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Properties

Label 24.0.729...625.1
Degree $24$
Signature $[0, 12]$
Discriminant $7.292\times 10^{38}$
Root discriminant \(41.62\)
Ramified primes $3,5,7,67$
Class number $99$ (GRH)
Class group [3, 33] (GRH)
Galois group $C_2^2\times D_6$ (as 24T30)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 2*x^23 + 8*x^22 - 16*x^21 + 22*x^20 - 64*x^19 + 86*x^18 - 358*x^17 + 696*x^16 - 2792*x^15 + 4514*x^14 - 7304*x^13 + 12417*x^12 + 14608*x^11 + 18056*x^10 + 22336*x^9 + 11136*x^8 + 11456*x^7 + 5504*x^6 + 8192*x^5 + 5632*x^4 + 8192*x^3 + 8192*x^2 + 4096*x + 4096)
 
gp: K = bnfinit(y^24 - 2*y^23 + 8*y^22 - 16*y^21 + 22*y^20 - 64*y^19 + 86*y^18 - 358*y^17 + 696*y^16 - 2792*y^15 + 4514*y^14 - 7304*y^13 + 12417*y^12 + 14608*y^11 + 18056*y^10 + 22336*y^9 + 11136*y^8 + 11456*y^7 + 5504*y^6 + 8192*y^5 + 5632*y^4 + 8192*y^3 + 8192*y^2 + 4096*y + 4096, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 2*x^23 + 8*x^22 - 16*x^21 + 22*x^20 - 64*x^19 + 86*x^18 - 358*x^17 + 696*x^16 - 2792*x^15 + 4514*x^14 - 7304*x^13 + 12417*x^12 + 14608*x^11 + 18056*x^10 + 22336*x^9 + 11136*x^8 + 11456*x^7 + 5504*x^6 + 8192*x^5 + 5632*x^4 + 8192*x^3 + 8192*x^2 + 4096*x + 4096);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 2*x^23 + 8*x^22 - 16*x^21 + 22*x^20 - 64*x^19 + 86*x^18 - 358*x^17 + 696*x^16 - 2792*x^15 + 4514*x^14 - 7304*x^13 + 12417*x^12 + 14608*x^11 + 18056*x^10 + 22336*x^9 + 11136*x^8 + 11456*x^7 + 5504*x^6 + 8192*x^5 + 5632*x^4 + 8192*x^3 + 8192*x^2 + 4096*x + 4096)
 

\( x^{24} - 2 x^{23} + 8 x^{22} - 16 x^{21} + 22 x^{20} - 64 x^{19} + 86 x^{18} - 358 x^{17} + 696 x^{16} + \cdots + 4096 \) Copy content Toggle raw display

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:   \(729239207533207895866388728730712890625\) \(\medspace = 3^{12}\cdot 5^{12}\cdot 7^{12}\cdot 67^{8}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(41.62\)
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:  $3^{1/2}5^{1/2}7^{1/2}67^{1/2}\approx 83.87490685538792$
Ramified primes:   \(3\), \(5\), \(7\), \(67\) Copy content Toggle raw display
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{3}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{4}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{5}$, $\frac{1}{16}a^{12}-\frac{1}{8}a^{10}+\frac{1}{16}a^{9}-\frac{1}{4}a^{7}+\frac{3}{16}a^{6}-\frac{1}{4}a^{5}-\frac{1}{8}a^{4}+\frac{3}{16}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{16}a^{13}-\frac{1}{8}a^{11}+\frac{1}{16}a^{10}+\frac{3}{16}a^{7}-\frac{1}{4}a^{6}-\frac{1}{8}a^{5}+\frac{3}{16}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{16}a^{14}+\frac{1}{16}a^{11}-\frac{1}{8}a^{9}-\frac{1}{16}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{3}{16}a^{5}-\frac{1}{4}a^{4}+\frac{3}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{16}a^{15}-\frac{1}{8}a^{9}-\frac{7}{16}a^{3}-\frac{1}{2}$, $\frac{1}{224}a^{16}+\frac{1}{56}a^{15}-\frac{3}{112}a^{14}+\frac{1}{56}a^{13}-\frac{1}{112}a^{12}+\frac{9}{112}a