Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 26*x^18 - 81*x^17 + 263*x^16 - 593*x^15 + 1335*x^14 - 1936*x^13 + 2256*x^12 + 1477*x^11 - 9342*x^10 + 26305*x^9 - 37870*x^8 + 37170*x^7 + 6725*x^6 - 83653*x^5 + 177709*x^4 - 231346*x^3 + 225474*x^2 - 137631*x + 41491)
gp: K = bnfinit(y^20 - 5*y^19 + 26*y^18 - 81*y^17 + 263*y^16 - 593*y^15 + 1335*y^14 - 1936*y^13 + 2256*y^12 + 1477*y^11 - 9342*y^10 + 26305*y^9 - 37870*y^8 + 37170*y^7 + 6725*y^6 - 83653*y^5 + 177709*y^4 - 231346*y^3 + 225474*y^2 - 137631*y + 41491, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 5*x^19 + 26*x^18 - 81*x^17 + 263*x^16 - 593*x^15 + 1335*x^14 - 1936*x^13 + 2256*x^12 + 1477*x^11 - 9342*x^10 + 26305*x^9 - 37870*x^8 + 37170*x^7 + 6725*x^6 - 83653*x^5 + 177709*x^4 - 231346*x^3 + 225474*x^2 - 137631*x + 41491);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 5*x^19 + 26*x^18 - 81*x^17 + 263*x^16 - 593*x^15 + 1335*x^14 - 1936*x^13 + 2256*x^12 + 1477*x^11 - 9342*x^10 + 26305*x^9 - 37870*x^8 + 37170*x^7 + 6725*x^6 - 83653*x^5 + 177709*x^4 - 231346*x^3 + 225474*x^2 - 137631*x + 41491)
\( x^{20} - 5 x^{19} + 26 x^{18} - 81 x^{17} + 263 x^{16} - 593 x^{15} + 1335 x^{14} - 1936 x^{13} + \cdots + 41491 \)
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: |
\(6138974057914386475157900390625\)
\(\medspace = 3^{10}\cdot 5^{10}\cdot 239^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(34.63\) | 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: | $3^{1/2}5^{1/2}239^{1/2}\approx 59.87486951969081$ | ||
| Ramified primes: |
\(3\), \(5\), \(239\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3}a^{14}+\frac{1}{3}a^{13}-\frac{1}{3}a^{11}-\frac{1}{3}a^{10}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{13}-\frac{1}{3}a^{12}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{15}a^{16}-\frac{2}{15}a^{15}-\frac{2}{15}a^{14}+\frac{2}{5}a^{13}-\frac{7}{15}a^{12}-\frac{1}{15}a^{11}+\frac{1}{5}a^{10}-\frac{2}{15}a^{9}-\frac{7}{15}a^{8}+\frac{1}{15}a^{7}-\frac{7}{15}a^{5}-\frac{4}{15}a^{4}+\frac{1}{5}a^{3}+\frac{4}{15}a^{2}-\frac{1}{3}a+\frac{4}{15}$, $\frac{1}{15}a^{17}-\frac{1}{15}a^{15}+\frac{2}{15}a^{14}-\frac{1}{3}a^{12}+\frac{1}{15}a^{11}-\frac{2}{5}a^{10}-\frac{2}{5}a^{9}+\frac{2}{15}a^{8}+\frac{2}{15}a^{7}+\frac{1}{5}a^{6}+\frac{7}{15}a^{5}-\frac{1}{3}a^{4}-\frac{7}{15}a^{2}-\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{8354107575}a^{18}+\frac{23710939}{928234175}a^{17}-\frac{16530958}{928234175}a^{16}-\frac{154019123}{928234175}a^{15}-\frac{33741131}{759464325}a^{14}-\frac{223905011}{759464325}a^{13}-\frac{33534694}{2784702525}a^{12}-\frac{2976961619}{8354107575}a^{11}-\frac{540301691}{1670821515}a^{10}-\frac{930180164}{2784702525}a^{9}+\frac{1365405416}{8354107575}a^{8}-\frac{2426448016}{8354107575}a^{7}-\frac{595325147}{1670821515}a^{6}+\frac{35951406}{84384925}a^{5}-\frac{3480624116}{8354107575}a^{4}-\frac{62228296}{1670821515}a^{3}-\frac{2150580107}{8354107575}a^{2}+\frac{597624107}{8354107575}a+\frac{2412501224}{8354107575}$, $\frac{1}{50\!\cdots\!25}a^{19}+\frac{25\!\cdots\!83}{50\!\cdots\!25}a^{18}-\frac{21\!\cdots\!04}{67\!\cdots\!11}a^{17}-\frac{45\!\cdots\!31}{19\!\cdots\!25}a^{16}-\frac{16\!\cdots\!39}{10\!\cdots\!65}a^{15}-\frac{71\!\cdots\!82}{51\!\cdots\!75}a^{14}-\frac{92\!\cdots\!59}{50\!\cdots\!25}a^{13}-\frac{14\!\cdots\!77}{17\!\cdots\!25}a^{12}-\frac{48\!\cdots\!93}{50\!\cdots\!25}a^{11}-\frac{10\!\cdots\!37}{46\!\cdots\!75}a^{10}-\frac{24\!\cdots\!83}{50\!\cdots\!25}a^{9}-\frac{12\!\cdots\!46}{56\!\cdots\!25}a^{8}+\frac{74\!\cdots\!76}{16\!\cdots\!75}a^{7}-\frac{11\!\cdots\!06}{50\!\cdots\!25}a^{6}+\frac{90\!\cdots\!32}{50\!\cdots\!25}a^{5}-\frac{17\!\cdots\!74}{16\!\cdots\!75}a^{4}+\frac{84\!\cdots\!56}{16\!\cdots\!75}a^{3}-\frac{98\!\cdots\!92}{50\!\cdots\!25}a^{2}-\frac{64\!\cdots\!89}{16\!\cdots\!75}a+\frac{11\!\cdots\!43}{50\!\cdots\!25}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
$C_{5}$, which has order $5$ (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: |
\( -\frac{42888313599281}{443715330654973575} a^{19} + \frac{59326944028577}{147905110218324525} a^{18} - \frac{320275703829502}{147905110218324525} a^{17} + \frac{293698848085067}{49301703406108175} a^{16} - \frac{8937566900405788}{443715330654973575} a^{15} + \frac{1599093722097094}{40337757332270325} a^{14} - \frac{918404524410863}{9860340681221635} a^{13} + \frac{9221677622460551}{88743066130994715} a^{12} - \frac{52312975825213423}{443715330654973575} a^{11} - \frac{37522416742729861}{147905110218324525} a^{10} + \frac{12604668517480697}{17748613226198943} a^{9} - \frac{845070572371436812}{443715330654973575} a^{8} + \frac{876416543745202118}{443715330654973575} a^{7} - \frac{238320418774658003}{147905110218324525} a^{6} - \frac{1006222003298799251}{443715330654973575} a^{5} + \frac{2934499512259845688}{443715330654973575} a^{4} - \frac{4989558997514662268}{443715330654973575} a^{3} + \frac{5360638210882002674}{443715330654973575} a^{2} - \frac{838349183816645342}{88743066130994715} a + \frac{544774709598063346}{147905110218324525} \)
(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: |
