73}{43\!\cdots\!72}a^{18}-\frac{26\!\cdots\!11}{43\!\cdots\!72}a^{17}+\frac{12\!\cdots\!05}{43\!\cdots\!72}a^{16}-\frac{49\!\cdots\!69}{10\!\cdots\!18}a^{15}+\frac{78\!\cdots\!49}{43\!\cdots\!72}a^{14}-\frac{20\!\cdots\!79}{21\!\cdots\!36}a^{13}+\frac{12\!\cdots\!67}{21\!\cdots\!36}a^{12}+\frac{96\!\cdots\!27}{43\!\cdots\!72}a^{11}+\frac{10\!\cdots\!91}{10\!\cdots\!18}a^{10}+\frac{75\!\cdots\!13}{21\!\cdots\!36}a^{9}-\frac{96\!\cdots\!87}{22\!\cdots\!88}a^{8}-\frac{49\!\cdots\!29}{21\!\cdots\!36}a^{7}-\frac{17\!\cdots\!81}{43\!\cdots\!72}a^{6}-\frac{10\!\cdots\!01}{43\!\cdots\!72}a^{5}+\frac{52\!\cdots\!91}{43\!\cdots\!72}a^{4}+\frac{16\!\cdots\!89}{43\!\cdots\!72}a^{3}+\frac{13\!\cdots\!93}{21\!\cdots\!36}a^{2}+\frac{30\!\cdots\!85}{54\!\cdots\!59}a+\frac{14\!\cdots\!37}{54\!\cdots\!59}$, $\frac{32\!\cdots\!95}{92\!\cdots\!84}a^{19}-\frac{55\!\cdots\!33}{46\!\cdots\!42}a^{18}+\frac{69\!\cdots\!69}{92\!\cdots\!84}a^{17}-\frac{68\!\cdots\!66}{23\!\cdots\!21}a^{16}+\frac{20\!\cdots\!09}{46\!\cdots\!42}a^{15}-\frac{17\!\cdots\!87}{92\!\cdots\!84}a^{14}+\frac{25\!\cdots\!53}{23\!\cdots\!21}a^{13}-\frac{10\!\cdots\!07}{23\!\cdots\!21}a^{12}-\frac{16\!\cdots\!87}{92\!\cdots\!84}a^{11}-\frac{80\!\cdots\!63}{92\!\cdots\!84}a^{10}-\frac{10\!\cdots\!31}{23\!\cdots\!21}a^{9}+\frac{31\!\cdots\!69}{92\!\cdots\!84}a^{8}+\frac{89\!\cdots\!53}{46\!\cdots\!42}a^{7}+\frac{32\!\cdots\!71}{92\!\cdots\!84}a^{6}+\frac{19\!\cdots\!31}{92\!\cdots\!84}a^{5}-\frac{46\!\cdots\!49}{46\!\cdots\!42}a^{4}-\frac{29\!\cdots\!89}{92\!\cdots\!84}a^{3}-\frac{47\!\cdots\!41}{92\!\cdots\!84}a^{2}-\frac{21\!\cdots\!57}{46\!\cdots\!42}a-\frac{50\!\cdots\!44}{23\!\cdots\!21}$, $\frac{26\!\cdots\!39}{18\!\cdots\!68}a^{19}-\frac{22\!\cdots\!51}{18\!\cdots\!68}a^{18}+\frac{10\!\cdots\!25}{18\!\cdots\!68}a^{17}-\frac{20\!\cdots\!63}{18\!\cdots\!68}a^{16}+\frac{23\!\cdots\!37}{92\!\cdots\!84}a^{15}-\frac{32\!\cdots\!17}{18\!\cdots\!68}a^{14}+\frac{22\!\cdots\!88}{23\!\cdots\!21}a^{13}+\frac{18\!\cdots\!15}{92\!\cdots\!84}a^{12}+\frac{98\!\cdots\!05}{18\!\cdots\!68}a^{11}-\frac{28\!\cdots\!63}{92\!\cdots\!84}a^{10}-\frac{10\!\cdots\!37}{92\!\cdots\!84}a^{9}-\frac{38\!\cdots\!71}{18\!\cdots\!68}a^{8}+\frac{64\!\cdots\!04}{23\!\cdots\!21}a^{7}+\frac{31\!\cdots\!61}{18\!\cdots\!68}a^{6}+\frac{70\!\cdots\!47}{18\!\cdots\!68}a^{5}+\frac{18\!\cdots\!25}{96\!\cdots\!72}a^{4}-\frac{14\!\cdots\!39}{18\!\cdots\!68}a^{3}-\frac{11\!\cdots\!91}{46\!\cdots\!42}a^{2}-\frac{70\!\cdots\!30}{23\!\cdots\!21}a-\frac{45\!\cdots\!28}{23\!\cdots\!21}$, $\frac{12\!\cdots\!85}{23\!\cdots\!21}a^{19}-\frac{75\!\cdots\!63}{46\!\cdots\!42}a^{18}+\frac{20\!\cdots\!03}{46\!\cdots\!42}a^{17}+\frac{21\!\cdots\!01}{92\!\cdots\!84}a^{16}+\frac{51\!\cdots\!81}{92\!\cdots\!84}a^{15}-\frac{26\!\cdots\!13}{92\!\cdots\!84}a^{14}+\frac{59\!\cdots\!65}{46\!\cdots\!42}a^{13}+\frac{19\!\cdots\!91}{92\!\cdots\!84}a^{12}+\frac{92\!\cdots\!71}{92\!\cdots\!84}a^{11}+\frac{50\!\cdots\!46}{23\!\cdots\!21}a^{10}-\frac{67\!\cdots\!41}{92\!\cdots\!84}a^{9}-\frac{47\!\cdots\!50}{23\!\cdots\!21}a^{8}-\frac{63\!\cdots\!53}{92\!\cdots\!84}a^{7}-\frac{97\!\cdots\!79}{92\!\cdots\!84}a^{6}+\frac{29\!\cdots\!91}{46\!\cdots\!42}a^{5}+\frac{40\!\cdots\!01}{92\!\cdots\!84}a^{4}+\frac{10\!\cdots\!83}{92\!\cdots\!84}a^{3}+\frac{14\!\cdots\!83}{92\!\cdots\!84}a^{2}+\frac{60\!\cdots\!27}{46\!\cdots\!42}a+\frac{11\!\cdots\!21}{23\!\cdots\!21}$, $\frac{22\!\cdots\!61}{48\!\cdots\!36}a^{19}-\frac{14\!\cdots\!03}{46\!\cdots\!42}a^{18}+\frac{18\!\cdots\!38}{12\!\cdots\!59}a^{17}-\frac{26\!\cdots\!89}{92\!\cdots\!84}a^{16}+\frac{51\!\cdots\!65}{48\!\cdots\!36}a^{15}-\frac{10\!\cdots\!87}{23\!\cdots\!21}a^{14}+\frac{23\!\cdots\!91}{92\!\cdots\!84}a^{13}+\frac{84\!\cdots\!13}{92\!\cdots\!84}a^{12}+\frac{42\!\cdots\!93}{92\!\cdots\!84}a^{11}+\frac{33\!\cdots\!89}{92\!\cdots\!84}a^{10}-\frac{62\!\cdots\!83}{46\!\cdots\!42}a^{9}-\frac{92\!\cdots\!69}{92\!\cdots\!84}a^{8}-\frac{20\!\cdots\!65}{92\!\cdots\!84}a^{7}-\frac{55\!\cdots\!56}{23\!\cdots\!21}a^{6}+\frac{83\!\cdots\!70}{23\!\cdots\!21}a^{5}+\frac{16\!\cdots\!95}{92\!\cdots\!84}a^{4}+\frac{85\!\cdots\!51}{23\!\cdots\!21}a^{3}+\frac{99\!\cdots\!96}{23\!\cdots\!21}a^{2}+\frac{64\!\cdots\!85}{23\!\cdots\!21}a+\frac{13\!\cdots\!85}{23\!\cdots\!21}$, $\frac{10\!\cdots\!99}{15\!\cdots\!76}a^{19}-\frac{21\!\cdots\!53}{15\!\cdots\!76}a^{18}+\frac{13\!\cdots\!77}{15\!\cdots\!76}a^{17}-\frac{30\!\cdots\!63}{78\!\cdots\!38}a^{16}+\frac{74\!\cdots\!07}{78\!\cdots\!38}a^{15}-\frac{29\!\cdots\!45}{78\!\cdots\!38}a^{14}+\frac{26\!\cdots\!28}{20\!\cdots\!01}a^{13}-\frac{61\!\cdots\!35}{15\!\cdots\!76}a^{12}-\frac{16\!\cdots\!19}{15\!\cdots\!76}a^{11}-\frac{40\!\cdots\!68}{39\!\cdots\!19}a^{10}-\frac{90\!\cdots\!48}{39\!\cdots\!19}a^{9}-\frac{65\!\cdots\!07}{15\!\cdots\!76}a^{8}+\frac{95\!\cdots\!46}{39\!\cdots\!19}a^{7}+\frac{11\!\cdots\!49}{39\!\cdots\!19}a^{6}+\frac{12\!\cdots\!51}{15\!\cdots\!76}a^{5}+\frac{54\!\cdots\!47}{39\!\cdots\!19}a^{4}+\frac{23\!\cdots\!97}{15\!\cdots\!76}a^{3}+\frac{17\!\cdots\!19}{15\!\cdots\!76}a^{2}+\frac{63\!\cdots\!47}{78\!\cdots\!38}a+\frac{22\!\cdots\!89}{39\!\cdots\!19}$, $\frac{10\!\cdots\!97}{21\!\cdots\!04}a^{19}-\frac{73\!\cdots\!97}{21\!\cdots\!04}a^{18}+\frac{33\!\cdots\!87}{21\!\cdots\!04}a^{17}-\frac{28\!\cdots\!27}{10\!\cdots\!02}a^{16}+\frac{47\!\cdots\!85}{54\!\cdots\!01}a^{15}-\frac{56\!\cdots\!35}{10\!\cdots\!02}a^{14}+\frac{30\!\cdots\!55}{10\!\cdots\!02}a^{13}+\frac{20\!\cdots\!85}{21\!\cdots\!04}a^{12}+\frac{84\!\cdots\!59}{21\!\cdots\!04}a^{11}-\frac{63\!\cdots\!04}{54\!\cdots\!01}a^{10}-\frac{12\!\cdots\!51}{54\!\cdots\!01}a^{9}-\frac{11\!\cdots\!77}{11\!\cdots\!16}a^{8}-\frac{65\!\cdots\!02}{54\!\cdots\!01}a^{7}+\frac{32\!\cdots\!01}{54\!\cdots\!01}a^{6}+\frac{16\!\cdots\!17}{21\!\cdots\!04}a^{5}+\frac{15\!\cdots\!57}{10\!\cdots\!02}a^{4}+\frac{31\!\cdots\!27}{21\!\cdots\!04}a^{3}-\frac{17\!\cdots\!67}{21\!\cdots\!04}a^{2}-\frac{85\!\cdots\!38}{54\!\cdots\!01}a-\frac{74\!\cdots\!65}{54\!\cdots\!01}$, $\frac{68\!\cdots\!23}{92\!\cdots\!84}a^{19}-\frac{17\!\cdots\!49}{92\!\cdots\!84}a^{18}-\frac{23\!\cdots\!47}{92\!\cdots\!84}a^{17}+\frac{10\!\cdots\!71}{92\!\cdots\!84}a^{16}-\frac{33\!\cdots\!45}{92\!\cdots\!84}a^{15}+\frac{10\!\cdots\!17}{92\!\cdots\!84}a^{14}-\frac{91\!\cdots\!73}{46\!\cdots\!42}a^{13}+\frac{18\!\cdots\!35}{46\!\cdots\!42}a^{12}+\frac{43\!\cdots\!69}{46\!\cdots\!42}a^{11}+\frac{14\!\cdots\!57}{46\!\cdots\!42}a^{10}-\frac{10\!\cdots\!03}{48\!\cdots\!36}a^{9}-\frac{20\!\cdots\!27}{92\!\cdots\!84}a^{8}-\frac{78\!\cdots\!69}{92\!\cdots\!84}a^{7}-\frac{11\!\cdots\!05}{92\!\cdots\!84}a^{6}-\frac{33\!\cdots\!11}{92\!\cdots\!84}a^{5}+\frac{45\!\cdots\!39}{92\!\cdots\!84}a^{4}+\frac{32\!\cdots\!21}{23\!\cdots\!21}a^{3}+\frac{52\!\cdots\!02}{23\!\cdots\!21}a^{2}+\frac{99\!\cdots\!79}{46\!\cdots\!42}a+\frac{24\!\cdots\!72}{23\!\cdots\!21}$
| 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: | \( 265705502.8157561 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 265705502.8157561 \cdot 2}{2\cdot\sqrt{9387703148876316845417022705078125}}\cr\approx \mathstrut & 0.262978084870350 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 25*x^18 - 35*x^17 + 270*x^16 - 919*x^15 + 4850*x^14 + 24400*x^13 + 158005*x^12 + 323200*x^11 + 240476*x^10 - 2451535*x^9 - 9806600*x^8 - 22103305*x^7 - 18499285*x^6 + 36486691*x^5 + 184536915*x^4 + 378693200*x^3 + 498508960*x^2 + 402067640*x + 168988496)
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^19 + 25*x^18 - 35*x^17 + 270*x^16 - 919*x^15 + 4850*x^14 + 24400*x^13 + 158005*x^12 + 323200*x^11 + 240476*x^10 - 2451535*x^9 - 9806600*x^8 - 22103305*x^7 - 18499285*x^6 + 36486691*x^5 + 184536915*x^4 + 378693200*x^3 + 498508960*x^2 + 402067640*x + 168988496, 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^19 + 25*x^18 - 35*x^17 + 270*x^16 - 919*x^15 + 4850*x^14 + 24400*x^13 + 158005*x^12 + 323200*x^11 + 240476*x^10 - 2451535*x^9 - 9806600*x^8 - 22103305*x^7 - 18499285*x^6 + 36486691*x^5 + 184536915*x^4 + 378693200*x^3 + 498508960*x^2 + 402067640*x + 168988496);
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^19 + 25*x^18 - 35*x^17 + 270*x^16 - 919*x^15 + 4850*x^14 + 24400*x^13 + 158005*x^12 + 323200*x^11 + 240476*x^10 - 2451535*x^9 - 9806600*x^8 - 22103305*x^7 - 18499285*x^6 + 36486691*x^5 + 184536915*x^4 + 378693200*x^3 + 498508960*x^2 + 402067640*x + 168988496);
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
$C_2\times F_5$ (as 20T9):
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 10 conjugacy class representatives for $C_2\times F_5$ |
| Character table for $C_2\times F_5$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.0.36125.1, 5.1.78125.1, 10.2.30517578125.1 |
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 10 siblings: | deg 10, 10.2.8666119384765625.2 |
| Degree 20 sibling: | deg 20 |
| Minimal sibling: | 10.2.8666119384765625.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{5}$ | ${\href{/padicField/3.4.0.1}{4} }^{5}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{5}$ | ${\href{/padicField/11.10.0.1}{10} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{5}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{10}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{5}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | ${\href{/padicField/47.4.0.1}{4} }^{5}$ | ${\href{/padicField/53.4.0.1}{4} }^{5}$ | ${\href{/padicField/59.2.0.1}{2} }^{10}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| Deg $20$ | $20$ | $1$ | $31$ | |||
|
\(17\)
| 17.4.2.2 | $x^{4} - 272 x^{2} + 867$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 17.8.4.1 | $x^{8} + 612 x^{7} + 140536 x^{6} + 14363966 x^{5} + 553913435 x^{4} + 345855654 x^{3} + 4032327212 x^{2} + 6379401496 x + 2294776272$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 17.8.4.1 | $x^{8} + 612 x^{7} + 140536 x^{6} + 14363966 x^{5} + 553913435 x^{4} + 345855654 x^{3} + 4032327212 x^{2} + 6379401496 x + 2294776272$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |