Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 93*x^19 - 62*x^18 + 3186*x^17 + 4248*x^16 - 45699*x^15 - 94230*x^14 + 151668*x^13 + 558008*x^12 + 1909683*x^11 + 4713258*x^10 - 6558322*x^9 - 46220760*x^8 - 64186983*x^7 + 10446906*x^6 + 132062292*x^5 + 178126920*x^4 + 124258480*x^3 + 50143968*x^2 + 11143104*x + 1061248)
gp: K = bnfinit(y^21 - 93*y^19 - 62*y^18 + 3186*y^17 + 4248*y^16 - 45699*y^15 - 94230*y^14 + 151668*y^13 + 558008*y^12 + 1909683*y^11 + 4713258*y^10 - 6558322*y^9 - 46220760*y^8 - 64186983*y^7 + 10446906*y^6 + 132062292*y^5 + 178126920*y^4 + 124258480*y^3 + 50143968*y^2 + 11143104*y + 1061248, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 93*x^19 - 62*x^18 + 3186*x^17 + 4248*x^16 - 45699*x^15 - 94230*x^14 + 151668*x^13 + 558008*x^12 + 1909683*x^11 + 4713258*x^10 - 6558322*x^9 - 46220760*x^8 - 64186983*x^7 + 10446906*x^6 + 132062292*x^5 + 178126920*x^4 + 124258480*x^3 + 50143968*x^2 + 11143104*x + 1061248);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 93*x^19 - 62*x^18 + 3186*x^17 + 4248*x^16 - 45699*x^15 - 94230*x^14 + 151668*x^13 + 558008*x^12 + 1909683*x^11 + 4713258*x^10 - 6558322*x^9 - 46220760*x^8 - 64186983*x^7 + 10446906*x^6 + 132062292*x^5 + 178126920*x^4 + 124258480*x^3 + 50143968*x^2 + 11143104*x + 1061248)
\( x^{21} - 93 x^{19} - 62 x^{18} + 3186 x^{17} + 4248 x^{16} - 45699 x^{15} - 94230 x^{14} + \cdots + 1061248 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $21$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[17, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(210302543645345736468562461003484609741021790208\)
\(\medspace = 2^{14}\cdot 3^{28}\cdot 71^{3}\cdot 8291^{2}\cdot 283583^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(179.25\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(3\), \(71\), \(8291\), \(283583\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{20134393}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{4}+\frac{1}{3}a$, $\frac{1}{3}a^{5}+\frac{1}{3}a^{2}$, $\frac{1}{9}a^{6}-\frac{1}{9}a^{3}-\frac{2}{9}$, $\frac{1}{9}a^{7}-\frac{1}{9}a^{4}-\frac{2}{9}a$, $\frac{1}{9}a^{8}-\frac{1}{9}a^{5}-\frac{2}{9}a^{2}$, $\frac{1}{27}a^{9}-\frac{1}{9}a^{3}-\frac{2}{27}$, $\frac{1}{27}a^{10}-\frac{1}{9}a^{4}-\frac{2}{27}a$, $\frac{1}{27}a^{11}-\frac{1}{9}a^{5}-\frac{2}{27}a^{2}$, $\frac{1}{81}a^{12}+\frac{1}{81}a^{9}-\frac{1}{27}a^{6}-\frac{5}{81}a^{3}-\frac{2}{81}$, $\frac{1}{81}a^{13}+\frac{1}{81}a^{10}-\frac{1}{27}a^{7}-\frac{5}{81}a^{4}-\frac{2}{81}a$, $\frac{1}{81}a^{14}+\frac{1}{81}a^{11}-\frac{1}{27}a^{8}-\frac{5}{81}a^{5}-\frac{2}{81}a^{2}$, $\frac{1}{1944}a^{15}-\frac{1}{162}a^{14}-\frac{1}{216}a^{13}-\frac{5}{972}a^{12}+\frac{1}{324}a^{11}+\frac{13}{1944}a^{9}-\frac{1}{108}a^{8}-\frac{10}{243}a^{6}-\frac{7}{648}a^{5}+\frac{7}{108}a^{4}+\frac{121}{972}a^{3}+\frac{31}{81}a^{2}-\frac{13}{72}a-\frac{313}{972}$, $\frac{1}{15552}a^{16}+\frac{1}{7776}a^{15}+\frac{7}{1728}a^{14}-\frac{1}{243}a^{13}-\frac{43}{7776}a^{12}-\frac{1}{432}a^{11}-\frac{203}{15552}a^{10}-\frac{19}{3888}a^{9}+\frac{7}{144}a^{8}+\frac{49}{972}a^{7}+\frac{227}{15552}a^{6}+\frac{13}{108}a^{5}-\frac{581}{7776}a^{4}+\frac{1}{3888}a^{3}+\frac{761}{1728}a^{2}-\frac{521}{3888}a+\frac{737}{3888}$, $\frac{1}{124416}a^{17}-\frac{5}{124416}a^{15}-\frac{191}{62208}a^{14}-\frac{89}{20736}a^{13}+\frac{49}{15552}a^{12}+\frac{1405}{124416}a^{11}-\frac{233}{20736}a^{10}+\frac{211}{31104}a^{9}+\frac{845}{15552}a^{8}+\frac{619}{13824}a^{7}+\frac{1253}{62208}a^{6}+\frac{7531}{62208}a^{5}-\frac{71}{5184}a^{4}-\frac{10183}{124416}a^{3}+\frac{30317}{62208}a^{2}+\frac{2177}{10368}a+\frac{6671}{15552}$, $\frac{1}{2985984}a^{18}+\frac{1}{497664}a^{17}-\frac{23}{995328}a^{16}-\frac{5}{27648}a^{15}+\frac{137}{497664}a^{14}+\frac{1259}{248832}a^{13}-\frac{2033}{995328}a^{12}+\frac{1093}{124416}a^{11}-\frac{2077}{124416}a^{10}+\frac{2675}{186624}a^{9}-\frac{3823}{995328}a^{8}-\frac{11555}{248832}a^{7}+\frac{1121}{165888}a^{6}+\frac{21853}{248832}a^{5}+\frac{133187}{995328}a^{4}-\frac{9647}{62208}a^{3}-\frac{20129}{124416}a^{2}-\frac{1955}{15552}a-\frac{10595}{186624}$, $\frac{1}{23887872}a^{19}-\frac{1}{5971968}a^{18}+\frac{7}{2654208}a^{17}+\frac{25}{3981312}a^{16}+\frac{877}{3981312}a^{15}-\frac{923}{331776}a^{14}+\frac{24023}{7962624}a^{13}-\frac{1213}{1327104}a^{12}-\frac{5197}{331776}a^{11}-\frac{5297}{746496}a^{10}+\frac{201811}{23887872}a^{9}+\frac{15479}{1327104}a^{8}+\frac{191551}{3981312}a^{7}+\frac{17959}{995328}a^{6}-\frac{146839}{2654208}a^{5}-\frac{134087}{3981312}a^{4}-\frac{2831}{331776}a^{3}-\frac{42581}{165888}a^{2}-\frac{672107}{1492992}a-\frac{237377}{746496}$, $\frac{1}{191102976}a^{20}+\frac{1}{95551488}a^{19}-\frac{25}{191102976}a^{18}+\frac{1}{1327104}a^{17}-\frac{95}{10616832}a^{16}-\frac{2075}{15925248}a^{15}+\frac{8743}{63700992}a^{14}+\frac{68327}{15925248}a^{13}-\frac{44387}{15925248}a^{12}+\frac{122015}{11943936}a^{11}-\frac{733805}{191102976}a^{10}-\frac{111899}{11943936}a^{9}-\frac{515201}{10616832}a^{8}-\frac{235367}{5308416}a^{7}+\frac{535819}{63700992}a^{6}+\frac{2321845}{15925248}a^{5}-\frac{942559}{15925248}a^{4}-\frac{106339}{1990656}a^{3}-\frac{1027577}{11943936}a^{2}-\frac{134785}{2985984}a+\frac{928109}{2985984}$
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
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: | $18$ | 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: |
$\frac{43046721}{262144}a^{20}-\frac{14348907}{131072}a^{19}-\frac{3984213177}{262144}a^{18}-\frac{1594323}{32768}a^{17}+\frac{68577678081}{131072}a^{16}+\frac{22856391675}{65536}a^{15}-\frac{2028142480779}{262144}a^{14}-\frac{676049383161}{65536}a^{13}+\frac{2082902108931}{65536}a^{12}+\frac{1154125566747}{16384}a^{11}+\frac{69894918587475}{262144}a^{10}+\frac{19536711216921}{32768}a^{9}-\frac{193255025259537}{131072}a^{8}-\frac{432994698278811}{65536}a^{7}-\frac{16\!\cdots\!47}{262144}a^{6}+\frac{380490698688681}{65536}a^{5}+\frac{11\!\cdots\!79}{65536}a^{4}+\frac{426966435410579}{24576}a^{3}+\frac{144544647343455}{16384}a^{2}+\frac{9636309822897}{4096}a+\frac{3212103290683}{12288}$, $\frac{3823180763}{7077888}a^{20}-\frac{3934644187}{10616832}a^{19}-\frac{9551397575377}{191102976}a^{18}+\frac{26271533}{32768}a^{17}+\frac{2029450453769}{1179648}a^{16}+\frac{1971609626489}{1769472}a^{15}-\frac{180124465218985}{7077888}a^{14}-\frac{177505806480073}{5308416}a^{13}+\frac{16\!\cdots\!09}{15925248}a^{12}+\frac{101507223346649}{442368}a^{11}+\frac{18\!\cdots\!67}{21233664}a^{10}+\frac{46\!\cdots\!01}{23887872}a^{9}-\frac{19\!\cdots\!95}{393216}a^{8}-\frac{38\!\cdots\!57}{1769472}a^{7}-\frac{42\!\cdots\!75}{21233664}a^{6}+\frac{34\!\cdots\!87}{1769472}a^{5}+\frac{30\!\cdots\!95}{5308416}a^{4}+\frac{11\!\cdots\!31}{1990656}a^{3}+\frac{466455915247967}{16384}a^{2}+\frac{278576022347065}{36864}a+\frac{24\!\cdots\!49}{2985984}$, $\frac{30687779435}{3538944}a^{20}-\frac{31394827489}{5308416}a^{19}-\frac{76671586738513}{95551488}a^{18}+\frac{2841709165}{294912}a^{17}+\frac{146621963737565}{5308416}a^{16}+\frac{143313704421059}{7962624}a^{15}-\frac{160649045773553}{393216}a^{14}-\frac{4962518311957}{9216}a^{13}+\frac{496167714185929}{294912}a^{12}+\frac{816451276144835}{221184}a^{11}+\frac{14\!\cdots\!07}{10616832}a^{10}+\frac{74\!\cdots\!97}{23887872}a^{9}-\frac{46\!\cdots\!81}{589824}a^{8}-\frac{92\!\cdots\!29}{2654208}a^{7}-\frac{10\!\cdots\!97}{31850496}a^{6}+\frac{45\!\cdots\!91}{147456}a^{5}+\frac{82\!\cdots\!43}{884736}a^{4}+\frac{15\!\cdots\!03}{165888}a^{3}+\frac{37\!\cdots\!63}{8192}a^{2}+\frac{11\!\cdots\!69}{9216}a+\frac{20\!\cdots\!23}{1492992}$, $\frac{43046721}{262144}a^{20}-\frac{14348907}{131072}a^{19}-\frac{3984213177}{262144}a^{18}-\frac{1594323}{32768}a^{17}+\frac{68577678081}{131072}a^{16}+\frac{22856391675}{65536}a^{15}-\frac{2028142480779}{262144}a^{14}-\frac{676049383161}{65536}a^{13}+\frac{2082902108931}{65536}a^{12}+\frac{1154125566747}{16384}a^{11}+\frac{69894918587475}{262144}a^{10}+\frac{19536711216921}{32768}a^{9}-\frac{193255025259537}{131072}a^{8}-\frac{432994698278811}{65536}a^{7}-\frac{16\!\cdots\!47}{262144}a^{6}+\frac{380490698688681}{65536}a^{5}+\frac{11\!\cdots\!79}{65536}a^{4}+\frac{426966435410579}{24576}a^{3}+\frac{144544647343455}{16384}a^{2}+\frac{9636309822897}{4096}a+\frac{3212103302971}{12288}$, $\frac{22436261321}{2359296}a^{20}-\frac{23052051337}{3538944}a^{19}-\frac{56053235183707}{63700992}a^{18}+\frac{3871083029}{294912}a^{17}+\frac{321573221365937}{10616832}a^{16}+\frac{312938175390785}{15925248}a^{15}-\frac{31\!\cdots\!93}{7077888}a^{14}-\frac{31\!\cdots\!09}{5308416}a^{13}+\frac{29\!\cdots\!61}{15925248}a^{12}+\frac{198678470104241}{49152}a^{11}+\frac{36\!\cdots\!07}{2359296}a^{10}+\frac{90\!\cdots\!77}{2654208}a^{9}-\frac{10\!\cdots\!45}{1179648}a^{8}-\frac{20\!\cdots\!91}{5308416}a^{7}-\frac{22\!\cdots\!85}{63700992}a^{6}+\frac{59\!\cdots\!99}{1769472}a^{5}+\frac{54\!\cdots\!95}{5308416}a^{4}+\frac{19\!\cdots\!43}{1990656}a^{3}+\frac{82\!\cdots\!71}{16384}a^{2}+\frac{16\!\cdots\!63}{12288}a+\frac{14\!\cdots\!87}{995328}$, $\frac{64439853467}{7077888}a^{20}-\frac{65799881779}{10616832}a^{19}-\frac{161002199331217}{191102976}a^{18}+\frac{2666381083}{294912}a^{17}+\frac{307892668365601}{10616832}a^{16}+\frac{301525306007881}{15925248}a^{15}-\frac{30\!\cdots\!65}{7077888}a^{14}-\frac{30\!\cdots\!25}{5308416}a^{13}+\frac{28\!\cdots\!57}{15925248}a^{12}+\frac{17\!\cdots\!33}{442368}a^{11}+\frac{31\!\cdots\!31}{21233664}a^{10}+\frac{78\!\cdots\!03}{23887872}a^{9}-\frac{96\!\cdots\!25}{1179648}a^{8}-\frac{19\!\cdots\!27}{5308416}a^{7}-\frac{21\!\cdots\!17}{63700992}a^{6}+\frac{57\!\cdots\!23}{1769472}a^{5}+\frac{52\!\cdots\!15}{5308416}a^{4}+\frac{18\!\cdots\!47}{1990656}a^{3}+\frac{79\!\cdots\!79}{16384}a^{2}+\frac{47\!\cdots\!41}{36864}a+\frac{42\!\cdots\!09}{2985984}$, $\frac{29472426791}{10616832}a^{20}-\frac{90135684169}{47775744}a^{19}-\frac{24545903831207}{95551488}a^{18}+\frac{102585247}{41472}a^{17}+\frac{140821944020125}{15925248}a^{16}+\frac{15348736236593}{2654208}a^{15}-\frac{13\!\cdots\!49}{10616832}a^{14}-\frac{13\!\cdots\!15}{7962624}a^{13}+\frac{42\!\cdots\!63}{7962624}a^{12}+\frac{785066152692497}{663552}a^{11}+\frac{42\!\cdots\!45}{95551488}a^{10}+\frac{14\!\cdots\!63}{1492992}a^{9}-\frac{13\!\cdots\!11}{5308416}a^{8}-\frac{88\!\cdots\!75}{7962624}a^{7}-\frac{12\!\cdots\!69}{1179648}a^{6}+\frac{26\!\cdots\!41}{2654208}a^{5}+\frac{23\!\cdots\!79}{7962624}a^{4}+\frac{28\!\cdots\!95}{995328}a^{3}+\frac{10\!\cdots\!97}{73728}a^{2}+\frac{58\!\cdots\!45}{1492992}a+\frac{64\!\cdots\!79}{1492992}$, $\frac{24622331998451}{191102976}a^{20}-\frac{8354541997621}{95551488}a^{19}-\frac{22\!\cdots\!31}{191102976}a^{18}+\frac{51213557521}{497664}a^{17}+\frac{43\!\cdots\!39}{10616832}a^{16}+\frac{42\!\cdots\!83}{15925248}a^{15}-\frac{14\!\cdots\!45}{2359296}a^{14}-\frac{12\!\cdots\!83}{15925248}a^{13}+\frac{39\!\cdots\!75}{15925248}a^{12}+\frac{65\!\cdots\!69}{11943936}a^{11}+\frac{39\!\cdots\!17}{191102976}a^{10}+\frac{17\!\cdots\!43}{373248}a^{9}-\frac{36\!\cdots\!97}{31850496}a^{8}-\frac{91\!\cdots\!59}{1769472}a^{7}-\frac{30\!\cdots\!31}{63700992}a^{6}+\frac{24\!\cdots\!17}{5308416}a^{5}+\frac{22\!\cdots\!31}{15925248}a^{4}+\frac{26\!\cdots\!79}{1990656}a^{3}+\frac{81\!\cdots\!05}{11943936}a^{2}+\frac{54\!\cdots\!85}{2985984}a+\frac{60\!\cdots\!15}{2985984}$, $\frac{39214864823093}{47775744}a^{20}-\frac{13424003046827}{23887872}a^{19}-\frac{12\!\cdots\!95}{15925248}a^{18}+\frac{1103300866793}{995328}a^{17}+\frac{69\!\cdots\!49}{2654208}a^{16}+\frac{67\!\cdots\!85}{3981312}a^{15}-\frac{20\!\cdots\!31}{5308416}a^{14}-\frac{20\!\cdots\!81}{3981312}a^{13}+\frac{63\!\cdots\!09}{3981312}a^{12}+\frac{10\!\cdots\!99}{2985984}a^{11}+\frac{63\!\cdots\!87}{47775744}a^{10}+\frac{29\!\cdots\!63}{995328}a^{9}-\frac{58\!\cdots\!03}{7962624}a^{8}-\frac{14\!\cdots\!89}{442368}a^{7}-\frac{48\!\cdots\!97}{15925248}a^{6}+\frac{12\!\cdots\!49}{442368}a^{5}+\frac{35\!\cdots\!25}{3981312}a^{4}+\frac{42\!\cdots\!13}{497664}a^{3}+\frac{12\!\cdots\!31}{2985984}a^{2}+\frac{85\!\cdots\!11}{746496}a+\frac{10\!\cdots\!45}{82944}$, $\frac{115878089695}{10616832}a^{20}-\frac{358498324559}{47775744}a^{19}-\frac{32165696556869}{31850496}a^{18}+\frac{140162393093}{7962624}a^{17}+\frac{553589652535609}{15925248}a^{16}+\frac{178890581877541}{7962624}a^{15}-\frac{16\!\cdots\!19}{31850496}a^{14}-\frac{149286856936373}{221184}a^{13}+\frac{16\!\cdots\!65}{7962624}a^{12}+\frac{92\!\cdots\!85}{1990656}a^{11}+\frac{16\!\cdots\!41}{95551488}a^{10}+\frac{10\!\cdots\!59}{2654208}a^{9}-\frac{15\!\cdots\!37}{15925248}a^{8}-\frac{34\!\cdots\!49}{7962624}a^{7}-\frac{12\!\cdots\!15}{31850496}a^{6}+\frac{15\!\cdots\!97}{3981312}a^{5}+\frac{31\!\cdots\!11}{2654208}a^{4}+\frac{56\!\cdots\!73}{497664}a^{3}+\frac{11\!\cdots\!67}{1990656}a^{2}+\frac{11\!\cdots\!09}{746496}a+\frac{83\!\cdots\!51}{497664}$, $\frac{8640680815279}{191102976}a^{20}-\frac{3023568421873}{95551488}a^{19}-\frac{799353463527575}{191102976}a^{18}+\frac{493839379289}{3981312}a^{17}+\frac{15\!\cdots\!79}{10616832}a^{16}+\frac{14\!\cdots\!47}{15925248}a^{15}-\frac{13\!\cdots\!67}{63700992}a^{14}-\frac{44\!\cdots\!47}{15925248}a^{13}+\frac{14\!\cdots\!23}{15925248}a^{12}+\frac{22\!\cdots\!97}{11943936}a^{11}+\frac{13\!\cdots\!85}{191102976}a^{10}+\frac{19\!\cdots\!47}{11943936}a^{9}-\frac{13\!\cdots\!01}{31850496}a^{8}-\frac{31\!\cdots\!19}{1769472}a^{7}-\frac{10\!\cdots\!55}{63700992}a^{6}+\frac{25\!\cdots\!27}{15925248}a^{5}+\frac{77\!\cdots\!19}{15925248}a^{4}+\frac{92\!\cdots\!19}{1990656}a^{3}+\frac{28\!\cdots\!85}{11943936}a^{2}+\frac{18\!\cdots\!01}{2985984}a+\frac{20\!\cdots\!15}{2985984}$, $\frac{8244535749865}{95551488}a^{20}-\frac{2832690743129}{47775744}a^{19}-\frac{254282914550651}{31850496}a^{18}+\frac{1087003683031}{7962624}a^{17}+\frac{43\!\cdots\!67}{15925248}a^{16}+\frac{14\!\cdots\!55}{7962624}a^{15}-\frac{43\!\cdots\!07}{10616832}a^{14}-\frac{53\!\cdots\!99}{995328}a^{13}+\frac{13\!\cdots\!43}{7962624}a^{12}+\frac{21\!\cdots\!17}{5971968}a^{11}+\frac{13\!\cdots\!95}{95551488}a^{10}+\frac{82\!\cdots\!93}{2654208}a^{9}-\frac{12\!\cdots\!63}{15925248}a^{8}-\frac{27\!\cdots\!67}{7962624}a^{7}-\frac{10\!\cdots\!81}{31850496}a^{6}+\frac{40\!\cdots\!27}{1327104}a^{5}+\frac{73\!\cdots\!31}{7962624}a^{4}+\frac{44\!\cdots\!13}{497664}a^{3}+\frac{27\!\cdots\!47}{5971968}a^{2}+\frac{89\!\cdots\!37}{746496}a+\frac{66\!\cdots\!77}{497664}$, $\frac{9995065748963}{15925248}a^{20}-\frac{10260660804755}{23887872}a^{19}-\frac{11418020275051}{196608}a^{18}+\frac{1657872446383}{1990656}a^{17}+\frac{17\!\cdots\!51}{884736}a^{16}+\frac{17\!\cdots\!43}{1327104}a^{15}-\frac{47\!\cdots\!13}{15925248}a^{14}-\frac{15\!\cdots\!83}{3981312}a^{13}+\frac{16\!\cdots\!83}{1327104}a^{12}+\frac{29\!\cdots\!81}{110592}a^{11}+\frac{48\!\cdots\!75}{47775744}a^{10}+\frac{50\!\cdots\!67}{221184}a^{9}-\frac{45\!\cdots\!43}{7962624}a^{8}-\frac{33\!\cdots\!75}{1327104}a^{7}-\frac{12\!\cdots\!79}{5308416}a^{6}+\frac{89\!\cdots\!67}{3981312}a^{5}+\frac{26\!\cdots\!97}{3981312}a^{4}+\frac{10\!\cdots\!73}{165888}a^{3}+\frac{32\!\cdots\!29}{995328}a^{2}+\frac{65\!\cdots\!33}{746496}a+\frac{26\!\cdots\!91}{27648}$, $\frac{1930068751}{5308416}a^{20}-\frac{6323916763}{23887872}a^{19}-\frac{535438038005}{15925248}a^{18}+\frac{2574479941}{1327104}a^{17}+\frac{9213401314609}{7962624}a^{16}+\frac{310521088993}{442368}a^{15}-\frac{90934791315013}{5308416}a^{14}-\frac{43372071883307}{1990656}a^{13}+\frac{31436160035201}{442368}a^{12}+\frac{16718943975281}{110592}a^{11}+\frac{27\!\cdots\!85}{47775744}a^{10}+\frac{51\!\cdots\!31}{3981312}a^{9}-\frac{88\!\cdots\!67}{2654208}a^{8}-\frac{57\!\cdots\!09}{3981312}a^{7}-\frac{25\!\cdots\!95}{196608}a^{6}+\frac{21\!\cdots\!79}{165888}a^{5}+\frac{15\!\cdots\!17}{3981312}a^{4}+\frac{169259078907263}{4608}a^{3}+\frac{60\!\cdots\!05}{331776}a^{2}+\frac{898386362999779}{186624}a+\frac{130955887587295}{248832}$, $\frac{11045221238503}{191102976}a^{20}-\frac{416810618281}{10616832}a^{19}-\frac{10\!\cdots\!55}{191102976}a^{18}+\frac{98704098439}{1990656}a^{17}+\frac{58\!\cdots\!13}{31850496}a^{16}+\frac{639478235863097}{5308416}a^{15}-\frac{17\!\cdots\!19}{63700992}a^{14}-\frac{57\!\cdots\!47}{15925248}a^{13}+\frac{17\!\cdots\!15}{15925248}a^{12}+\frac{29\!\cdots\!21}{11943936}a^{11}+\frac{59\!\cdots\!83}{63700992}a^{10}+\frac{12\!\cdots\!37}{5971968}a^{9}-\frac{16\!\cdots\!73}{31850496}a^{8}-\frac{36\!\cdots\!95}{15925248}a^{7}-\frac{15\!\cdots\!55}{7077888}a^{6}+\frac{32\!\cdots\!51}{15925248}a^{5}+\frac{99\!\cdots\!83}{15925248}a^{4}+\frac{12\!\cdots\!91}{1990656}a^{3}+\frac{36\!\cdots\!53}{11943936}a^{2}+\frac{81\!\cdots\!75}{995328}a+\frac{26\!\cdots\!51}{2985984}$, $\frac{143625324099437}{95551488}a^{20}-\frac{48785084612117}{47775744}a^{19}-\frac{44\!\cdots\!99}{31850496}a^{18}+\frac{10352232206039}{7962624}a^{17}+\frac{76\!\cdots\!15}{15925248}a^{16}+\frac{83\!\cdots\!33}{2654208}a^{15}-\frac{25\!\cdots\!97}{3538944}a^{14}-\frac{20\!\cdots\!19}{221184}a^{13}+\frac{77\!\cdots\!09}{2654208}a^{12}+\frac{38\!\cdots\!49}{5971968}a^{11}+\frac{23\!\cdots\!67}{95551488}a^{10}+\frac{43\!\cdots\!59}{7962624}a^{9}-\frac{21\!\cdots\!59}{15925248}a^{8}-\frac{47\!\cdots\!71}{7962624}a^{7}-\frac{58\!\cdots\!27}{10616832}a^{6}+\frac{70\!\cdots\!49}{1327104}a^{5}+\frac{43\!\cdots\!93}{2654208}a^{4}+\frac{26\!\cdots\!93}{165888}a^{3}+\frac{47\!\cdots\!95}{5971968}a^{2}+\frac{15\!\cdots\!75}{746496}a+\frac{11\!\cdots\!09}{497664}$, $\frac{4165454380103}{95551488}a^{20}-\frac{155791394845}{5308416}a^{19}-\frac{385499125356815}{95551488}a^{18}+\frac{17571272201}{1327104}a^{17}+\frac{737236719270047}{5308416}a^{16}+\frac{243364804948813}{2654208}a^{15}-\frac{65\!\cdots\!51}{31850496}a^{14}-\frac{72\!\cdots\!95}{2654208}a^{13}+\frac{67\!\cdots\!35}{7962624}a^{12}+\frac{11\!\cdots\!01}{5971968}a^{11}+\frac{27\!\cdots\!59}{393216}a^{10}+\frac{18\!\cdots\!29}{11943936}a^{9}-\frac{69\!\cdots\!37}{1769472}a^{8}-\frac{46\!\cdots\!57}{2654208}a^{7}-\frac{17\!\cdots\!13}{10616832}a^{6}+\frac{12\!\cdots\!85}{7962624}a^{5}+\frac{12\!\cdots\!81}{2654208}a^{4}+\frac{45\!\cdots\!53}{995328}a^{3}+\frac{13\!\cdots\!77}{5971968}a^{2}+\frac{10\!\cdots\!83}{165888}a+\frac{10\!\cdots\!39}{1492992}$, $\frac{2573730077323}{10616832}a^{20}-\frac{7875858979177}{47775744}a^{19}-\frac{714500137996985}{31850496}a^{18}+\frac{895167180971}{3981312}a^{17}+\frac{12\!\cdots\!49}{15925248}a^{16}+\frac{40\!\cdots\!03}{7962624}a^{15}-\frac{36\!\cdots\!27}{31850496}a^{14}-\frac{13\!\cdots\!13}{884736}a^{13}+\frac{37\!\cdots\!95}{7962624}a^{12}+\frac{20\!\cdots\!07}{1990656}a^{11}+\frac{37\!\cdots\!73}{95551488}a^{10}+\frac{34\!\cdots\!23}{3981312}a^{9}-\frac{34\!\cdots\!41}{15925248}a^{8}-\frac{77\!\cdots\!07}{7962624}a^{7}-\frac{28\!\cdots\!23}{31850496}a^{6}+\frac{68\!\cdots\!29}{7962624}a^{5}+\frac{69\!\cdots\!73}{2654208}a^{4}+\frac{25\!\cdots\!33}{995328}a^{3}+\frac{25\!\cdots\!11}{1990656}a^{2}+\frac{51\!\cdots\!51}{1492992}a+\frac{20\!\cdots\!77}{55296}$
| 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: | \( 38671611908100000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{17}\cdot(2\pi)^{2}\cdot 38671611908100000 \cdot 1}{2\cdot\sqrt{210302543645345736468562461003484609741021790208}}\cr\approx \mathstrut & 0.218177358422109 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 93*x^19 - 62*x^18 + 3186*x^17 + 4248*x^16 - 45699*x^15 - 94230*x^14 + 151668*x^13 + 558008*x^12 + 1909683*x^11 + 4713258*x^10 - 6558322*x^9 - 46220760*x^8 - 64186983*x^7 + 10446906*x^6 + 132062292*x^5 + 178126920*x^4 + 124258480*x^3 + 50143968*x^2 + 11143104*x + 1061248)
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^21 - 93*x^19 - 62*x^18 + 3186*x^17 + 4248*x^16 - 45699*x^15 - 94230*x^14 + 151668*x^13 + 558008*x^12 + 1909683*x^11 + 4713258*x^10 - 6558322*x^9 - 46220760*x^8 - 64186983*x^7 + 10446906*x^6 + 132062292*x^5 + 178126920*x^4 + 124258480*x^3 + 50143968*x^2 + 11143104*x + 1061248, 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^21 - 93*x^19 - 62*x^18 + 3186*x^17 + 4248*x^16 - 45699*x^15 - 94230*x^14 + 151668*x^13 + 558008*x^12 + 1909683*x^11 + 4713258*x^10 - 6558322*x^9 - 46220760*x^8 - 64186983*x^7 + 10446906*x^6 + 132062292*x^5 + 178126920*x^4 + 124258480*x^3 + 50143968*x^2 + 11143104*x + 1061248);
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^21 - 93*x^19 - 62*x^18 + 3186*x^17 + 4248*x^16 - 45699*x^15 - 94230*x^14 + 151668*x^13 + 558008*x^12 + 1909683*x^11 + 4713258*x^10 - 6558322*x^9 - 46220760*x^8 - 64186983*x^7 + 10446906*x^6 + 132062292*x^5 + 178126920*x^4 + 124258480*x^3 + 50143968*x^2 + 11143104*x + 1061248);
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
$C_3^7.(C_2^6.S_7)$ (as 21T149):
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 705438720 |
| The 261 conjugacy class representatives for $C_3^7.(C_2^6.S_7)$ |
| Character table for $C_3^7.(C_2^6.S_7)$ |
Intermediate fields
| 7.7.20134393.1 |
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 42 siblings: | data not computed |
| 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 | R | R | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{3}$ | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $21$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{3}$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $21$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{5}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $21$ | ${\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | $15{,}\,{\href{/padicField/59.1.0.1}{1} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.7.0.1 | $x^{7} + x + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |
| 2.14.14.31 | $x^{14} - 12 x^{13} + 50 x^{12} + 120 x^{11} - 820 x^{10} - 144 x^{9} + 8472 x^{8} + 9920 x^{7} - 30672 x^{6} - 59456 x^{5} + 106336 x^{4} + 342912 x^{3} + 180800 x^{2} - 207616 x - 301952$ | $2$ | $7$ | $14$ | 14T21 | $[2, 2, 2, 2, 2, 2]^{7}$ | |
|
\(3\)
| Deg $21$ | $3$ | $7$ | $28$ | |||
|
\(71\)
| 71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 71.2.1.1 | $x^{2} + 497$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.3.0.1 | $x^{3} + 4 x + 64$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 71.4.2.1 | $x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 71.6.0.1 | $x^{6} + x^{4} + 10 x^{3} + 13 x^{2} + 29 x + 7$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(8291\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $3$ | $3$ | $1$ | $2$ | ||||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
|
\(283583\)
| $\Q_{283583}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{283583}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| 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 $6$ | $2$ | $3$ | $3$ |