^{11}+\frac{13}{112}a^{10}+\frac{13}{112}a^{9}+\frac{3}{112}a^{8}+\frac{1}{56}a^{7}+\frac{17}{112}a^{6}-\frac{9}{112}a^{5}-\frac{11}{224}a^{4}+\frac{41}{112}a^{3}+\frac{2}{7}a^{2}+\frac{13}{28}a-\frac{5}{14}$, $\frac{1}{448}a^{17}-\frac{1}{56}a^{15}+\frac{5}{224}a^{13}+\frac{3}{112}a^{12}+\frac{19}{224}a^{11}-\frac{11}{224}a^{10}-\frac{1}{16}a^{9}-\frac{3}{28}a^{8}-\frac{33}{224}a^{7}-\frac{3}{16}a^{6}-\frac{79}{448}a^{5}-\frac{7}{32}a^{4}-\frac{45}{112}a^{3}+\frac{23}{56}a^{2}-\frac{3}{28}a+\frac{3}{14}$, $\frac{1}{1792}a^{18}-\frac{1}{896}a^{17}+\frac{3}{112}a^{15}-\frac{5}{896}a^{14}-\frac{1}{56}a^{13}+\frac{27}{896}a^{12}+\frac{93}{896}a^{11}-\frac{1}{32}a^{10}-\frac{1}{224}a^{9}-\frac{31}{896}a^{8}+\frac{3}{224}a^{7}-\frac{31}{1792}a^{6}+\frac{1}{16}a^{5}-\frac{1}{14}a^{4}+\frac{13}{56}a^{3}+\frac{3}{7}a^{2}-\frac{3}{7}a-\frac{13}{28}$, $\frac{1}{3584}a^{19}-\frac{1}{896}a^{17}-\frac{53}{1792}a^{15}+\frac{3}{896}a^{14}-\frac{45}{1792}a^{13}-\frac{29}{1792}a^{12}-\frac{25}{896}a^{11}+\frac{11}{448}a^{10}+\frac{9}{1792}a^{9}+\frac{71}{896}a^{8}+\frac{23}{512}a^{7}+\frac{15}{256}a^{6}-\frac{13}{56}a^{5}-\frac{5}{56}a^{4}+\frac{11}{112}a^{3}-\frac{11}{28}a^{2}+\frac{11}{56}a-\frac{11}{28}$, $\frac{1}{7168}a^{20}-\frac{1}{896}a^{17}-\frac{5}{3584}a^{16}-\frac{3}{256}a^{15}+\frac{95}{3584}a^{14}-\frac{13}{3584}a^{13}+\frac{37}{1792}a^{12}+\frac{5}{112}a^{11}+\frac{361}{3584}a^{10}+\frac{183}{1792}a^{9}-\frac{407}{7168}a^{8}-\frac{439}{3584}a^{7}-\frac{375}{1792}a^{6}-\frac{9}{56}a^{5}+\frac{11}{224}a^{4}+\frac{1}{112}a^{3}+\frac{15}{112}a^{2}-\frac{27}{56}a+\frac{13}{28}$, $\frac{1}{57344}a^{21}+\frac{1}{28672}a^{20}-\frac{1}{7168}a^{19}-\frac{1}{7168}a^{18}-\frac{13}{28672}a^{17}+\frac{11}{7168}a^{16}-\frac{845}{28672}a^{15}-\frac{895}{28672}a^{14}-\frac{89}{3584}a^{13}-\frac{73}{7168}a^{12}+\frac{3545}{28672}a^{11}+\frac{7}{64}a^{10}-\frac{4847}{57344}a^{9}+\frac{801}{14336}a^{8}-\frac{3}{28}a^{7}+\frac{535}{7168}a^{6}-\frac{207}{896}a^{5}-\frac{71}{448}a^{4}-\frac{349}{896}a^{3}+\frac{11}{32}a^{2}+\frac{5}{56}a+\frac{13}{112}$, $\frac{1}{814094057684992}a^{22}+\frac{1775510551}{407047028842496}a^{21}-\frac{28709617}{1496496429568}a^{20}+\frac{14074926959}{101761757210624}a^{19}-\frac{59663354237}{407047028842496}a^{18}-\frac{2754381617}{25440439302656}a^{17}-\frac{466438113677}{407047028842496}a^{16}+\frac{6967917853765}{407047028842496}a^{15}-\frac{2592536834523}{101761757210624}a^{14}-\frac{692878779149}{101761757210624}a^{13}+\frac{8523465174633}{407047028842496}a^{12}+\frac{11287331065299}{101761757210624}a^{11}-\frac{31261624291791}{814094057684992}a^{10}+\frac{5052433972815}{50880878605312}a^{9}+\frac{846345967209}{12720219651328}a^{8}-\frac{14654464707209}{101761757210624}a^{7}+\frac{556664373827}{25440439302656}a^{6}+\frac{687058590605}{6360109825664}a^{5}+\frac{2318466116635}{12720219651328}a^{4}+\frac{63398044695}{1590027456416}a^{3}+\frac{20084167541}{46765513424}a^{2}-\frac{369109811323}{1590027456416}a+\frac{28393347435}{397506864104}$, $\frac{1}{13\!\cdots\!76}a^{23}+\frac{1095}{65\!\cdots\!88}a^{22}-\frac{2612549950383}{47\!\cdots\!92}a^{21}-\frac{4064650331143}{82\!\cdots\!36}a^{20}+\frac{887004027488659}{65\!\cdots\!88}a^{19}-\frac{20762738067931}{20\!\cdots\!84}a^{18}-\frac{48\!\cdots\!29}{65\!\cdots\!88}a^{17}-\frac{219954126765931}{65\!\cdots\!88}a^{16}+\frac{12\!\cdots\!49}{82\!\cdots\!36}a^{15}+\frac{24\!\cdots\!33}{82\!\cdots\!36}a^{14}+\frac{17\!\cdots\!77}{65\!\cdots\!88}a^{13}+\frac{30\!\cdots\!23}{16\!\cdots\!72}a^{12}-\frac{95\!\cdots\!11}{13\!\cdots\!76}a^{11}+\frac{37\!\cdots\!41}{82\!\cdots\!36}a^{10}-\frac{98\!\cdots\!37}{32\!\cdots\!44}a^{9}-\frac{67\!\cdots\!11}{16\!\cdots\!72}a^{8}+\frac{14\!\cdots\!07}{58\!\cdots\!24}a^{7}+\frac{17\!\cdots\!25}{41\!\cdots\!68}a^{6}-\frac{58\!\cdots\!59}{29\!\cdots\!12}a^{5}-\frac{253835586653695}{32\!\cdots\!56}a^{4}-\frac{46\!\cdots\!15}{51\!\cdots\!96}a^{3}+\frac{27585818625221}{67871937176512}a^{2}-\frac{513959437404725}{64\!\cdots\!12}a-\frac{10\!\cdots\!73}{64\!\cdots\!12}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

$C_{3}\times C_{33}$, which has order $99$ (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:   \( -\frac{274458456837}{5329993389408256} a^{23} + \frac{449324546867}{1332498347352064} a^{22} - \frac{664498192911}{666249173676032} a^{21} + \frac{117548945007}{41640573354752} a^{20} - \frac{14766692509127}{2664996694704128} a^{19} + \frac{12619805416027}{1332498347352064} a^{18} - \frac{53647066266203}{2664996694704128} a^{17} + \frac{117023891769581}{2664996694704128} a^{16} - \frac{165350004194675}{1332498347352064} a^{15} + \frac{112998011645907}{333124586838016} a^{14} - \frac{2486612789901141}{2664996694704128} a^{13} + \frac{2236470968295595}{1332498347352064} a^{12} - \frac{13781805248793133}{5329993389408256} a^{11} + \frac{6588815859659377}{2664996694704128} a^{10} + \frac{2247837502747731}{1332498347352064} a^{9} + \frac{237634144141339}{666249173676032} a^{8} + \frac{180609337041665}{333124586838016} a^{7} + \frac{28521921620527}{166562293419008} a^{6} + \frac{103495845920345}{83281146709504} a^{5} + \frac{20455726720939}{41640573354752} a^{4} + \frac{21574760126641}{20820286677376} a^{3} + \frac{4677368391185}{10410143338688} a^{2} + \frac{106653275331}{306180686432} a + \frac{2980511871181}{2602535834672} \)  (order $6$) Copy content Toggle raw display
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:   $\frac{68581169275159}{82\!\cdots\!36}a^{23}-\frac{17\!\cdots\!05}{32\!\cdots\!44}a^{22}+\frac{53\!\cdots\!55}{32\!\cdots\!44}a^{21}-\frac{75\!\cdots\!23}{16\!\cdots\!72}a^{20}+\frac{36\!\cdots\!21}{41\!\cdots\!68}a^{19}-\frac{25\!\cdots\!71}{16\!\cdots\!72}a^{18}+\frac{53\!\cdots\!63}{16\!\cdots\!72}a^{17}-\frac{68\!\cdots\!71}{96\!\cdots\!16}a^{16}+\frac{97\!\cdots\!55}{48\!\cdots\!08}a^{15}-\frac{90\!\cdots\!35}{16\!\cdots\!72}a^{14}+\frac{77\!\cdots\!63}{51\!\cdots\!96}a^{13}-\frac{44\!\cdots\!81}{16\!\cdots\!72}a^{12}+\frac{68\!\cdots\!15}{16\!\cdots\!72}a^{11}-\frac{13\!\cdots\!57}{32\!\cdots\!44}a^{10}-\frac{89\!\cdots\!71}{32\!\cdots\!44}a^{9}-\frac{67\!\cdots\!75}{11\!\cdots\!48}a^{8}-\frac{51\!\cdots\!39}{58\!\cdots\!24}a^{7}-\frac{11\!\cdots\!29}{41\!\cdots\!68}a^{6}-\frac{10\!\cdots\!61}{51\!\cdots\!96}a^{5}-\frac{343622065060991}{432327129241984}a^{4}-\frac{86\!\cdots\!21}{51\!\cdots\!96}a^{3}-\frac{93\!\cdots\!73}{12\!\cdots\!24}a^{2}-\frac{36\!\cdots\!61}{64\!\cdots\!12}a-\frac{15\!\cdots\!83}{64\!\cdots\!12}$, $\frac{27\!\cdots\!61}{13\!\cdots\!76}a^{23}-\frac{50075977276239}{23\!\cdots\!96}a^{22}+\frac{857158566664737}{82\!\cdots\!36}a^{21}-\frac{41050134644373}{29\!\cdots\!12}a^{20}-\frac{821125035878465}{65\!\cdots\!88}a^{19}-\frac{21\!\cdots\!93}{32\!\cdots\!44}a^{18}+\frac{20\!\cdots\!63}{65\!\cdots\!88}a^{17}-\frac{31\!\cdots\!89}{65\!\cdots\!88}a^{16}+\frac{21\!\cdots\!85}{32\!\cdots\!44}a^{15}-\frac{31\!\cdots\!21}{82\!\cdots\!36}a^{14}+\frac{17\!\cdots\!65}{65\!\cdots\!88}a^{13}-\frac{32\!\cdots\!69}{32\!\cdots\!44}a^{12}+\frac{73\!\cdots\!81}{13\!\cdots\!76}a^{11}+\frac{41\!\cdots\!37}{65\!\cdots\!88}a^{10}+\frac{18\!\cdots\!89}{32\!\cdots\!44}a^{9}+\frac{40\!\cdots\!13}{16\!\cdots\!72}a^{8}+\frac{15\!\cdots\!41}{82\!\cdots\!36}a^{7}-\frac{24\!\cdots\!95}{41\!\cdots\!68}a^{6}+\frac{48\!\cdots\!91}{20\!\cdots\!84}a^{5}+\frac{27\!\cdots\!13}{14\!\cdots\!56}a^{4}+\frac{46\!\cdots\!35}{51\!\cdots\!96}a^{3}+\frac{53\!\cdots\!77}{36\!\cdots\!64}a^{2}+\frac{853310026060039}{756572476173472}a+\frac{869037789912149}{918695149639216}$, $\frac{804256469724767}{65\!\cdots\!88}a^{23}+\frac{20\!\cdots\!13}{65\!\cdots\!88}a^{22}-\frac{627865516166451}{16\!\cdots\!72}a^{21}+\frac{636386313060897}{23\!\cdots\!96}a^{20}-\frac{24\!\cdots\!11}{32\!\cdots\!44}a^{19}+\frac{17\!\cdots\!17}{32\!\cdots\!44}a^{18}-\frac{78\!\cdots\!53}{32\!\cdots\!44}a^{17}+\frac{473955597970223}{51\!\cdots\!96}a^{16}-\frac{52\!\cdots\!25}{47\!\cdots\!92}a^{15}+\frac{15\!\cdots\!37}{16\!\cdots\!72}a^{14}-\frac{49\!\cdots\!59}{47\!\cdots\!92}a^{13}+\frac{65\!\cdots\!47}{32\!\cdots\!44}a^{12}-\frac{17\!\cdots\!85}{65\!\cdots\!88}a^{11}+\frac{55\!\cdots\!75}{65\!\cdots\!88}a^{10}+\frac{35\!\cdots\!57}{32\!\cdots\!44}a^{9}+\frac{42\!\cdots\!07}{10\!\cdots\!92}a^{8}+\frac{64\!\cdots\!13}{82\!\cdots\!36}a^{7}+\frac{86\!\cdots\!33}{41\!\cdots\!68}a^{6}+\frac{17\!\cdots\!97}{10\!\cdots\!92}a^{5}+\frac{12\!\cdots\!45}{10\!\cdots\!92}a^{4}+\frac{23\!\cdots\!99}{51\!\cdots\!96}a^{3}+\frac{57\!\cdots\!51}{32\!\cdots\!56}a^{2}+\frac{23\!\cdots\!63}{756572476173472}a+\frac{20\!\cdots\!47}{64\!\cdots\!12}$, $\frac{248033333300663}{16\!\cdots\!72}a^{23}-\frac{33381367084035}{10\!\cdots\!92}a^{22}+\frac{40\!\cdots\!85}{32\!\cdots\!44}a^{21}-\frac{41\!\cdots\!71}{16\!\cdots\!72}a^{20}+\frac{387333606242709}{11\!\cdots\!48}a^{19}-\frac{948471205944233}{10\!\cdots\!92}a^{18}+\frac{19\!\cdots\!45}{16\!\cdots\!72}a^{17}-\frac{41\!\cdots\!21}{82\!\cdots\!36}a^{16}+\frac{17\!\cdots\!27}{16\!\cdots\!72}a^{15}-\frac{67\!\cdots\!87}{16\!\cdots\!72}a^{14}+\frac{57\!\cdots\!51}{82\!\cdots\!36}a^{13}-\frac{54\!\cdots\!61}{51\!\cdots\!96}a^{12}+\frac{71\!\cdots\!45}{41\!\cdots\!68}a^{11}+\frac{22\!\cdots\!19}{82\!\cdots\!36}a^{10}+\frac{26\!\cdots\!05}{32\!\cdots\!44}a^{9}+\frac{37\!\cdots\!11}{82\!\cdots\!36}a^{8}-\frac{53\!\cdots\!17}{20\!\cdots\!84}a^{7}-\frac{12\!\cdots\!85}{41\!\cdots\!68}a^{6}+\frac{199047940857847}{73\!\cdots\!28}a^{5}-\frac{23\!\cdots\!63}{25\!\cdots\!48}a^{4}+\frac{12\!\cdots\!07}{51\!\cdots\!96}a^{3}-\frac{32340403228719}{18\!\cdots\!32}a^{2}+\frac{28511700411883}{47285779760842}a-\frac{348102126466727}{64\!\cdots\!12}$, $\frac{17\!\cdots\!97}{32\!\cdots\!44}a^{23}-\frac{12\!\cdots\!35}{65\!\cdots\!88}a^{22}+\frac{27\!\cdots\!45}{47\!\cdots\!92}a^{21}-\frac{825631085239091}{58\!\cdots\!24}a^{20}+\frac{36\!\cdots\!37}{16\!\cdots\!72}a^{19}-\frac{15\!\cdots\!77}{32\!\cdots\!44}a^{18}+\frac{14\!\cdots\!47}{16\!\cdots\!72}a^{17}-\frac{80\!\cdots\!27}{32\!\cdots\!44}a^{16}+\frac{20\!\cdots\!77}{32\!\cdots\!44}a^{15}-\frac{23\!\cdots\!19}{11\!\cdots\!48}a^{14}+\frac{71\!\cdots\!31}{16\!\cdots\!72}a^{13}-\frac{22\!\cdots\!91}{32\!\cdots\!44}a^{12}+\frac{34\!\cdots\!97}{32\!\cdots\!44}a^{11}+\frac{11\!\cdots\!79}{94\!\cdots\!84}a^{10}-\frac{27\!\cdots\!35}{51\!\cdots\!96}a^{9}+\frac{13\!\cdots\!59}{41\!\cdots\!68}a^{8}-\frac{42\!\cdots\!03}{11\!\cdots\!48}a^{7}+\frac{52\!\cdots\!73}{20\!\cdots\!84}a^{6}-\frac{41\!\cdots\!35}{25\!\cdots\!48}a^{5}+\frac{44\!\cdots\!27}{10\!\cdots\!92}a^{4}-\frac{470880092418711}{18\!\cdots\!32}a^{3}+\frac{242053004349251}{803858255934314}a^{2}+\frac{15\!\cdots\!37}{756572476173472}a-\frac{82\!\cdots\!47}{32\!\cdots\!56}$, $\frac{28\!\cdots\!59}{32\!\cdots\!44}a^{23}-\frac{12\!\cdots\!53}{65\!\cdots\!88}a^{22}+\frac{56\!\cdots\!81}{82\!\cdots\!36}a^{21}-\frac{23\!\cdots\!21}{16\!\cdots\!72}a^{20}+\frac{28\!\cdots\!15}{16\!\cdots\!72}a^{19}-\frac{16\!\cdots\!55}{32\!\cdots\!44}a^{18}+\frac{83\!\cdots\!09}{11\!\cdots\!48}a^{17}-\frac{96\!\cdots\!41}{32\!\cdots\!44}a^{16}+\frac{19\!\cdots\!77}{32\!\cdots\!44}a^{15}-\frac{38\!\cdots\!11}{16\!\cdots\!72}a^{14}+\frac{65\!\cdots\!69}{16\!\cdots\!72}a^{13}-\frac{18\!\cdots\!29}{32\!\cdots\!44}a^{12}+\frac{30\!\cdots\!49}{32\!\cdots\!44}a^{11}+\frac{10\!\cdots\!63}{65\!\cdots\!88}a^{10}+\frac{21\!\cdots\!59}{32\!\cdots\!44}a^{9}+\frac{98\!\cdots\!07}{82\!\cdots\!36}a^{8}+\frac{16\!\cdots\!77}{82\!\cdots\!36}a^{7}+\frac{22\!\cdots\!43}{41\!\cdots\!68}a^{6}+\frac{16\!\cdots\!13}{51\!\cdots\!96}a^{5}+\frac{25\!\cdots\!37}{10\!\cdots\!92}a^{4}+\frac{27\!\cdots\!69}{51\!\cdots\!96}a^{3}+\frac{35\!\cdots\!85}{12\!\cdots\!24}a^{2}+\frac{38\!\cdots\!11}{756572476173472}a+\frac{81\!\cdots\!61}{64\!\cdots\!12}$, $\frac{23\!\cdots\!27}{13\!\cdots\!76}a^{23}-\frac{13\!\cdots\!85}{65\!\cdots\!88}a^{22}+\frac{45\!\cdots\!49}{82\!\cdots\!36}a^{21}-\frac{39\!\cdots\!15}{23\!\cdots\!96}a^{20}+\frac{22\!\cdots\!41}{65\!\cdots\!88}a^{19}-\frac{82\!\cdots\!45}{16\!\cdots\!72}a^{18}+\frac{77\!\cdots\!61}{65\!\cdots\!88}a^{17}-\frac{14\!\cdots\!33}{65\!