$\frac{25\!\cdots\!72}{13\!\cdots\!35}a^{19}-\frac{85\!\cdots\!72}{67\!\cdots\!75}a^{18}+\frac{12\!\cdots\!71}{22\!\cdots\!25}a^{17}-\frac{44\!\cdots\!77}{22\!\cdots\!25}a^{16}+\frac{36\!\cdots\!79}{67\!\cdots\!75}a^{15}-\frac{97\!\cdots\!37}{67\!\cdots\!25}a^{14}+\frac{17\!\cdots\!62}{67\!\cdots\!75}a^{13}-\frac{31\!\cdots\!86}{67\!\cdots\!75}a^{12}+\frac{22\!\cdots\!13}{67\!\cdots\!75}a^{11}+\frac{21\!\cdots\!26}{12\!\cdots\!85}a^{10}-\frac{17\!\cdots\!06}{67\!\cdots\!75}a^{9}+\frac{10\!\cdots\!46}{22\!\cdots\!25}a^{8}-\frac{20\!\cdots\!16}{22\!\cdots\!25}a^{7}+\frac{72\!\cdots\!46}{13\!\cdots\!35}a^{6}-\frac{71\!\cdots\!83}{67\!\cdots\!75}a^{5}-\frac{44\!\cdots\!26}{22\!\cdots\!25}a^{4}+\frac{16\!\cdots\!43}{44\!\cdots\!45}a^{3}-\frac{33\!\cdots\!46}{67\!\cdots\!75}a^{2}+\frac{11\!\cdots\!92}{22\!\cdots\!25}a-\frac{18\!\cdots\!58}{67\!\cdots\!75}$, $\frac{16\!\cdots\!71}{56\!\cdots\!25}a^{19}-\frac{62\!\cdots\!13}{51\!\cdots\!75}a^{18}+\frac{11\!\cdots\!12}{16\!\cdots\!75}a^{17}-\frac{32\!\cdots\!72}{16\!\cdots\!75}a^{16}+\frac{36\!\cdots\!02}{56\!\cdots\!25}a^{15}-\frac{19\!\cdots\!51}{15\!\cdots\!25}a^{14}+\frac{51\!\cdots\!61}{16\!\cdots\!75}a^{13}-\frac{20\!\cdots\!06}{56\!\cdots\!25}a^{12}+\frac{65\!\cdots\!03}{16\!\cdots\!75}a^{11}+\frac{11\!\cdots\!59}{16\!\cdots\!75}a^{10}-\frac{41\!\cdots\!58}{16\!\cdots\!75}a^{9}+\frac{70\!\cdots\!04}{11\!\cdots\!85}a^{8}-\frac{11\!\cdots\!78}{15\!\cdots\!25}a^{7}+\frac{86\!\cdots\!37}{16\!\cdots\!75}a^{6}+\frac{48\!\cdots\!03}{56\!\cdots\!25}a^{5}-\frac{40\!\cdots\!73}{16\!\cdots\!75}a^{4}+\frac{65\!\cdots\!79}{16\!\cdots\!75}a^{3}-\frac{44\!\cdots\!61}{11\!\cdots\!85}a^{2}+\frac{13\!\cdots\!26}{56\!\cdots\!25}a-\frac{38\!\cdots\!11}{56\!\cdots\!25}$, $\frac{44\!\cdots\!93}{10\!\cdots\!65}a^{19}-\frac{37\!\cdots\!08}{50\!\cdots\!25}a^{18}+\frac{17\!\cdots\!23}{56\!\cdots\!25}a^{17}-\frac{23\!\cdots\!73}{16\!\cdots\!75}a^{16}+\frac{19\!\cdots\!01}{50\!\cdots\!25}a^{15}-\frac{54\!\cdots\!67}{46\!\cdots\!75}a^{14}+\frac{11\!\cdots\!58}{50\!\cdots\!25}a^{13}-\frac{23\!\cdots\!59}{50\!\cdots\!25}a^{12}+\frac{25\!\cdots\!38}{56\!\cdots\!25}a^{11}-\frac{18\!\cdots\!02}{10\!\cdots\!65}a^{10}-\frac{99\!\cdots\!39}{50\!\cdots\!25}a^{9}+\frac{21\!\cdots\!57}{46\!\cdots\!75}a^{8}-\frac{46\!\cdots\!82}{50\!\cdots\!25}a^{7}+\frac{76\!\cdots\!49}{10\!\cdots\!65}a^{6}-\frac{77\!\cdots\!42}{50\!\cdots\!25}a^{5}-\frac{92\!\cdots\!77}{50\!\cdots\!25}a^{4}+\frac{70\!\cdots\!83}{20\!\cdots\!33}a^{3}-\frac{85\!\cdots\!28}{16\!\cdots\!75}a^{2}+\frac{21\!\cdots\!14}{50\!\cdots\!25}a-\frac{20\!\cdots\!52}{50\!\cdots\!25}$, $\frac{48\!\cdots\!83}{33\!\cdots\!55}a^{19}-\frac{83\!\cdots\!26}{16\!\cdots\!75}a^{18}+\frac{42\!\cdots\!19}{16\!\cdots\!75}a^{17}-\frac{34\!\cdots\!16}{56\!\cdots\!25}a^{16}+\frac{34\!\cdots\!97}{16\!\cdots\!75}a^{15}-\frac{17\!\cdots\!83}{51\!\cdots\!75}a^{14}+\frac{12\!\cdots\!86}{16\!\cdots\!75}a^{13}-\frac{99\!\cdots\!58}{16\!\cdots\!75}a^{12}+\frac{23\!\cdots\!89}{16\!\cdots\!75}a^{11}+\frac{22\!\cdots\!88}{61\!\cdots\!01}a^{10}-\frac{41\!\cdots\!46}{56\!\cdots\!25}a^{9}+\frac{77\!\cdots\!58}{56\!\cdots\!25}a^{8}-\frac{61\!\cdots\!53}{56\!\cdots\!25}a^{7}-\frac{10\!\cdots\!32}{33\!\cdots\!55}a^{6}+\frac{47\!\cdots\!76}{16\!\cdots\!75}a^{5}-\frac{32\!\cdots\!13}{56\!\cdots\!25}a^{4}+\frac{76\!\cdots\!26}{11\!\cdots\!85}a^{3}-\frac{11\!\cdots\!18}{16\!\cdots\!75}a^{2}+\frac{48\!\cdots\!08}{16\!\cdots\!75}a-\frac{13\!\cdots\!94}{16\!\cdots\!75}$, $\frac{74\!\cdots\!21}{50\!\cdots\!25}a^{19}-\frac{26\!\cdots\!12}{56\!\cdots\!25}a^{18}+\frac{47\!\cdots\!48}{16\!\cdots\!75}a^{17}-\frac{10\!\cdots\!08}{16\!\cdots\!75}a^{16}+\frac{12\!\cdots\!62}{50\!\cdots\!25}a^{15}-\frac{17\!\cdots\!82}{46\!\cdots\!75}a^{14}+\frac{17\!\cdots\!74}{16\!\cdots\!75}a^{13}-\frac{33\!\cdots\!41}{50\!\cdots\!25}a^{12}+\frac{54\!\cdots\!61}{46\!\cdots\!75}a^{11}+\frac{77\!\cdots\!86}{16\!\cdots\!75}a^{10}-\frac{33\!\cdots\!76}{50\!\cdots\!25}a^{9}+\frac{44\!\cdots\!22}{20\!\cdots\!33}a^{8}-\frac{57\!\cdots\!66}{50\!\cdots\!25}a^{7}+\frac{75\!\cdots\!66}{56\!\cdots\!25}a^{6}+\frac{20\!\cdots\!38}{50\!\cdots\!25}a^{5}-\frac{29\!\cdots\!46}{46\!\cdots\!75}a^{4}+\frac{53\!\cdots\!03}{50\!\cdots\!25}a^{3}-\frac{92\!\cdots\!66}{92\!\cdots\!15}a^{2}+\frac{30\!\cdots\!16}{50\!\cdots\!25}a-\frac{18\!\cdots\!17}{16\!\cdots\!75}$, $\frac{79\!\cdots\!04}{50\!\cdots\!25}a^{19}-\frac{30\!\cdots\!91}{50\!\cdots\!25}a^{18}+\frac{52\!\cdots\!24}{16\!\cdots\!75}a^{17}-\frac{13\!\cdots\!41}{16\!\cdots\!75}a^{16}+\frac{13\!\cdots\!16}{50\!\cdots\!25}a^{15}-\frac{21\!\cdots\!44}{46\!\cdots\!75}a^{14}+\frac{53\!\cdots\!32}{50\!\cdots\!25}a^{13}-\frac{40\!\cdots\!41}{50\!\cdots\!25}a^{12}+\frac{11\!\cdots\!94}{22\!\cdots\!37}a^{11}+\frac{25\!\cdots\!97}{50\!\cdots\!25}a^{10}-\frac{49\!\cdots\!56}{50\!\cdots\!25}a^{9}+\frac{10\!\cdots\!71}{50\!\cdots\!25}a^{8}-\frac{12\!\cdots\!59}{10\!\cdots\!65}a^{7}-\frac{69\!\cdots\!84}{50\!\cdots\!25}a^{6}+\frac{21\!\cdots\!86}{50\!\cdots\!25}a^{5}-\frac{76\!\cdots\!32}{10\!\cdots\!65}a^{4}+\frac{17\!\cdots\!03}{21\!\cdots\!75}a^{3}-\frac{44\!\cdots\!48}{56\!\cdots\!25}a^{2}+\frac{30\!\cdots\!26}{50\!\cdots\!25}a-\frac{19\!\cdots\!21}{10\!\cdots\!65}$, $\frac{40\!\cdots\!61}{50\!\cdots\!25}a^{19}-\frac{68\!\cdots\!86}{17\!\cdots\!25}a^{18}+\frac{10\!\cdots\!47}{56\!\cdots\!25}a^{17}-\frac{32\!\cdots\!13}{56\!\cdots\!25}a^{16}+\frac{90\!\cdots\!39}{50\!\cdots\!25}a^{15}-\frac{19\!\cdots\!39}{51\!\cdots\!75}a^{14}+\frac{41\!\cdots\!53}{50\!\cdots\!25}a^{13}-\frac{48\!\cdots\!84}{46\!\cdots\!75}a^{12}+\frac{86\!\cdots\!77}{10\!\cdots\!65}a^{11}+\frac{10\!\cdots\!93}{50\!\cdots\!25}a^{10}-\frac{34\!\cdots\!44}{46\!\cdots\!75}a^{9}+\frac{94\!\cdots\!26}{56\!\cdots\!25}a^{8}-\frac{65\!\cdots\!81}{33\!\cdots\!55}a^{7}+\frac{55\!\cdots\!69}{50\!\cdots\!25}a^{6}+\frac{11\!\cdots\!44}{50\!\cdots\!25}a^{5}-\frac{44\!\cdots\!98}{67\!\cdots\!11}a^{4}+\frac{57\!\cdots\!47}{56\!\cdots\!25}a^{3}-\frac{52\!\cdots\!03}{50\!\cdots\!25}a^{2}+\frac{38\!\cdots\!76}{56\!\cdots\!25}a-\frac{22\!\cdots\!87}{10\!\cdots\!65}$, $\frac{22\!\cdots\!52}{89\!\cdots\!25}a^{19}-\frac{11\!\cdots\!16}{89\!\cdots\!25}a^{18}+\frac{49\!\cdots\!13}{89\!\cdots\!25}a^{17}-\frac{50\!\cdots\!46}{29\!\cdots\!75}a^{16}+\frac{14\!\cdots\!78}{29\!\cdots\!75}a^{15}-\frac{87\!\cdots\!99}{81\!\cdots\!75}a^{14}+\frac{16\!\cdots\!44}{89\!\cdots\!25}a^{13}-\frac{76\!\cdots\!74}{29\!\cdots\!75}a^{12}+\frac{31\!\cdots\!59}{29\!\cdots\!75}a^{11}+\frac{65\!\cdots\!83}{10\!\cdots\!75}a^{10}-\frac{22\!\cdots\!47}{89\!\cdots\!25}a^{9}+\frac{62\!\cdots\!89}{17\!\cdots\!45}a^{8}-\frac{39\!\cdots\!77}{89\!\cdots\!25}a^{7}-\frac{50\!\cdots\!07}{89\!\cdots\!25}a^{6}+\frac{15\!\cdots\!22}{29\!\cdots\!75}a^{5}-\frac{51\!\cdots\!73}{30\!\cdots\!25}a^{4}+\frac{63\!\cdots\!47}{29\!\cdots\!75}a^{3}-\frac{34\!\cdots\!73}{17\!\cdots\!45}a^{2}+\frac{12\!\cdots\!97}{89\!\cdots\!25}a-\frac{60\!\cdots\!62}{89\!\cdots\!25}$, $\frac{11\!\cdots\!77}{50\!\cdots\!25}a^{19}-\frac{82\!\cdots\!08}{30\!\cdots\!05}a^{18}+\frac{43\!\cdots\!88}{16\!\cdots\!75}a^{17}-\frac{33\!\cdots\!12}{11\!\cdots\!95}a^{16}+\frac{90\!\cdots\!22}{50\!\cdots\!25}a^{15}-\frac{12\!\cdots\!06}{18\!\cdots\!03}a^{14}+\frac{10\!\cdots\!61}{16\!\cdots\!75}a^{13}+\frac{69\!\cdots\!99}{17\!\cdots\!25}a^{12}+\frac{91\!\cdots\!53}{50\!\cdots\!25}a^{11}+\frac{20\!\cdots\!14}{56\!\cdots\!25}a^{10}-\frac{30\!\cdots\!44}{50\!\cdots\!25}a^{9}+\frac{38\!\cdots\!06}{50\!\cdots\!25}a^{8}+\frac{67\!\cdots\!32}{46\!\cdots\!75}a^{7}-\frac{15\!\cdots\!59}{16\!\cdots\!75}a^{6}+\frac{11\!\cdots\!28}{10\!\cdots\!65}a^{5}-\frac{12\!\cdots\!73}{50\!\cdots\!25}a^{4}+\frac{52\!\cdots\!01}{50\!\cdots\!25}a^{3}+\frac{12\!\cdots\!98}{50\!\cdots\!25}a^{2}+\frac{52\!\cdots\!14}{50\!\cdots\!25}a-\frac{47\!\cdots\!47}{56\!\cdots\!25}$
| 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: | \( 10123076.9997 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 10123076.9997 \cdot 5}{6\cdot\sqrt{6138974057914386475157900390625}}\cr\approx \mathstrut & 0.326499167848 \end{aligned}\] (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 + 26*x^18 - 81*x^17 + 263*x^16 - 593*x^15 + 1335*x^14 - 1936*x^13 + 2256*x^12 + 1477*x^11 - 9342*x^10 + 26305*x^9 - 37870*x^8 + 37170*x^7 + 6725*x^6 - 83653*x^5 + 177709*x^4 - 231346*x^3 + 225474*x^2 - 137631*x + 41491)
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 + 26*x^18 - 81*x^17 + 263*x^16 - 593*x^15 + 1335*x^14 - 1936*x^13 + 2256*x^12 + 1477*x^11 - 9342*x^10 + 26305*x^9 - 37870*x^8 + 37170*x^7 + 6725*x^6 - 83653*x^5 + 177709*x^4 - 231346*x^3 + 225474*x^2 - 137631*x + 41491, 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 + 26*x^18 - 81*x^17 + 263*x^16 - 593*x^15 + 1335*x^14 - 1936*x^13 + 2256*x^12 + 1477*x^11 - 9342*x^10 + 26305*x^9 - 37870*x^8 + 37170*x^7 + 6725*x^6 - 83653*x^5 + 177709*x^4 - 231346*x^3 + 225474*x^2 - 137631*x + 41491);
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 + 26*x^18 - 81*x^17 + 263*x^16 - 593*x^15 + 1335*x^14 - 1936*x^13 + 2256*x^12 + 1477*x^11 - 9342*x^10 + 26305*x^9 - 37870*x^8 + 37170*x^7 + 6725*x^6 - 83653*x^5 + 177709*x^4 - 231346*x^3 + 225474*x^2 - 137631*x + 41491);
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
$C_2\times D_{10}$ (as 20T8):
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 16 conjugacy class representatives for $C_2\times D_{10}$ |
| Character table for $C_2\times D_{10}$ |
Intermediate fields
| \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), 5.5.12852225.1, 10.0.2477695311759375.1, 10.10.825898437253125.1, 10.0.495539062351875.1 |
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 siblings: | deg 20, deg 20, deg 20 |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | R | R | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{10}$ | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{10}$ | ${\href{/padicField/29.2.0.1}{2} }^{10}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{10}$ | ${\href{/padicField/53.2.0.1}{2} }^{10}$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(5\)
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(239\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |