Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 8*x^22 + 67*x^20 - 320*x^18 + 2315*x^16 + 7064*x^14 - 14367*x^12 - 58928*x^10 + 82856*x^8 - 58880*x^6 + 282768*x^4 + 17664*x^2 + 1024)
gp: K = bnfinit(y^24 - 8*y^22 + 67*y^20 - 320*y^18 + 2315*y^16 + 7064*y^14 - 14367*y^12 - 58928*y^10 + 82856*y^8 - 58880*y^6 + 282768*y^4 + 17664*y^2 + 1024, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 8*x^22 + 67*x^20 - 320*x^18 + 2315*x^16 + 7064*x^14 - 14367*x^12 - 58928*x^10 + 82856*x^8 - 58880*x^6 + 282768*x^4 + 17664*x^2 + 1024);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 8*x^22 + 67*x^20 - 320*x^18 + 2315*x^16 + 7064*x^14 - 14367*x^12 - 58928*x^10 + 82856*x^8 - 58880*x^6 + 282768*x^4 + 17664*x^2 + 1024)
\( x^{24} - 8 x^{22} + 67 x^{20} - 320 x^{18} + 2315 x^{16} + 7064 x^{14} - 14367 x^{12} - 58928 x^{10} + \cdots + 1024 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $24$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(5910619501477475486901757182719574409216\)
\(\medspace = 2^{48}\cdot 7^{12}\cdot 79^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(45.41\) | 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: | $2^{2}7^{1/2}79^{1/2}\approx 94.06380813043877$ | ||
| Ramified primes: |
\(2\), \(7\), \(79\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{2048}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{7}-\frac{1}{2}a^{5}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{8}-\frac{1}{4}a^{6}+\frac{1}{8}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{9}-\frac{3}{8}a^{5}-\frac{1}{2}a$, $\frac{1}{8}a^{10}+\frac{1}{8}a^{6}$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{7}-\frac{1}{2}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{16}a^{12}-\frac{1}{16}a^{10}-\frac{1}{16}a^{8}-\frac{1}{16}a^{6}-\frac{1}{8}a^{4}$, $\frac{1}{16}a^{13}-\frac{1}{16}a^{11}-\frac{1}{16}a^{9}-\frac{1}{16}a^{7}-\frac{1}{8}a^{5}$, $\frac{1}{128}a^{14}+\frac{3}{64}a^{10}-\frac{1}{16}a^{8}+\frac{1}{128}a^{6}-\frac{5}{16}a^{4}-\frac{1}{32}a^{2}-\frac{1}{4}$, $\frac{1}{256}a^{15}-\frac{5}{128}a^{11}-\frac{1}{32}a^{9}-\frac{15}{256}a^{7}+\frac{11}{32}a^{5}+\frac{31}{64}a^{3}-\frac{1}{8}a$, $\frac{1}{256}a^{16}+\frac{3}{128}a^{12}+\frac{1}{32}a^{10}+\frac{1}{256}a^{8}+\frac{5}{32}a^{6}+\frac{31}{64}a^{4}-\frac{3}{8}a^{2}$, $\frac{1}{256}a^{17}+\frac{3}{128}a^{13}+\frac{1}{32}a^{11}+\frac{1}{256}a^{9}-\frac{3}{32}a^{7}-\frac{1}{64}a^{5}+\frac{3}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{256}a^{18}-\frac{1}{32}a^{12}+\frac{13}{256}a^{10}+\frac{1}{32}a^{8}-\frac{13}{128}a^{6}+\frac{5}{16}a^{4}+\frac{11}{32}a^{2}-\frac{1}{4}$, $\frac{1}{256}a^{19}-\frac{1}{32}a^{13}+\frac{13}{256}a^{11}+\frac{1}{32}a^{9}-\frac{13}{128}a^{7}+\frac{5}{16}a^{5}+\frac{11}{32}a^{3}-\frac{1}{4}a$, $\frac{1}{256}a^{20}-\frac{3}{256}a^{12}+\frac{1}{32}a^{10}-\frac{5}{128}a^{8}+\frac{5}{32}a^{6}+\frac{15}{32}a^{4}-\frac{3}{8}a^{2}$, $\frac{1}{1024}a^{21}-\frac{1}{512}a^{19}-\frac{1}{1024}a^{17}-\frac{1}{512}a^{15}-\frac{25}{1024}a^{13}-\frac{3}{512}a^{11}-\frac{11}{1024}a^{9}+\frac{9}{512}a^{7}+\frac{115}{256}a^{5}-\frac{61}{128}a^{3}-\frac{1}{16}a$, $\frac{1}{16\!\cdots\!52}a^{22}+\frac{42\!\cdots\!47}{84\!\cdots\!76}a^{20}+\frac{22\!\cdots\!35}{16\!\cdots\!52}a^{18}-\frac{32\!\cdots\!79}{84\!\cdots\!76}a^{16}-\frac{62\!\cdots\!73}{16\!\cdots\!52}a^{14}+\frac{13\!\cdots\!13}{84\!\cdots\!76}a^{12}+\frac{34\!\cdots\!77}{16\!\cdots\!52}a^{10}+\frac{16\!\cdots\!13}{36\!\cdots\!12}a^{8}-\frac{33\!\cdots\!81}{42\!\cdots\!88}a^{6}+\frac{10\!\cdots\!73}{21\!\cdots\!44}a^{4}-\frac{52\!\cdots\!27}{26\!\cdots\!68}a^{2}-\frac{28\!\cdots\!52}{82\!\cdots\!49}$, $\frac{1}{33\!\cdots\!04}a^{23}+\frac{42\!\cdots\!47}{16\!\cdots\!52}a^{21}-\frac{43\!\cdots\!57}{33\!\cdots\!04}a^{19}+\frac{29\!\cdots\!17}{16\!\cdots\!52}a^{17}-\frac{62\!\cdots\!73}{33\!\cdots\!04}a^{15}+\frac{65\!\cdots\!21}{16\!\cdots\!52}a^{13}+\frac{10\!\cdots\!89}{33\!\cdots\!04}a^{11}+\frac{29\!\cdots\!81}{73\!\cdots\!24}a^{9}+\frac{10\!\cdots\!55}{84\!\cdots\!76}a^{7}-\frac{44\!\cdots\!47}{42\!\cdots\!88}a^{5}-\frac{12\!\cdots\!27}{26\!\cdots\!68}a^{3}-\frac{33\!\cdots\!59}{65\!\cdots\!92}a$
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
$C_{2}\times C_{2}\times C_{66}$, which has order $264$ (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: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{3105049468239422335}{11626724522800123252736} a^{23} - \frac{12445077880699859953}{5813362261400061626368} a^{21} + \frac{208457443476351879505}{11626724522800123252736} a^{19} - \frac{498554541254020026163}{5813362261400061626368} a^{17} + \frac{7205475387506275557617}{11626724522800123252736} a^{15} + \frac{10906406029845280837537}{5813362261400061626368} a^{13} - \frac{44915483409356816233525}{11626724522800123252736} a^{11} - \frac{3958396803089225559679}{252754880930437462016} a^{9} + \frac{65001002369943334042531}{2906681130700030813184} a^{7} - \frac{23485028518664799519859}{1453340565350015406592} a^{5} + \frac{27683735910655354098479}{363335141337503851648} a^{3} + \frac{157064446818803100197}{45416892667187981456} a \)
(order $8$)
| 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: |
$\frac{83\!\cdots\!75}{11\!\cdots\!36}a^{23}-\frac{60199823505}{910755767601152}a^{22}-\frac{34\!\cdots\!05}{58\!\cdots\!68}a^{21}+\frac{242020228773}{455377883800576}a^{20}+\frac{57\!\cdots\!21}{11\!\cdots\!36}a^{19}-\frac{4052814999383}{910755767601152}a^{18}-\frac{13\!\cdots\!63}{58\!\cdots\!68}a^{17}+\frac{9717378708231}{455377883800576}a^{16}+\frac{19\!\cdots\!65}{11\!\cdots\!36}a^{15}-\frac{140178514460487}{910755767601152}a^{14}+\frac{27\!\cdots\!73}{58\!\cdots\!68}a^{13}-\frac{209633543396005}{455377883800576}a^{12}-\frac{13\!\cdots\!61}{11\!\cdots\!36}a^{11}+\frac{881355432760307}{910755767601152}a^{10}-\frac{10\!\cdots\!19}{25\!\cdots\!16}a^{9}+\frac{76621153387411}{19799038426112}a^{8}+\frac{19\!\cdots\!39}{29\!\cdots\!84}a^{7}-\frac{12\!\cdots\!07}{227688941900288}a^{6}-\frac{76\!\cdots\!11}{14\!\cdots\!92}a^{5}+\frac{472826319130191}{113844470950144}a^{4}+\frac{77\!\cdots\!55}{36\!\cdots\!48}a^{3}-\frac{275417139667829}{14230558868768}a^{2}-\frac{14\!\cdots\!15}{45\!\cdots\!56}a-\frac{709970649554}{444704964649}$, $\frac{19\!\cdots\!73}{33\!\cdots\!04}a^{23}-\frac{13\!\cdots\!05}{82\!\cdots\!64}a^{22}-\frac{79\!\cdots\!73}{16\!\cdots\!52}a^{21}+\frac{52\!\cdots\!09}{41\!\cdots\!32}a^{20}+\frac{13\!\cdots\!71}{33\!\cdots\!04}a^{19}-\frac{87\!\cdots\!95}{82\!\cdots\!64}a^{18}-\frac{31\!\cdots\!55}{16\!\cdots\!52}a^{17}+\frac{21\!\cdots\!07}{41\!\cdots\!32}a^{16}+\frac{45\!\cdots\!99}{33\!\cdots\!04}a^{15}-\frac{30\!\cdots\!35}{82\!\cdots\!64}a^{14}+\frac{69\!\cdots\!57}{16\!\cdots\!52}a^{13}-\frac{45\!\cdots\!49}{41\!\cdots\!32}a^{12}-\frac{28\!\cdots\!23}{33\!\cdots\!04}a^{11}+\frac{19\!\cdots\!83}{82\!\cdots\!64}a^{10}-\frac{25\!\cdots\!31}{73\!\cdots\!24}a^{9}+\frac{16\!\cdots\!63}{17\!\cdots\!84}a^{8}+\frac{41\!\cdots\!43}{84\!\cdots\!76}a^{7}-\frac{27\!\cdots\!99}{20\!\cdots\!16}a^{6}-\frac{14\!\cdots\!75}{42\!\cdots\!88}a^{5}+\frac{99\!\cdots\!67}{10\!\cdots\!08}a^{4}+\frac{87\!\cdots\!97}{52\!\cdots\!36}a^{3}-\frac{57\!\cdots\!61}{12\!\cdots\!76}a^{2}+\frac{79\!\cdots\!05}{82\!\cdots\!49}a-\frac{35\!\cdots\!67}{40\!\cdots\!93}$, $\frac{57\!\cdots\!45}{16\!\cdots\!52}a^{23}-\frac{13\!\cdots\!05}{82\!\cdots\!64}a^{22}-\frac{12\!\cdots\!23}{42\!\cdots\!88}a^{21}+\frac{52\!\cdots\!09}{41\!\cdots\!32}a^{20}+\frac{43\!\cdots\!95}{16\!\cdots\!52}a^{19}-\frac{87\!\cdots\!95}{82\!\cdots\!64}a^{18}-\frac{55\!\cdots\!11}{42\!\cdots\!88}a^{17}+\frac{21\!\cdots\!07}{41\!\cdots\!32}a^{16}+\frac{15\!\cdots\!35}{16\!\cdots\!52}a^{15}-\frac{30\!\cdots\!35}{82\!\cdots\!64}a^{14}+\frac{67\!\cdots\!11}{42\!\cdots\!88}a^{13}-\frac{45\!\cdots\!49}{41\!\cdots\!32}a^{12}-\frac{12\!\cdots\!95}{16\!\cdots\!52}a^{11}+\frac{19\!\cdots\!83}{82\!\cdots\!64}a^{10}-\frac{27\!\cdots\!23}{18\!\cdots\!56}a^{9}+\frac{16\!\cdots\!63}{17\!\cdots\!84}a^{8}+\frac{10\!\cdots\!03}{21\!\cdots\!44}a^{7}-\frac{27\!\cdots\!99}{20\!\cdots\!16}a^{6}-\frac{53\!\cdots\!93}{10\!\cdots\!72}a^{5}+\frac{99\!\cdots\!67}{10\!\cdots\!08}a^{4}+\frac{12\!\cdots\!53}{10\!\cdots\!72}a^{3}-\frac{57\!\cdots\!61}{12\!\cdots\!76}a^{2}-\frac{12\!\cdots\!83}{13\!\cdots\!84}a-\frac{75\!\cdots\!60}{40\!\cdots\!93}$, $\frac{22\!\cdots\!09}{30\!\cdots\!64}a^{23}+\frac{7084463552871}{22\!\cdots\!48}a^{22}-\frac{89\!\cdots\!19}{15\!\cdots\!32}a^{21}-\frac{11143753317775}{11\!\cdots\!24}a^{20}+\frac{14\!\cdots\!11}{30\!\cdots\!64}a^{19}+\frac{191511310038205}{22\!\cdots\!48}a^{18}-\frac{35\!\cdots\!29}{15\!\cdots\!32}a^{17}+\frac{41397624285959}{11\!\cdots\!24}a^{16}+\frac{51\!\cdots\!27}{30\!\cdots\!64}a^{15}+\frac{50\!\cdots\!69}{22\!\cdots\!48}a^{14}+\frac{76\!\cdots\!63}{15\!\cdots\!32}a^{13}+\frac{65\!\cdots\!07}{11\!\cdots\!24}a^{12}-\frac{32\!\cdots\!51}{30\!\cdots\!64}a^{11}+\frac{12\!\cdots\!31}{22\!\cdots\!48}a^{10}-\frac{27\!\cdots\!77}{66\!\cdots\!84}a^{9}-\frac{22\!\cdots\!97}{48\!\cdots\!88}a^{8}+\frac{47\!\cdots\!45}{76\!\cdots\!16}a^{7}-\frac{35\!\cdots\!63}{55\!\cdots\!12}a^{6}-\frac{17\!\cdots\!05}{38\!\cdots\!08}a^{5}+\frac{42\!\cdots\!83}{27\!\cdots\!56}a^{4}+\frac{19\!\cdots\!79}{95\!\cdots\!52}a^{3}+\frac{33\!\cdots\!99}{34\!\cdots\!32}a^{2}+\frac{45\!\cdots\!69}{11\!\cdots\!44}a+\frac{82\!\cdots\!64}{21\!\cdots\!27}$, $\frac{59\!\cdots\!63}{16\!\cdots\!52}a^{23}-\frac{36\!\cdots\!65}{26\!\cdots\!52}a^{22}-\frac{23\!\cdots\!99}{84\!\cdots\!76}a^{21}+\frac{13\!\cdots\!83}{13\!\cdots\!76}a^{20}+\frac{39\!\cdots\!97}{16\!\cdots\!52}a^{19}-\frac{22\!\cdots\!51}{26\!\cdots\!52}a^{18}-\frac{95\!\cdots\!29}{84\!\cdots\!76}a^{17}+\frac{51\!\cdots\!31}{13\!\cdots\!76}a^{16}+\frac{13\!\cdots\!77}{16\!\cdots\!52}a^{15}-\frac{78\!\cdots\!55}{26\!\cdots\!52}a^{14}+\frac{20\!\cdots\!91}{84\!\cdots\!76}a^{13}-\frac{15\!\cdots\!71}{13\!\cdots\!76}a^{12}-\frac{85\!\cdots\!17}{16\!\cdots\!52}a^{11}+\frac{35\!\cdots\!15}{26\!\cdots\!52}a^{10}-\frac{74\!\cdots\!01}{36\!\cdots\!12}a^{9}+\frac{50\!\cdots\!55}{58\!\cdots\!12}a^{8}+\frac{12\!\cdots\!63}{42\!\cdots\!88}a^{7}-\frac{58\!\cdots\!93}{67\!\cdots\!88}a^{6}-\frac{43\!\cdots\!13}{21\!\cdots\!44}a^{5}+\frac{85\!\cdots\!35}{33\!\cdots\!44}a^{4}+\frac{53\!\cdots\!53}{52\!\cdots\!36}a^{3}-\frac{20\!\cdots\!29}{83\!\cdots\!36}a^{2}-\frac{10\!\cdots\!01}{65\!\cdots\!92}a-\frac{20\!\cdots\!59}{10\!\cdots\!92}$, $\frac{32\!\cdots\!35}{26\!\cdots\!68}a^{23}-\frac{52\!\cdots\!03}{53\!\cdots\!04}a^{22}-\frac{41\!\cdots\!69}{42\!\cdots\!88}a^{21}+\frac{22\!\cdots\!87}{26\!\cdots\!52}a^{20}+\frac{43\!\cdots\!19}{52\!\cdots\!36}a^{19}-\frac{37\!\cdots\!21}{53\!\cdots\!04}a^{18}-\frac{16\!\cdots\!81}{42\!\cdots\!88}a^{17}+\frac{93\!\cdots\!53}{26\!\cdots\!52}a^{16}+\frac{15\!\cdots\!87}{52\!\cdots\!36}a^{15}-\frac{13\!\cdots\!37}{53\!\cdots\!04}a^{14}+\frac{36\!\cdots\!97}{42\!\cdots\!88}a^{13}-\frac{15\!\cdots\!03}{26\!\cdots\!52}a^{12}-\frac{92\!\cdots\!49}{52\!\cdots\!36}a^{11}+\frac{94\!\cdots\!25}{53\!\cdots\!04}a^{10}-\frac{13\!\cdots\!29}{18\!\cdots\!56}a^{9}+\frac{54\!\cdots\!29}{11\!\cdots\!24}a^{8}+\frac{53\!\cdots\!37}{52\!\cdots\!36}a^{7}-\frac{18\!\cdots\!33}{13\!\cdots\!76}a^{6}-\frac{81\!\cdots\!13}{10\!\cdots\!72}a^{5}+\frac{62\!\cdots\!57}{67\!\cdots\!88}a^{4}+\frac{47\!\cdots\!85}{13\!\cdots\!84}a^{3}-\frac{10\!\cdots\!71}{41\!\cdots\!68}a^{2}+\frac{26\!\cdots\!65}{16\!\cdots\!98}a+\frac{54\!\cdots\!61}{10\!\cdots\!92}$, $\frac{29\!\cdots\!97}{13\!\cdots\!76}a^{22}-\frac{11\!\cdots\!29}{67\!\cdots\!88}a^{20}+\frac{20\!\cdots\!49}{13\!\cdots\!76}a^{18}-\frac{23\!\cdots\!97}{33\!\cdots\!44}a^{16}+\frac{69\!\cdots\!11}{13\!\cdots\!76}a^{14}+\frac{10\!\cdots\!75}{67\!\cdots\!88}a^{12}-\frac{43\!\cdots\!93}{13\!\cdots\!76}a^{10}-\frac{11\!\cdots\!35}{91\!\cdots\!08}a^{8}+\frac{63\!\cdots\!59}{33\!\cdots\!44}a^{6}-\frac{57\!\cdots\!03}{41\!\cdots\!68}a^{4}+\frac{26\!\cdots\!85}{41\!\cdots\!68}a^{2}+\frac{10\!\cdots\!59}{52\!\cdots\!46}$, $\frac{49\!\cdots\!07}{21\!\cdots\!44}a^{23}+\frac{57\!\cdots\!63}{16\!\cdots\!52}a^{22}-\frac{16\!\cdots\!03}{84\!\cdots\!76}a^{21}-\frac{22\!\cdots\!87}{84\!\cdots\!76}a^{20}+\frac{68\!\cdots\!51}{42\!\cdots\!88}a^{19}+\frac{36\!\cdots\!61}{16\!\cdots\!52}a^{18}-\frac{67\!\cdots\!29}{84\!\cdots\!76}a^{17}-\frac{84\!\cdots\!53}{84\!\cdots\!76}a^{16}+\frac{24\!\cdots\!85}{42\!\cdots\!88}a^{15}+\frac{12\!\cdots\!81}{16\!\cdots\!52}a^{14}+\frac{12\!\cdots\!91}{84\!\cdots\!76}a^{13}+\frac{23\!\cdots\!79}{84\!\cdots\!76}a^{12}-\frac{14\!\cdots\!91}{42\!\cdots\!88}a^{11}-\frac{71\!\cdots\!17}{16\!\cdots\!52}a^{10}-\frac{46\!\cdots\!97}{36\!\cdots\!12}a^{9}-\frac{85\!\cdots\!97}{36\!\cdots\!12}a^{8}+\frac{78\!\cdots\!17}{42\!\cdots\!88}a^{7}+\frac{87\!\cdots\!97}{42\!\cdots\!88}a^{6}-\frac{54\!\cdots\!53}{21\!\cdots\!44}a^{5}+\frac{18\!\cdots\!23}{21\!\cdots\!44}a^{4}+\frac{10\!\cdots\!51}{10\!\cdots\!72}a^{3}+\frac{64\!\cdots\!33}{65\!\cdots\!92}a^{2}-\frac{22\!\cdots\!53}{13\!\cdots\!84}a+\frac{10\!\cdots\!37}{32\!\cdots\!96}$, $\frac{41\!\cdots\!75}{21\!\cdots\!44}a^{22}-\frac{32\!\cdots\!67}{21\!\cdots\!44}a^{20}+\frac{68\!\cdots\!71}{52\!\cdots\!36}a^{18}-\frac{16\!\cdots\!29}{26\!\cdots\!68}a^{16}+\frac{95\!\cdots\!87}{21\!\cdots\!44}a^{14}+\frac{29\!\cdots\!01}{21\!\cdots\!44}a^{12}-\frac{28\!\cdots\!53}{10\!\cdots\!72}a^{10}-\frac{50\!\cdots\!47}{45\!\cdots\!64}a^{8}+\frac{21\!\cdots\!23}{13\!\cdots\!84}a^{6}-\frac{31\!\cdots\!25}{26\!\cdots\!68}a^{4}+\frac{35\!\cdots\!25}{65\!\cdots\!92}a^{2}+\frac{13\!\cdots\!54}{82\!\cdots\!49}$, $\frac{49\!\cdots\!07}{21\!\cdots\!44}a^{23}-\frac{57\!\cdots\!63}{16\!\cdots\!52}a^{22}-\frac{16\!\cdots\!03}{84\!\cdots\!76}a^{21}+\frac{22\!\cdots\!87}{84\!\cdots\!76}a^{20}+\frac{68\!\cdots\!51}{42\!\cdots\!88}a^{19}-\frac{36\!\cdots\!61}{16\!\cdots\!52}a^{18}-\frac{67\!\cdots\!29}{84\!\cdots\!76}a^{17}+\frac{84\!\cdots\!53}{84\!\cdots\!76}a^{16}+\frac{24\!\cdots\!85}{42\!\cdots\!88}a^{15}-\frac{12\!\cdots\!81}{16\!\cdots\!52}a^{14}+\frac{12\!\cdots\!91}{84\!\cdots\!76}a^{13}-\frac{23\!\cdots\!79}{84\!\cdots\!76}a^{12}-\frac{14\!\cdots\!91}{42\!\cdots\!88}a^{11}+\frac{71\!\cdots\!17}{16\!\cdots\!52}a^{10}-\frac{46\!\cdots\!97}{36\!\cdots\!12}a^{9}+\frac{85\!\cdots\!97}{36\!\cdots\!12}a^{8}+\frac{78\!\cdots\!17}{42\!\cdots\!88}a^{7}-\frac{87\!\cdots\!97}{42\!\cdots\!88}a^{6}-\frac{54\!\cdots\!53}{21\!\cdots\!44}a^{5}-\frac{18\!\cdots\!23}{21\!\cdots\!44}a^{4}+\frac{10\!\cdots\!51}{10\!\cdots\!72}a^{3}-\frac{64\!\cdots\!33}{65\!\cdots\!92}a^{2}-\frac{22\!\cdots\!53}{13\!\cdots\!84}a-\frac{10\!\cdots\!37}{32\!\cdots\!96}$, $\frac{10\!\cdots\!43}{33\!\cdots\!04}a^{23}-\frac{43\!\cdots\!45}{21\!\cdots\!44}a^{22}-\frac{41\!\cdots\!93}{16\!\cdots\!52}a^{21}+\frac{34\!\cdots\!87}{21\!\cdots\!44}a^{20}+\frac{69\!\cdots\!97}{33\!\cdots\!04}a^{19}-\frac{18\!\cdots\!89}{13\!\cdots\!84}a^{18}-\frac{16\!\cdots\!03}{16\!\cdots\!52}a^{17}+\frac{17\!\cdots\!43}{26\!\cdots\!68}a^{16}+\frac{24\!\cdots\!69}{33\!\cdots\!04}a^{15}-\frac{99\!\cdots\!13}{21\!\cdots\!44}a^{14}+\frac{33\!\cdots\!41}{16\!\cdots\!52}a^{13}-\frac{30\!\cdots\!41}{21\!\cdots\!44}a^{12}-\frac{15\!\cdots\!25}{33\!\cdots\!04}a^{11}+\frac{31\!\cdots\!09}{10\!\cdots\!72}a^{10}-\frac{12\!\cdots\!83}{73\!\cdots\!24}a^{9}+\frac{55\!\cdots\!27}{45\!\cdots\!64}a^{8}+\frac{24\!\cdots\!75}{84\!\cdots\!76}a^{7}-\frac{44\!\cdots\!09}{26\!\cdots\!68}a^{6}-\frac{96\!\cdots\!71}{42\!\cdots\!88}a^{5}+\frac{31\!\cdots\!57}{26\!\cdots\!68}a^{4}+\frac{93\!\cdots\!13}{10\!\cdots\!72}a^{3}-\frac{19\!\cdots\!63}{32\!\cdots\!96}a^{2}-\frac{17\!\cdots\!01}{13\!\cdots\!84}a-\frac{39\!\cdots\!32}{82\!\cdots\!49}$
| 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: | \( 5260470710.85801 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 5260470710.85801 \cdot 264}{8\cdot\sqrt{5910619501477475486901757182719574409216}}\cr\approx \mathstrut & 8.54831398860747 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^24 - 8*x^22 + 67*x^20 - 320*x^18 + 2315*x^16 + 7064*x^14 - 14367*x^12 - 58928*x^10 + 82856*x^8 - 58880*x^6 + 282768*x^4 + 17664*x^2 + 1024)
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^24 - 8*x^22 + 67*x^20 - 320*x^18 + 2315*x^16 + 7064*x^14 - 14367*x^12 - 58928*x^10 + 82856*x^8 - 58880*x^6 + 282768*x^4 + 17664*x^2 + 1024, 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^24 - 8*x^22 + 67*x^20 - 320*x^18 + 2315*x^16 + 7064*x^14 - 14367*x^12 - 58928*x^10 + 82856*x^8 - 58880*x^6 + 282768*x^4 + 17664*x^2 + 1024);
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^24 - 8*x^22 + 67*x^20 - 320*x^18 + 2315*x^16 + 7064*x^14 - 14367*x^12 - 58928*x^10 + 82856*x^8 - 58880*x^6 + 282768*x^4 + 17664*x^2 + 1024);
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
$C_2^2\times D_6$ (as 24T30):
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 48 |
| The 24 conjugacy class representatives for $C_2^2\times D_6$ |
| Character table for $C_2^2\times D_6$ is not computed |
Intermediate fields
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 24 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 | ${\href{/padicField/3.6.0.1}{6} }^{4}$ | ${\href{/padicField/5.6.0.1}{6} }^{4}$ | R | ${\href{/padicField/11.2.0.1}{2} }^{12}$ | ${\href{/padicField/13.6.0.1}{6} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{12}$ | ${\href{/padicField/19.2.0.1}{2} }^{12}$ | ${\href{/padicField/23.2.0.1}{2} }^{12}$ | ${\href{/padicField/29.2.0.1}{2} }^{12}$ | ${\href{/padicField/31.2.0.1}{2} }^{12}$ | ${\href{/padicField/37.2.0.1}{2} }^{12}$ | ${\href{/padicField/41.2.0.1}{2} }^{12}$ | ${\href{/padicField/43.2.0.1}{2} }^{12}$ | ${\href{/padicField/47.6.0.1}{6} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{12}$ | ${\href{/padicField/59.6.0.1}{6} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.8.2 | $x^{4} + 2 x^{2} + 4 x + 2$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ |
| 2.4.8.2 | $x^{4} + 2 x^{2} + 4 x + 2$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ | |
| 2.4.8.2 | $x^{4} + 2 x^{2} + 4 x + 2$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ | |
| 2.4.8.2 | $x^{4} + 2 x^{2} + 4 x + 2$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ | |
| 2.4.8.2 | $x^{4} + 2 x^{2} + 4 x + 2$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ | |
| 2.4.8.2 | $x^{4} + 2 x^{2} + 4 x + 2$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ | |
|
\(7\)
| 7.12.6.1 | $x^{12} + 44 x^{10} + 10 x^{9} + 786 x^{8} + 22 x^{7} + 6899 x^{6} - 3434 x^{5} + 31050 x^{4} - 28440 x^{3} + 84557 x^{2} - 48082 x + 107648$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
| 7.12.6.1 | $x^{12} + 44 x^{10} + 10 x^{9} + 786 x^{8} + 22 x^{7} + 6899 x^{6} - 3434 x^{5} + 31050 x^{4} - 28440 x^{3} + 84557 x^{2} - 48082 x + 107648$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
|
\(79\)
| 79.2.0.1 | $x^{2} + 78 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 79.2.0.1 | $x^{2} + 78 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 79.2.0.1 | $x^{2} + 78 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 79.2.0.1 | $x^{2} + 78 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 79.4.2.1 | $x^{4} + 156 x^{3} + 6248 x^{2} + 12792 x + 486412$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 79.4.2.1 | $x^{4} + 156 x^{3} + 6248 x^{2} + 12792 x + 486412$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 79.4.2.1 | $x^{4} + 156 x^{3} + 6248 x^{2} + 12792 x + 486412$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 79.4.2.1 | $x^{4} + 156 x^{3} + 6248 x^{2} + 12792 x + 486412$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |