Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 33*x^18 - 56*x^17 + 264*x^16 - 132*x^15 + 1608*x^14 + 355*x^13 + 6315*x^12 + 5561*x^11 + 19739*x^10 + 18482*x^9 + 33800*x^8 + 32094*x^7 + 40751*x^6 + 29511*x^5 + 19191*x^4 + 6594*x^3 + 1781*x^2 + 44*x + 1)
gp: K = bnfinit(y^20 - 5*y^19 + 33*y^18 - 56*y^17 + 264*y^16 - 132*y^15 + 1608*y^14 + 355*y^13 + 6315*y^12 + 5561*y^11 + 19739*y^10 + 18482*y^9 + 33800*y^8 + 32094*y^7 + 40751*y^6 + 29511*y^5 + 19191*y^4 + 6594*y^3 + 1781*y^2 + 44*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 5*x^19 + 33*x^18 - 56*x^17 + 264*x^16 - 132*x^15 + 1608*x^14 + 355*x^13 + 6315*x^12 + 5561*x^11 + 19739*x^10 + 18482*x^9 + 33800*x^8 + 32094*x^7 + 40751*x^6 + 29511*x^5 + 19191*x^4 + 6594*x^3 + 1781*x^2 + 44*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 5*x^19 + 33*x^18 - 56*x^17 + 264*x^16 - 132*x^15 + 1608*x^14 + 355*x^13 + 6315*x^12 + 5561*x^11 + 19739*x^10 + 18482*x^9 + 33800*x^8 + 32094*x^7 + 40751*x^6 + 29511*x^5 + 19191*x^4 + 6594*x^3 + 1781*x^2 + 44*x + 1)
\( x^{20} - 5 x^{19} + 33 x^{18} - 56 x^{17} + 264 x^{16} - 132 x^{15} + 1608 x^{14} + 355 x^{13} + \cdots + 1 \)
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: |
\(22416185558603163911567862205995264\)
\(\medspace = 2^{8}\cdot 3^{10}\cdot 33769^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(52.18\) | 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: | $2^{2/3}3^{1/2}33769^{3/4}\approx 6849.138386497971$ | ||
| Ramified primes: |
\(2\), \(3\), \(33769\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{512}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{7972}a^{18}+\frac{1943}{7972}a^{17}+\frac{129}{3986}a^{16}+\frac{3885}{7972}a^{15}-\frac{727}{3986}a^{14}+\frac{2761}{7972}a^{13}-\frac{592}{1993}a^{12}+\frac{313}{3986}a^{11}+\frac{2103}{7972}a^{10}-\frac{3477}{7972}a^{9}+\frac{643}{3986}a^{8}-\frac{1865}{7972}a^{7}-\frac{577}{3986}a^{6}+\frac{2201}{7972}a^{5}-\frac{3429}{7972}a^{4}-\frac{1105}{3986}a^{3}-\frac{413}{3986}a^{2}-\frac{555}{3986}a+\frac{2429}{7972}$, $\frac{1}{38\!\cdots\!12}a^{19}-\frac{10\!\cdots\!01}{38\!\cdots\!12}a^{18}+\frac{13\!\cdots\!37}{19\!\cdots\!06}a^{17}+\frac{17\!\cdots\!73}{38\!\cdots\!12}a^{16}-\frac{72\!\cdots\!29}{19\!\cdots\!06}a^{15}-\frac{12\!\cdots\!67}{38\!\cdots\!12}a^{14}-\frac{29\!\cdots\!45}{95\!\cdots\!03}a^{13}+\frac{71\!\cdots\!69}{19\!\cdots\!06}a^{12}+\frac{40\!\cdots\!51}{38\!\cdots\!12}a^{11}+\frac{10\!\cdots\!47}{38\!\cdots\!12}a^{10}-\frac{24\!\cdots\!75}{19\!\cdots\!06}a^{9}-\frac{72\!\cdots\!29}{38\!\cdots\!12}a^{8}-\frac{94\!\cdots\!23}{19\!\cdots\!06}a^{7}+\frac{40\!\cdots\!77}{38\!\cdots\!12}a^{6}-\frac{16\!\cdots\!25}{38\!\cdots\!12}a^{5}+\frac{19\!\cdots\!21}{19\!\cdots\!06}a^{4}+\frac{43\!\cdots\!53}{19\!\cdots\!06}a^{3}+\frac{19\!\cdots\!35}{19\!\cdots\!06}a^{2}+\frac{14\!\cdots\!93}{38\!\cdots\!12}a-\frac{23\!\cdots\!68}{95\!\cdots\!03}$
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
$C_{3}\times C_{3}\times C_{72}$, which has order $648$ (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: |
\( -\frac{48569551905713541770200495545}{1912537800206691741795571009484} a^{19} + \frac{244332412916720027960652321881}{1912537800206691741795571009484} a^{18} - \frac{402644978967003547115985352512}{478134450051672935448892752371} a^{17} + \frac{2770766017554862041123667635995}{1912537800206691741795571009484} a^{16} - \frac{6458946703171702578621608950083}{956268900103345870897785504742} a^{15} + \frac{6826249952625365064485179655637}{1912537800206691741795571009484} a^{14} - \frac{39199408159863311612647873508107}{956268900103345870897785504742} a^{13} - \frac{7391224629096244765072209803015}{956268900103345870897785504742} a^{12} - \frac{306808872521130513711234896578435}{1912537800206691741795571009484} a^{11} - \frac{260718717976897157319330729433059}{1912537800206691741795571009484} a^{10} - \frac{238222910690067438379870790654418}{478134450051672935448892752371} a^{9} - \frac{869940860274606054431729284201495}{1912537800206691741795571009484} a^{8} - \frac{810807153708523048078039452209161}{956268900103345870897785504742} a^{7} - \frac{1514345972867944977232909244613259}{1912537800206691741795571009484} a^{6} - \frac{1944504980740163627753113996116269}{1912537800206691741795571009484} a^{5} - \frac{691685588675157176768611807571727}{956268900103345870897785504742} a^{4} - \frac{225865276778598312105673341920643}{478134450051672935448892752371} a^{3} - \frac{75161372546187744210548503493162}{478134450051672935448892752371} a^{2} - \frac{84373185894742738649201920631727}{1912537800206691741795571009484} a - \frac{86001197015747151453837956449}{956268900103345870897785504742} \)
(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: |
$\frac{23\!\cdots\!69}{15\!\cdots\!16}a^{19}-\frac{31\!\cdots\!89}{38\!\cdots\!29}a^{18}+\frac{80\!\cdots\!97}{15\!\cdots\!16}a^{17}-\frac{15\!\cdots\!21}{15\!\cdots\!16}a^{16}+\frac{65\!\cdots\!67}{15\!\cdots\!16}a^{15}-\frac{51\!\cdots\!85}{15\!\cdots\!16}a^{14}+\frac{37\!\cdots\!85}{15\!\cdots\!16}a^{13}-\frac{23\!\cdots\!07}{76\!\cdots\!58}a^{12}+\frac{13\!\cdots\!37}{15\!\cdots\!16}a^{11}+\frac{39\!\cdots\!83}{76\!\cdots\!58}a^{10}+\frac{39\!\cdots\!65}{15\!\cdots\!16}a^{9}+\frac{26\!\cdots\!17}{15\!\cdots\!16}a^{8}+\frac{58\!\cdots\!53}{15\!\cdots\!16}a^{7}+\frac{46\!\cdots\!47}{15\!\cdots\!16}a^{6}+\frac{15\!\cdots\!72}{38\!\cdots\!29}a^{5}+\frac{34\!\cdots\!57}{15\!\cdots\!16}a^{4}+\frac{41\!\cdots\!79}{38\!\cdots\!29}a^{3}+\frac{62\!\cdots\!18}{38\!\cdots\!29}a^{2}+\frac{61\!\cdots\!23}{15\!\cdots\!16}a+\frac{28\!\cdots\!73}{15\!\cdots\!16}$, $\frac{21\!\cdots\!99}{76\!\cdots\!58}a^{19}-\frac{11\!\cdots\!17}{76\!\cdots\!58}a^{18}+\frac{73\!\cdots\!27}{76\!\cdots\!58}a^{17}-\frac{14\!\cdots\!95}{76\!\cdots\!58}a^{16}+\frac{29\!\cdots\!68}{38\!\cdots\!29}a^{15}-\frac{23\!\cdots\!30}{38\!\cdots\!29}a^{14}+\frac{34\!\cdots\!73}{76\!\cdots\!58}a^{13}-\frac{21\!\cdots\!81}{38\!\cdots\!29}a^{12}+\frac{12\!\cdots\!23}{76\!\cdots\!58}a^{11}+\frac{70\!\cdots\!83}{76\!\cdots\!58}a^{10}+\frac{35\!\cdots\!17}{76\!\cdots\!58}a^{9}+\frac{23\!\cdots\!77}{76\!\cdots\!58}a^{8}+\frac{26\!\cdots\!62}{38\!\cdots\!29}a^{7}+\frac{21\!\cdots\!20}{38\!\cdots\!29}a^{6}+\frac{28\!\cdots\!63}{38\!\cdots\!29}a^{5}+\frac{15\!\cdots\!36}{38\!\cdots\!29}a^{4}+\frac{15\!\cdots\!23}{76\!\cdots\!58}a^{3}+\frac{22\!\cdots\!05}{76\!\cdots\!58}a^{2}+\frac{27\!\cdots\!83}{38\!\cdots\!29}a-\frac{13\!\cdots\!65}{76\!\cdots\!58}$, $\frac{27\!\cdots\!23}{38\!\cdots\!12}a^{19}-\frac{13\!\cdots\!51}{38\!\cdots\!12}a^{18}+\frac{45\!\cdots\!67}{19\!\cdots\!06}a^{17}-\frac{15\!\cdots\!01}{38\!\cdots\!12}a^{16}+\frac{36\!\cdots\!19}{19\!\cdots\!06}a^{15}-\frac{35\!\cdots\!73}{38\!\cdots\!12}a^{14}+\frac{11\!\cdots\!56}{95\!\cdots\!03}a^{13}+\frac{52\!\cdots\!63}{19\!\cdots\!06}a^{12}+\frac{17\!\cdots\!69}{38\!\cdots\!12}a^{11}+\frac{15\!\cdots\!17}{38\!\cdots\!12}a^{10}+\frac{27\!\cdots\!81}{19\!\cdots\!06}a^{9}+\frac{52\!\cdots\!49}{38\!\cdots\!12}a^{8}+\frac{47\!\cdots\!71}{19\!\cdots\!06}a^{7}+\frac{90\!\cdots\!27}{38\!\cdots\!12}a^{6}+\frac{11\!\cdots\!13}{38\!\cdots\!12}a^{5}+\frac{41\!\cdots\!81}{19\!\cdots\!06}a^{4}+\frac{27\!\cdots\!05}{19\!\cdots\!06}a^{3}+\frac{94\!\cdots\!71}{19\!\cdots\!06}a^{2}+\frac{50\!\cdots\!31}{38\!\cdots\!12}a+\frac{30\!\cdots\!29}{95\!\cdots\!03}$, $\frac{42\!\cdots\!21}{38\!\cdots\!12}a^{19}-\frac{21\!\cdots\!85}{38\!\cdots\!12}a^{18}+\frac{35\!\cdots\!82}{95\!\cdots\!03}a^{17}-\frac{24\!\cdots\!43}{38\!\cdots\!12}a^{16}+\frac{57\!\cdots\!99}{19\!\cdots\!06}a^{15}-\frac{62\!\cdots\!57}{38\!\cdots\!12}a^{14}+\frac{34\!\cdots\!09}{19\!\cdots\!06}a^{13}+\frac{58\!\cdots\!91}{19\!\cdots\!06}a^{12}+\frac{27\!\cdots\!87}{38\!\cdots\!12}a^{11}+\frac{22\!\cdots\!79}{38\!\cdots\!12}a^{10}+\frac{21\!\cdots\!76}{95\!\cdots\!03}a^{9}+\frac{75\!\cdots\!23}{38\!\cdots\!12}a^{8}+\frac{71\!\cdots\!63}{19\!\cdots\!06}a^{7}+\frac{13\!\cdots\!99}{38\!\cdots\!12}a^{6}+\frac{17\!\cdots\!13}{38\!\cdots\!12}a^{5}+\frac{60\!\cdots\!65}{19\!\cdots\!06}a^{4}+\frac{20\!\cdots\!59}{95\!\cdots\!03}a^{3}+\frac{67\!\cdots\!46}{95\!\cdots\!03}a^{2}+\frac{74\!\cdots\!79}{38\!\cdots\!12}a+\frac{76\!\cdots\!93}{19\!\cdots\!06}$, $\frac{47\!\cdots\!43}{38\!\cdots\!12}a^{19}-\frac{25\!\cdots\!33}{38\!\cdots\!12}a^{18}+\frac{41\!\cdots\!56}{95\!\cdots\!03}a^{17}-\frac{31\!\cdots\!49}{38\!\cdots\!12}a^{16}+\frac{33\!\cdots\!12}{95\!\cdots\!03}a^{15}-\frac{99\!\cdots\!85}{38\!\cdots\!12}a^{14}+\frac{39\!\cdots\!97}{19\!\cdots\!06}a^{13}-\frac{26\!\cdots\!57}{19\!\cdots\!06}a^{12}+\frac{29\!\cdots\!77}{38\!\cdots\!12}a^{11}+\frac{17\!\cdots\!23}{38\!\cdots\!12}a^{10}+\frac{21\!\cdots\!00}{95\!\cdots\!03}a^{9}+\frac{61\!\cdots\!09}{38\!\cdots\!12}a^{8}+\frac{33\!\cdots\!71}{95\!\cdots\!03}a^{7}+\frac{10\!\cdots\!67}{38\!\cdots\!12}a^{6}+\frac{15\!\cdots\!95}{38\!\cdots\!12}a^{5}+\frac{21\!\cdots\!26}{95\!\cdots\!03}a^{4}+\frac{25\!\cdots\!17}{19\!\cdots\!06}a^{3}+\frac{31\!\cdots\!49}{19\!\cdots\!06}a^{2}+\frac{15\!\cdots\!75}{38\!\cdots\!12}a-\frac{67\!\cdots\!55}{19\!\cdots\!06}$, $\frac{55\!\cdots\!75}{38\!\cdots\!12}a^{19}-\frac{29\!\cdots\!33}{38\!\cdots\!12}a^{18}+\frac{47\!\cdots\!01}{95\!\cdots\!03}a^{17}-\frac{37\!\cdots\!49}{38\!\cdots\!12}a^{16}+\frac{38\!\cdots\!62}{95\!\cdots\!03}a^{15}-\frac{12\!\cdots\!73}{38\!\cdots\!12}a^{14}+\frac{44\!\cdots\!55}{19\!\cdots\!06}a^{13}-\frac{57\!\cdots\!09}{19\!\cdots\!06}a^{12}+\frac{32\!\cdots\!61}{38\!\cdots\!12}a^{11}+\frac{18\!\cdots\!43}{38\!\cdots\!12}a^{10}+\frac{23\!\cdots\!30}{95\!\cdots\!03}a^{9}+\frac{61\!\cdots\!65}{38\!\cdots\!12}a^{8}+\frac{34\!\cdots\!42}{95\!\cdots\!03}a^{7}+\frac{11\!\cdots\!11}{38\!\cdots\!12}a^{6}+\frac{14\!\cdots\!03}{38\!\cdots\!12}a^{5}+\frac{20\!\cdots\!65}{95\!\cdots\!03}a^{4}+\frac{19\!\cdots\!93}{19\!\cdots\!06}a^{3}+\frac{29\!\cdots\!65}{19\!\cdots\!06}a^{2}+\frac{14\!\cdots\!47}{38\!\cdots\!12}a-\frac{23\!\cdots\!99}{19\!\cdots\!06}$, $\frac{21\!\cdots\!41}{19\!\cdots\!06}a^{19}-\frac{21\!\cdots\!69}{38\!\cdots\!12}a^{18}+\frac{14\!\cdots\!29}{38\!\cdots\!12}a^{17}-\frac{59\!\cdots\!73}{95\!\cdots\!03}a^{16}+\frac{11\!\cdots\!45}{38\!\cdots\!12}a^{15}-\frac{27\!\cdots\!35}{19\!\cdots\!06}a^{14}+\frac{68\!\cdots\!27}{38\!\cdots\!12}a^{13}+\frac{39\!\cdots\!31}{95\!\cdots\!03}a^{12}+\frac{67\!\cdots\!56}{95\!\cdots\!03}a^{11}+\frac{24\!\cdots\!41}{38\!\cdots\!12}a^{10}+\frac{84\!\cdots\!57}{38\!\cdots\!12}a^{9}+\frac{20\!\cdots\!72}{95\!\cdots\!03}a^{8}+\frac{14\!\cdots\!43}{38\!\cdots\!12}a^{7}+\frac{69\!\cdots\!71}{19\!\cdots\!06}a^{6}+\frac{17\!\cdots\!89}{38\!\cdots\!12}a^{5}+\frac{12\!\cdots\!27}{38\!\cdots\!12}a^{4}+\frac{20\!\cdots\!85}{95\!\cdots\!03}a^{3}+\frac{71\!\cdots\!51}{95\!\cdots\!03}a^{2}+\frac{38\!\cdots\!37}{19\!\cdots\!06}a+\frac{19\!\cdots\!55}{38\!\cdots\!12}$, $\frac{23\!\cdots\!99}{19\!\cdots\!06}a^{19}-\frac{22\!\cdots\!15}{38\!\cdots\!12}a^{18}+\frac{14\!\cdots\!01}{38\!\cdots\!12}a^{17}-\frac{59\!\cdots\!71}{95\!\cdots\!03}a^{16}+\frac{11\!\cdots\!23}{38\!\cdots\!12}a^{15}-\frac{11\!\cdots\!72}{95\!\cdots\!03}a^{14}+\frac{73\!\cdots\!11}{38\!\cdots\!12}a^{13}+\frac{63\!\cdots\!50}{95\!\cdots\!03}a^{12}+\frac{72\!\cdots\!88}{95\!\cdots\!03}a^{11}+\frac{28\!\cdots\!39}{38\!\cdots\!12}a^{10}+\frac{93\!\cdots\!65}{38\!\cdots\!12}a^{9}+\frac{23\!\cdots\!89}{95\!\cdots\!03}a^{8}+\frac{16\!\cdots\!65}{38\!\cdots\!12}a^{7}+\frac{40\!\cdots\!95}{95\!\cdots\!03}a^{6}+\frac{19\!\cdots\!09}{38\!\cdots\!12}a^{5}+\frac{15\!\cdots\!81}{38\!\cdots\!12}a^{4}+\frac{48\!\cdots\!39}{19\!\cdots\!06}a^{3}+\frac{17\!\cdots\!99}{19\!\cdots\!06}a^{2}+\frac{22\!\cdots\!92}{95\!\cdots\!03}a+\frac{21\!\cdots\!59}{38\!\cdots\!12}$, $\frac{51\!\cdots\!39}{19\!\cdots\!06}a^{19}-\frac{55\!\cdots\!79}{38\!\cdots\!12}a^{18}+\frac{35\!\cdots\!03}{38\!\cdots\!12}a^{17}-\frac{17\!\cdots\!33}{95\!\cdots\!03}a^{16}+\frac{28\!\cdots\!83}{38\!\cdots\!12}a^{15}-\frac{11\!\cdots\!09}{19\!\cdots\!06}a^{14}+\frac{16\!\cdots\!37}{38\!\cdots\!12}a^{13}-\frac{54\!\cdots\!47}{95\!\cdots\!03}a^{12}+\frac{15\!\cdots\!31}{95\!\cdots\!03}a^{11}+\frac{34\!\cdots\!43}{38\!\cdots\!12}a^{10}+\frac{17\!\cdots\!03}{38\!\cdots\!12}a^{9}+\frac{28\!\cdots\!56}{95\!\cdots\!03}a^{8}+\frac{25\!\cdots\!33}{38\!\cdots\!12}a^{7}+\frac{10\!\cdots\!71}{19\!\cdots\!06}a^{6}+\frac{27\!\cdots\!51}{38\!\cdots\!12}a^{5}+\frac{15\!\cdots\!69}{38\!\cdots\!12}a^{4}+\frac{18\!\cdots\!24}{95\!\cdots\!03}a^{3}+\frac{27\!\cdots\!24}{95\!\cdots\!03}a^{2}+\frac{13\!\cdots\!55}{19\!\cdots\!06}a-\frac{84\!\cdots\!19}{38\!\cdots\!12}$
| 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: | \( 5520444.6153 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 5520444.6153 \cdot 648}{6\cdot\sqrt{22416185558603163911567862205995264}}\cr\approx \mathstrut & 0.38187008659 \end{aligned}\] (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 + 33*x^18 - 56*x^17 + 264*x^16 - 132*x^15 + 1608*x^14 + 355*x^13 + 6315*x^12 + 5561*x^11 + 19739*x^10 + 18482*x^9 + 33800*x^8 + 32094*x^7 + 40751*x^6 + 29511*x^5 + 19191*x^4 + 6594*x^3 + 1781*x^2 + 44*x + 1)
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 + 33*x^18 - 56*x^17 + 264*x^16 - 132*x^15 + 1608*x^14 + 355*x^13 + 6315*x^12 + 5561*x^11 + 19739*x^10 + 18482*x^9 + 33800*x^8 + 32094*x^7 + 40751*x^6 + 29511*x^5 + 19191*x^4 + 6594*x^3 + 1781*x^2 + 44*x + 1, 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 + 33*x^18 - 56*x^17 + 264*x^16 - 132*x^15 + 1608*x^14 + 355*x^13 + 6315*x^12 + 5561*x^11 + 19739*x^10 + 18482*x^9 + 33800*x^8 + 32094*x^7 + 40751*x^6 + 29511*x^5 + 19191*x^4 + 6594*x^3 + 1781*x^2 + 44*x + 1);
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 + 33*x^18 - 56*x^17 + 264*x^16 - 132*x^15 + 1608*x^14 + 355*x^13 + 6315*x^12 + 5561*x^11 + 19739*x^10 + 18482*x^9 + 33800*x^8 + 32094*x^7 + 40751*x^6 + 29511*x^5 + 19191*x^4 + 6594*x^3 + 1781*x^2 + 44*x + 1);
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
$C_2\wr S_5$ (as 20T288):
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 3840 |
| The 36 conjugacy class representatives for $C_2\wr S_5$ |
| Character table for $C_2\wr S_5$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 5.5.135076.1, 10.0.4433662763568.1, 10.0.149720357862927792.1, 10.10.616133159929744.1 |
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 10 siblings: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 10.0.149720357862927792.1 |
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 | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{10}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(3\)
| 3.10.5.2 | $x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
| 3.10.5.2 | $x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
|
\(33769\)
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $4$ | $4$ | $1$ | $3$ | ||||
| Deg $4$ | $4$ | $1$ | $3$ |