\cdots\!88}a^{16}+\frac{84\!\cdots\!83}{11\!\cdots\!48}a^{15}-\frac{29\!\cdots\!69}{16\!\cdots\!72}a^{14}+\frac{37\!\cdots\!79}{65\!\cdots\!88}a^{13}-\frac{81\!\cdots\!25}{82\!\cdots\!36}a^{12}+\frac{19\!\cdots\!39}{13\!\cdots\!76}a^{11}-\frac{55\!\cdots\!67}{32\!\cdots\!44}a^{10}-\frac{39\!\cdots\!05}{16\!\cdots\!72}a^{9}-\frac{28\!\cdots\!27}{16\!\cdots\!72}a^{8}-\frac{33\!\cdots\!63}{20\!\cdots\!84}a^{7}-\frac{10\!\cdots\!37}{12\!\cdots\!52}a^{6}-\frac{57\!\cdots\!65}{29\!\cdots\!12}a^{5}+\frac{60\!\cdots\!77}{51\!\cdots\!96}a^{4}-\frac{37\!\cdots\!79}{36\!\cdots\!64}a^{3}-\frac{90\!\cdots\!49}{25\!\cdots\!48}a^{2}-\frac{35\!\cdots\!93}{16\!\cdots\!28}a-\frac{39\!\cdots\!47}{459347574819608}$, $\frac{323113789647099}{38\!\cdots\!64}a^{23}-\frac{742825812775499}{38\!\cdots\!64}a^{22}+\frac{13\!\cdots\!95}{19\!\cdots\!32}a^{21}-\frac{345018603833503}{24\!\cdots\!04}a^{20}+\frac{36\!\cdots\!13}{19\!\cdots\!32}a^{19}-\frac{27084467520143}{548675790086144}a^{18}+\frac{13\!\cdots\!37}{19\!\cdots\!32}a^{17}-\frac{983061429345075}{34\!\cdots\!72}a^{16}+\frac{11\!\cdots\!35}{19\!\cdots\!32}a^{15}-\frac{11\!\cdots\!11}{48\!\cdots\!08}a^{14}+\frac{78\!\cdots\!87}{19\!\cdots\!32}a^{13}-\frac{11\!\cdots\!57}{19\!\cdots\!32}a^{12}+\frac{35\!\cdots\!71}{38\!\cdots\!64}a^{11}+\frac{56\!\cdots\!67}{38\!\cdots\!64}a^{10}+\frac{22\!\cdots\!33}{96\!\cdots\!16}a^{9}+\frac{91\!\cdots\!27}{48\!\cdots\!08}a^{8}+\frac{87\!\cdots\!05}{48\!\cdots\!08}a^{7}+\frac{14\!\cdots\!59}{30\!\cdots\!88}a^{6}+\frac{57\!\cdots\!75}{60\!\cdots\!76}a^{5}-\frac{16\!\cdots\!65}{864654258483968}a^{4}+\frac{11\!\cdots\!77}{15\!\cdots\!44}a^{3}+\frac{24\!\cdots\!23}{756572476173472}a^{2}+\frac{18\!\cdots\!11}{756572476173472}a+\frac{256047901213675}{94571559521684}$, $\frac{763437269576245}{18\!\cdots\!68}a^{23}-\frac{11\!\cdots\!09}{16\!\cdots\!72}a^{22}+\frac{50\!\cdots\!41}{16\!\cdots\!72}a^{21}-\frac{46\!\cdots\!13}{82\!\cdots\!36}a^{20}+\frac{47\!\cdots\!33}{65\!\cdots\!88}a^{19}-\frac{44\!\cdots\!43}{19\!\cdots\!32}a^{18}+\frac{25\!\cdots\!67}{94\!\cdots\!84}a^{17}-\frac{89\!\cdots\!99}{65\!\cdots\!88}a^{16}+\frac{80\!\cdots\!51}{32\!\cdots\!44}a^{15}-\frac{39\!\cdots\!79}{36\!\cdots\!64}a^{14}+\frac{10\!\cdots\!43}{65\!\cdots\!88}a^{13}-\frac{81\!\cdots\!83}{32\!\cdots\!44}a^{12}+\frac{52\!\cdots\!47}{13\!\cdots\!76}a^{11}+\frac{70\!\cdots\!49}{94\!\cdots\!84}a^{10}+\frac{26\!\cdots\!77}{32\!\cdots\!44}a^{9}+\frac{19\!\cdots\!99}{16\!\cdots\!72}a^{8}+\frac{12\!\cdots\!61}{11\!\cdots\!48}a^{7}+\frac{37\!\cdots\!61}{41\!\cdots\!68}a^{6}+\frac{86\!\cdots\!53}{20\!\cdots\!84}a^{5}+\frac{11\!\cdots\!49}{10\!\cdots\!92}a^{4}-\frac{98\!\cdots\!77}{51\!\cdots\!96}a^{3}-\frac{64\!\cdots\!95}{25\!\cdots\!48}a^{2}-\frac{21\!\cdots\!31}{12\!\cdots\!24}a-\frac{14\!\cdots\!75}{918695149639216}$, $\frac{21\!\cdots\!75}{32\!\cdots\!44}a^{23}-\frac{26\!\cdots\!73}{16\!\cdots\!72}a^{22}+\frac{17\!\cdots\!67}{32\!\cdots\!44}a^{21}-\frac{18\!\cdots\!89}{16\!\cdots\!72}a^{20}+\frac{21\!\cdots\!45}{16\!\cdots\!72}a^{19}-\frac{18\!\cdots\!11}{51\!\cdots\!96}a^{18}+\frac{46\!\cdots\!81}{82\!\cdots\!36}a^{17}-\frac{36\!\cdots\!97}{16\!\cdots\!72}a^{16}+\frac{80\!\cdots\!01}{16\!\cdots\!72}a^{15}-\frac{29\!\cdots\!21}{16\!\cdots\!72}a^{14}+\frac{74\!\cdots\!21}{23\!\cdots\!96}a^{13}-\frac{16\!\cdots\!93}{41\!\cdots\!68}a^{12}+\frac{20\!\cdots\!29}{32\!\cdots\!44}a^{11}+\frac{97\!\cdots\!53}{82\!\cdots\!36}a^{10}-\frac{14\!\cdots\!87}{47\!\cdots\!92}a^{9}+\frac{19\!\cdots\!67}{82\!\cdots\!36}a^{8}+\frac{55\!\cdots\!37}{10\!\cdots\!92}a^{7}+\frac{48\!\cdots\!61}{24\!\cdots\!04}a^{6}-\frac{14\!\cdots\!17}{25\!\cdots\!48}a^{5}+\frac{62\!\cdots\!01}{25\!\cdots\!48}a^{4}-\frac{14\!\cdots\!75}{51\!\cdots\!96}a^{3}+\frac{54\!\cdots\!91}{12\!\cdots\!24}a^{2}+\frac{15\!\cdots\!29}{32\!\cdots\!56}a+\frac{11\!\cdots\!55}{64\!\cdots\!12}$, $\frac{16\!\cdots\!57}{94\!\cdots\!84}a^{23}-\frac{39\!\cdots\!79}{82\!\cdots\!36}a^{22}+\frac{61\!\cdots\!09}{32\!\cdots\!44}a^{21}-\frac{74\!\cdots\!87}{16\!\cdots\!72}a^{20}+\frac{39\!\cdots\!71}{47\!\cdots\!92}a^{19}-\frac{33\!\cdots\!23}{16\!\cdots\!72}a^{18}+\frac{11\!\cdots\!79}{32\!\cdots\!44}a^{17}-\frac{18\!\cdots\!79}{19\!\cdots\!32}a^{16}+\frac{10\!\cdots\!99}{48\!\cdots\!08}a^{15}-\frac{11\!\cdots\!67}{16\!\cdots\!72}a^{14}+\frac{46\!\cdots\!87}{32\!\cdots\!44}a^{13}-\frac{45\!\cdots\!11}{16\!\cdots\!72}a^{12}+\frac{33\!\cdots\!87}{65\!\cdots\!88}a^{11}-\frac{85\!\cdots\!25}{32\!\cdots\!44}a^{10}+\frac{34\!\cdots\!53}{47\!\cdots\!92}a^{9}-\frac{57\!\cdots\!79}{58\!\cdots\!24}a^{8}+\frac{19\!\cdots\!11}{58\!\cdots\!24}a^{7}+\frac{13\!\cdots\!65}{41\!\cdots\!68}a^{6}+\frac{11\!\cdots\!87}{10\!\cdots\!92}a^{5}+\frac{26\!\cdots\!03}{30\!\cdots\!88}a^{4}+\frac{444130691529479}{135743874353024}a^{3}+\frac{66\!\cdots\!83}{32\!\cdots\!56}a^{2}-\frac{17\!\cdots\!01}{918695149639216}a+\frac{90\!\cdots\!79}{64\!\cdots\!12}$ Copy content Toggle raw display (assuming GRH)
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:  \( 2094070987.821792 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 2094070987.821792 \cdot 99}{6\cdot\sqrt{729239207533207895866388728730712890625}}\cr\approx \mathstrut & 4.84394004629176 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^24 - 2*x^23 + 8*x^22 - 16*x^21 + 22*x^20 - 64*x^19 + 86*x^18 - 358*x^17 + 696*x^16 - 2792*x^15 + 4514*x^14 - 7304*x^13 + 12417*x^12 + 14608*x^11 + 18056*x^10 + 22336*x^9 + 11136*x^8 + 11456*x^7 + 5504*x^6 + 8192*x^5 + 5632*x^4 + 8192*x^3 + 8192*x^2 + 4096*x + 4096)
 
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 - 2*x^23 + 8*x^22 - 16*x^21 + 22*x^20 - 64*x^19 + 86*x^18 - 358*x^17 + 696*x^16 - 2792*x^15 + 4514*x^14 - 7304*x^13 + 12417*x^12 + 14608*x^11 + 18056*x^10 + 22336*x^9 + 11136*x^8 + 11456*x^7 + 5504*x^6 + 8192*x^5 + 5632*x^4 + 8192*x^3 + 8192*x^2 + 4096*x + 4096, 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 - 2*x^23 + 8*x^22 - 16*x^21 + 22*x^20 - 64*x^19 + 86*x^18 - 358*x^17 + 696*x^16 - 2792*x^15 + 4514*x^14 - 7304*x^13 + 12417*x^12 + 14608*x^11 + 18056*x^10 + 22336*x^9 + 11136*x^8 + 11456*x^7 + 5504*x^6 + 8192*x^5 + 5632*x^4 + 8192*x^3 + 8192*x^2 + 4096*x + 4096);
 
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 - 2*x^23 + 8*x^22 - 16*x^21 + 22*x^20 - 64*x^19 + 86*x^18 - 358*x^17 + 696*x^16 - 2792*x^15 + 4514*x^14 - 7304*x^13 + 12417*x^12 + 14608*x^11 + 18056*x^10 + 22336*x^9 + 11136*x^8 + 11456*x^7 + 5504*x^6 + 8192*x^5 + 5632*x^4 + 8192*x^3 + 8192*x^2 + 4096*x + 4096);
 
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 $C_2^2\times D_6$

Intermediate fields

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{21}) \), 3.3.469.1, \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{-35})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{-7}, \sqrt{-15})\), \(\Q(\sqrt{-15}, \sqrt{21})\), \(\Q(\sqrt{5}, \sqrt{21})\), 6.6.27495125.1, 6.0.742368375.2, 6.0.192465875.3, 6.6.5196578625.1, 6.0.5938947.1, 6.0.1539727.2, 6.6.41572629.1, 8.0.121550625.1, 12.0.1728283481971641.1, 12.0.551110804200140625.1, 12.0.27004429405806890625.3, 12.0.37043113039515625.1, 12.0.27004429405806890625.1, 12.0.27004429405806890625.2, 12.12.27004429405806890625.1

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 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 ${\href{/padicField/2.2.0.1}{2} }^{12}$ R R R ${\href{/padicField/11.2.0.1}{2} }^{12}$ ${\href{/padicField/13.6.0.1}{6} }^{4}$ ${\href{/padicField/17.2.0.1}{2} }^{12}$ ${\href{/padicField/19.2.0.1}{2} }^{12}$ ${\href{/padicField/23.6.0.1}{6} }^{4}$ ${\href{/padicField/29.6.0.1}{6} }^{4}$ ${\href{/padicField/31.6.0.1}{6} }^{4}$ ${\href{/padicField/37.6.0.1}{6} }^{4}$ ${\href{/padicField/41.6.0.1}{6} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{12}$ ${\href{/padicField/47.2.0.1}{2} }^{12}$ ${\href{/padicField/53.2.0.1}{2} }^{12}$ ${\href{/padicField/59.2.0.1}{2} }^{12}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display 3.12.6.2$x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
3.12.6.2$x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
\(5\) Copy content Toggle raw display 5.12.6.1$x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
5.12.6.1$x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
\(7\) Copy content Toggle raw display 7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(67\) Copy content Toggle raw display 67.2.0.1$x^{2} + 63 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} + 63 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} + 63 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} + 63 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
67.4.2.1$x^{4} + 126 x^{3} + 4107 x^{2} + 8694 x + 270148$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.4.2.1$x^{4} + 126 x^{3} + 4107 x^{2} + 8694 x + 270148$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.4.2.1$x^{4} + 126 x^{3} + 4107 x^{2} + 8694 x + 270148$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.4.2.1$x^{4} + 126 x^{3} + 4107 x^{2} + 8694 x + 270148$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$