Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^19 + 43*x^18 - 133*x^17 + 293*x^16 - 525*x^15 + 893*x^14 - 1457*x^13 + 2102*x^12 - 2770*x^11 + 3616*x^10 - 4338*x^9 + 4157*x^8 - 3013*x^7 + 1702*x^6 - 747*x^5 + 166*x^4 + 36*x^3 + 32*x^2 - 80*x + 32)
gp: K = bnfinit(y^20 - 9*y^19 + 43*y^18 - 133*y^17 + 293*y^16 - 525*y^15 + 893*y^14 - 1457*y^13 + 2102*y^12 - 2770*y^11 + 3616*y^10 - 4338*y^9 + 4157*y^8 - 3013*y^7 + 1702*y^6 - 747*y^5 + 166*y^4 + 36*y^3 + 32*y^2 - 80*y + 32, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 9*x^19 + 43*x^18 - 133*x^17 + 293*x^16 - 525*x^15 + 893*x^14 - 1457*x^13 + 2102*x^12 - 2770*x^11 + 3616*x^10 - 4338*x^9 + 4157*x^8 - 3013*x^7 + 1702*x^6 - 747*x^5 + 166*x^4 + 36*x^3 + 32*x^2 - 80*x + 32);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 9*x^19 + 43*x^18 - 133*x^17 + 293*x^16 - 525*x^15 + 893*x^14 - 1457*x^13 + 2102*x^12 - 2770*x^11 + 3616*x^10 - 4338*x^9 + 4157*x^8 - 3013*x^7 + 1702*x^6 - 747*x^5 + 166*x^4 + 36*x^3 + 32*x^2 - 80*x + 32)
\( x^{20} - 9 x^{19} + 43 x^{18} - 133 x^{17} + 293 x^{16} - 525 x^{15} + 893 x^{14} - 1457 x^{13} + \cdots + 32 \)
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: |
\(1759014386056767111350986161\)
\(\medspace = 3^{8}\cdot 401^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(23.03\) | 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: | $3^{1/2}401^{1/2}\approx 34.68429039204925$ | ||
| Ramified primes: |
\(3\), \(401\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{401}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{12}+\frac{1}{3}a^{11}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{6}a^{16}-\frac{1}{6}a^{15}-\frac{1}{6}a^{14}-\frac{1}{6}a^{13}-\frac{1}{6}a^{12}+\frac{1}{6}a^{11}-\frac{1}{2}a^{10}+\frac{1}{6}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{2}a^{4}+\frac{1}{6}a^{3}+\frac{1}{6}a$, $\frac{1}{12}a^{17}-\frac{1}{12}a^{16}-\frac{1}{12}a^{15}-\frac{1}{12}a^{14}+\frac{5}{12}a^{13}-\frac{5}{12}a^{12}-\frac{1}{4}a^{11}-\frac{5}{12}a^{10}-\frac{1}{2}a^{9}+\frac{1}{6}a^{8}+\frac{1}{3}a^{7}+\frac{1}{6}a^{6}-\frac{1}{4}a^{5}-\frac{5}{12}a^{4}-\frac{1}{2}a^{3}+\frac{1}{12}a^{2}-\frac{1}{2}a$, $\frac{1}{1992}a^{18}+\frac{25}{664}a^{17}-\frac{23}{664}a^{16}-\frac{39}{664}a^{15}-\frac{35}{1992}a^{14}-\frac{87}{664}a^{13}+\frac{205}{1992}a^{12}+\frac{743}{1992}a^{11}-\frac{149}{996}a^{10}-\frac{247}{996}a^{9}-\frac{31}{249}a^{8}+\frac{155}{996}a^{7}+\frac{71}{664}a^{6}+\frac{189}{664}a^{5}-\frac{63}{332}a^{4}-\frac{277}{664}a^{3}+\frac{109}{996}a^{2}-\frac{59}{249}a-\frac{67}{249}$, $\frac{1}{56\!\cdots\!72}a^{19}-\frac{2593601285403}{18\!\cdots\!24}a^{18}-\frac{131517172248745}{56\!\cdots\!72}a^{17}-\frac{18\!\cdots\!41}{56\!\cdots\!72}a^{16}-\frac{209120805001409}{14\!\cdots\!48}a^{15}+\frac{23\!\cdots\!45}{18\!\cdots\!24}a^{14}+\frac{10\!\cdots\!33}{56\!\cdots\!72}a^{13}+\frac{19\!\cdots\!35}{56\!\cdots\!72}a^{12}+\frac{18\!\cdots\!71}{94\!\cdots\!12}a^{11}+\frac{87\!\cdots\!17}{28\!\cdots\!36}a^{10}-\frac{17\!\cdots\!35}{70\!\cdots\!34}a^{9}+\frac{317566849292689}{94\!\cdots\!12}a^{8}-\frac{14\!\cdots\!39}{56\!\cdots\!72}a^{7}-\frac{32\!\cdots\!57}{56\!\cdots\!72}a^{6}+\frac{33\!\cdots\!39}{94\!\cdots\!12}a^{5}-\frac{26\!\cdots\!67}{56\!\cdots\!72}a^{4}+\frac{33\!\cdots\!63}{28\!\cdots\!36}a^{3}-\frac{843954194227}{6571510604646}a^{2}+\frac{10\!\cdots\!99}{23\!\cdots\!78}a-\frac{262575187935147}{11\!\cdots\!39}$
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
$C_{2}$, which has order $2$
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: |
$\frac{12\!\cdots\!69}{18\!\cdots\!24}a^{19}-\frac{10\!\cdots\!31}{18\!\cdots\!24}a^{18}+\frac{15\!\cdots\!79}{56\!\cdots\!72}a^{17}-\frac{45\!\cdots\!37}{56\!\cdots\!72}a^{16}+\frac{73\!\cdots\!65}{43\!\cdots\!44}a^{15}-\frac{16\!\cdots\!97}{56\!\cdots\!72}a^{14}+\frac{27\!\cdots\!65}{56\!\cdots\!72}a^{13}-\frac{44\!\cdots\!25}{56\!\cdots\!72}a^{12}+\frac{38\!\cdots\!13}{35\!\cdots\!17}a^{11}-\frac{39\!\cdots\!59}{28\!\cdots\!36}a^{10}+\frac{25\!\cdots\!73}{14\!\cdots\!68}a^{9}-\frac{60\!\cdots\!35}{28\!\cdots\!36}a^{8}+\frac{10\!\cdots\!27}{56\!\cdots\!72}a^{7}-\frac{21\!\cdots\!75}{18\!\cdots\!24}a^{6}+\frac{18\!\cdots\!64}{35\!\cdots\!17}a^{5}-\frac{10\!\cdots\!57}{56\!\cdots\!72}a^{4}-\frac{81\!\cdots\!32}{35\!\cdots\!17}a^{3}+\frac{20651469763739}{4381007069764}a^{2}+\frac{29\!\cdots\!61}{70\!\cdots\!34}a-\frac{15\!\cdots\!96}{35\!\cdots\!17}$, $\frac{39\!\cdots\!65}{56\!\cdots\!72}a^{19}-\frac{42\!\cdots\!59}{56\!\cdots\!72}a^{18}+\frac{21\!\cdots\!61}{56\!\cdots\!72}a^{17}-\frac{23\!\cdots\!73}{18\!\cdots\!24}a^{16}+\frac{39\!\cdots\!81}{14\!\cdots\!48}a^{15}-\frac{89\!\cdots\!45}{18\!\cdots\!24}a^{14}+\frac{44\!\cdots\!79}{56\!\cdots\!72}a^{13}-\frac{24\!\cdots\!97}{18\!\cdots\!24}a^{12}+\frac{22\!\cdots\!50}{11\!\cdots\!39}a^{11}-\frac{67\!\cdots\!11}{28\!\cdots\!36}a^{10}+\frac{44\!\cdots\!97}{14\!\cdots\!68}a^{9}-\frac{10\!\cdots\!77}{28\!\cdots\!36}a^{8}+\frac{19\!\cdots\!37}{56\!\cdots\!72}a^{7}-\frac{42\!\cdots\!93}{18\!\cdots\!24}a^{6}+\frac{81\!\cdots\!67}{70\!\cdots\!34}a^{5}-\frac{25\!\cdots\!91}{56\!\cdots\!72}a^{4}+\frac{18\!\cdots\!85}{14\!\cdots\!68}a^{3}+\frac{54109993108471}{6571510604646}a^{2}+\frac{44\!\cdots\!07}{70\!\cdots\!34}a-\frac{95\!\cdots\!78}{11\!\cdots\!39}$, $\frac{98\!\cdots\!31}{56\!\cdots\!72}a^{19}-\frac{78\!\cdots\!67}{56\!\cdots\!72}a^{18}+\frac{34\!\cdots\!01}{56\!\cdots\!72}a^{17}-\frac{96\!\cdots\!59}{56\!\cdots\!72}a^{16}+\frac{48\!\cdots\!05}{14\!\cdots\!48}a^{15}-\frac{32\!\cdots\!07}{56\!\cdots\!72}a^{14}+\frac{54\!\cdots\!75}{56\!\cdots\!72}a^{13}-\frac{28\!\cdots\!29}{18\!\cdots\!24}a^{12}+\frac{58\!\cdots\!73}{28\!\cdots\!36}a^{11}-\frac{25\!\cdots\!65}{94\!\cdots\!12}a^{10}+\frac{24\!\cdots\!29}{70\!\cdots\!34}a^{9}-\frac{36\!\cdots\!01}{94\!\cdots\!12}a^{8}+\frac{58\!\cdots\!57}{18\!\cdots\!24}a^{7}-\frac{10\!\cdots\!27}{56\!\cdots\!72}a^{6}+\frac{79\!\cdots\!67}{94\!\cdots\!12}a^{5}-\frac{13\!\cdots\!65}{56\!\cdots\!72}a^{4}-\frac{19\!\cdots\!67}{28\!\cdots\!36}a^{3}+\frac{56420344932713}{13143021209292}a^{2}+\frac{55\!\cdots\!89}{70\!\cdots\!34}a-\frac{65\!\cdots\!64}{11\!\cdots\!39}$, $\frac{405580811451687}{94\!\cdots\!12}a^{19}-\frac{24\!\cdots\!73}{70\!\cdots\!34}a^{18}+\frac{73\!\cdots\!91}{47\!\cdots\!56}a^{17}-\frac{21\!\cdots\!41}{47\!\cdots\!56}a^{16}+\frac{16\!\cdots\!07}{181811793395206}a^{15}-\frac{11\!\cdots\!23}{70\!\cdots\!34}a^{14}+\frac{65\!\cdots\!99}{23\!\cdots\!78}a^{13}-\frac{15\!\cdots\!43}{35\!\cdots\!17}a^{12}+\frac{17\!\cdots\!41}{28\!\cdots\!36}a^{11}-\frac{57\!\cdots\!39}{70\!\cdots\!34}a^{10}+\frac{14\!\cdots\!21}{14\!\cdots\!68}a^{9}-\frac{17\!\cdots\!33}{14\!\cdots\!68}a^{8}+\frac{31\!\cdots\!67}{28\!\cdots\!36}a^{7}-\frac{17\!\cdots\!85}{23\!\cdots\!78}a^{6}+\frac{34\!\cdots\!19}{94\!\cdots\!12}a^{5}-\frac{15\!\cdots\!75}{94\!\cdots\!12}a^{4}+\frac{32\!\cdots\!13}{94\!\cdots\!12}a^{3}+\frac{28095977642825}{13143021209292}a^{2}+\frac{55\!\cdots\!07}{35\!\cdots\!17}a-\frac{11\!\cdots\!52}{35\!\cdots\!17}$, $\frac{238555267040071}{683437102883184}a^{19}-\frac{661724467445131}{227812367627728}a^{18}+\frac{29\!\cdots\!85}{227812367627728}a^{17}-\frac{84\!\cdots\!59}{227812367627728}a^{16}+\frac{39\!\cdots\!49}{52572084837168}a^{15}-\frac{87\!\cdots\!09}{683437102883184}a^{14}+\frac{14\!\cdots\!33}{683437102883184}a^{13}-\frac{79\!\cdots\!07}{227812367627728}a^{12}+\frac{20\!\cdots\!58}{42714818930199}a^{11}-\frac{20\!\cdots\!43}{341718551441592}a^{10}+\frac{45\!\cdots\!53}{56953091906932}a^{9}-\frac{10\!\cdots\!37}{113906183813864}a^{8}+\frac{51\!\cdots\!43}{683437102883184}a^{7}-\frac{31\!\cdots\!25}{683437102883184}a^{6}+\frac{18\!\cdots\!17}{85429637860398}a^{5}-\frac{47\!\cdots\!13}{683437102883184}a^{4}-\frac{11\!\cdots\!19}{85429637860398}a^{3}+\frac{156100183441405}{13143021209292}a^{2}+\frac{764912573908562}{42714818930199}a-\frac{542888611496381}{42714818930199}$, $\frac{30\!\cdots\!57}{56\!\cdots\!72}a^{19}-\frac{24\!\cdots\!03}{56\!\cdots\!72}a^{18}+\frac{10\!\cdots\!65}{56\!\cdots\!72}a^{17}-\frac{96\!\cdots\!49}{18\!\cdots\!24}a^{16}+\frac{14\!\cdots\!53}{14\!\cdots\!48}a^{15}-\frac{94\!\cdots\!99}{56\!\cdots\!72}a^{14}+\frac{16\!\cdots\!03}{56\!\cdots\!72}a^{13}-\frac{25\!\cdots\!63}{56\!\cdots\!72}a^{12}+\frac{42\!\cdots\!41}{70\!\cdots\!34}a^{11}-\frac{22\!\cdots\!31}{28\!\cdots\!36}a^{10}+\frac{14\!\cdots\!11}{14\!\cdots\!68}a^{9}-\frac{10\!\cdots\!75}{94\!\cdots\!12}a^{8}+\frac{16\!\cdots\!39}{18\!\cdots\!24}a^{7}-\frac{30\!\cdots\!79}{56\!\cdots\!72}a^{6}+\frac{18\!\cdots\!95}{70\!\cdots\!34}a^{5}-\frac{46\!\cdots\!95}{56\!\cdots\!72}a^{4}-\frac{20\!\cdots\!59}{14\!\cdots\!68}a^{3}+\frac{14294058311297}{3285755302323}a^{2}+\frac{30\!\cdots\!89}{11\!\cdots\!39}a-\frac{43\!\cdots\!52}{35\!\cdots\!17}$, $\frac{10\!\cdots\!73}{28\!\cdots\!36}a^{19}-\frac{83\!\cdots\!37}{28\!\cdots\!36}a^{18}+\frac{37\!\cdots\!21}{28\!\cdots\!36}a^{17}-\frac{10\!\cdots\!75}{28\!\cdots\!36}a^{16}+\frac{16\!\cdots\!87}{21\!\cdots\!72}a^{15}-\frac{35\!\cdots\!91}{28\!\cdots\!36}a^{14}+\frac{20\!\cdots\!97}{94\!\cdots\!12}a^{13}-\frac{32\!\cdots\!69}{94\!\cdots\!12}a^{12}+\frac{16\!\cdots\!95}{35\!\cdots\!17}a^{11}-\frac{14\!\cdots\!97}{23\!\cdots\!78}a^{10}+\frac{27\!\cdots\!63}{35\!\cdots\!17}a^{9}-\frac{12\!\cdots\!35}{14\!\cdots\!68}a^{8}+\frac{68\!\cdots\!15}{94\!\cdots\!12}a^{7}-\frac{12\!\cdots\!13}{28\!\cdots\!36}a^{6}+\frac{48\!\cdots\!21}{23\!\cdots\!78}a^{5}-\frac{19\!\cdots\!97}{28\!\cdots\!36}a^{4}-\frac{46\!\cdots\!19}{47\!\cdots\!56}a^{3}+\frac{99614062650191}{13143021209292}a^{2}+\frac{77\!\cdots\!21}{35\!\cdots\!17}a-\frac{16\!\cdots\!87}{11\!\cdots\!39}$, $\frac{727684314847523}{28\!\cdots\!36}a^{19}-\frac{31\!\cdots\!75}{94\!\cdots\!12}a^{18}+\frac{49\!\cdots\!27}{28\!\cdots\!36}a^{17}-\frac{16\!\cdots\!17}{28\!\cdots\!36}a^{16}+\frac{93\!\cdots\!79}{727247173580824}a^{15}-\frac{62\!\cdots\!01}{28\!\cdots\!36}a^{14}+\frac{10\!\cdots\!77}{28\!\cdots\!36}a^{13}-\frac{59\!\cdots\!47}{94\!\cdots\!12}a^{12}+\frac{31\!\cdots\!54}{35\!\cdots\!17}a^{11}-\frac{16\!\cdots\!39}{14\!\cdots\!68}a^{10}+\frac{36\!\cdots\!43}{23\!\cdots\!78}a^{9}-\frac{26\!\cdots\!11}{14\!\cdots\!68}a^{8}+\frac{15\!\cdots\!09}{94\!\cdots\!12}a^{7}-\frac{30\!\cdots\!53}{28\!\cdots\!36}a^{6}+\frac{22\!\cdots\!24}{35\!\cdots\!17}a^{5}-\frac{81\!\cdots\!29}{28\!\cdots\!36}a^{4}-\frac{22\!\cdots\!63}{35\!\cdots\!17}a^{3}+\frac{38441881725883}{6571510604646}a^{2}+\frac{14\!\cdots\!26}{35\!\cdots\!17}a-\frac{14\!\cdots\!59}{35\!\cdots\!17}$, $\frac{85\!\cdots\!51}{18\!\cdots\!24}a^{19}-\frac{19\!\cdots\!29}{56\!\cdots\!72}a^{18}+\frac{80\!\cdots\!71}{56\!\cdots\!72}a^{17}-\frac{21\!\cdots\!21}{56\!\cdots\!72}a^{16}+\frac{31\!\cdots\!45}{43\!\cdots\!44}a^{15}-\frac{67\!\cdots\!33}{56\!\cdots\!72}a^{14}+\frac{11\!\cdots\!33}{56\!\cdots\!72}a^{13}-\frac{18\!\cdots\!25}{56\!\cdots\!72}a^{12}+\frac{39\!\cdots\!87}{94\!\cdots\!12}a^{11}-\frac{51\!\cdots\!03}{94\!\cdots\!12}a^{10}+\frac{25\!\cdots\!23}{35\!\cdots\!17}a^{9}-\frac{71\!\cdots\!19}{94\!\cdots\!12}a^{8}+\frac{10\!\cdots\!87}{18\!\cdots\!24}a^{7}-\frac{18\!\cdots\!29}{56\!\cdots\!72}a^{6}+\frac{14\!\cdots\!15}{94\!\cdots\!12}a^{5}-\frac{23\!\cdots\!67}{56\!\cdots\!72}a^{4}-\frac{11\!\cdots\!25}{94\!\cdots\!12}a^{3}-\frac{31409651066747}{13143021209292}a^{2}+\frac{13\!\cdots\!25}{70\!\cdots\!34}a-\frac{24\!\cdots\!00}{35\!\cdots\!17}$
| 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: | \( 110300.954595 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 110300.954595 \cdot 2}{2\cdot\sqrt{1759014386056767111350986161}}\cr\approx \mathstrut & 0.252198941199 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^19 + 43*x^18 - 133*x^17 + 293*x^16 - 525*x^15 + 893*x^14 - 1457*x^13 + 2102*x^12 - 2770*x^11 + 3616*x^10 - 4338*x^9 + 4157*x^8 - 3013*x^7 + 1702*x^6 - 747*x^5 + 166*x^4 + 36*x^3 + 32*x^2 - 80*x + 32)
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 - 9*x^19 + 43*x^18 - 133*x^17 + 293*x^16 - 525*x^15 + 893*x^14 - 1457*x^13 + 2102*x^12 - 2770*x^11 + 3616*x^10 - 4338*x^9 + 4157*x^8 - 3013*x^7 + 1702*x^6 - 747*x^5 + 166*x^4 + 36*x^3 + 32*x^2 - 80*x + 32, 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 - 9*x^19 + 43*x^18 - 133*x^17 + 293*x^16 - 525*x^15 + 893*x^14 - 1457*x^13 + 2102*x^12 - 2770*x^11 + 3616*x^10 - 4338*x^9 + 4157*x^8 - 3013*x^7 + 1702*x^6 - 747*x^5 + 166*x^4 + 36*x^3 + 32*x^2 - 80*x + 32);
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 - 9*x^19 + 43*x^18 - 133*x^17 + 293*x^16 - 525*x^15 + 893*x^14 - 1457*x^13 + 2102*x^12 - 2770*x^11 + 3616*x^10 - 4338*x^9 + 4157*x^8 - 3013*x^7 + 1702*x^6 - 747*x^5 + 166*x^4 + 36*x^3 + 32*x^2 - 80*x + 32);
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
$C_2^4:D_5$ (as 20T38):
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 160 |
| The 10 conjugacy class representatives for $C_2^4:D_5$ |
| Character table for $C_2^4:D_5$ |
Intermediate fields
| 5.5.160801.1, 10.6.2094413889681.1 |
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 16 sibling: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 32 sibling: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 10.6.232712654409.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{4}$ | R | ${\href{/padicField/5.5.0.1}{5} }^{4}$ | ${\href{/padicField/7.5.0.1}{5} }^{4}$ | ${\href{/padicField/11.5.0.1}{5} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{5}$ | ${\href{/padicField/19.4.0.1}{4} }^{5}$ | ${\href{/padicField/23.2.0.1}{2} }^{9}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.5.0.1}{5} }^{4}$ | ${\href{/padicField/47.5.0.1}{5} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{9}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(401\)
| $\Q_{401}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{401}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |