Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 39*x^22 + 648*x^20 - 5829*x^18 + 30363*x^16 - 94896*x^14 + 196451*x^12 - 293637*x^10 + 327912*x^8 - 264279*x^6 + 203193*x^4 - 165600*x^2 + 92416)
gp: K = bnfinit(y^24 - 39*y^22 + 648*y^20 - 5829*y^18 + 30363*y^16 - 94896*y^14 + 196451*y^12 - 293637*y^10 + 327912*y^8 - 264279*y^6 + 203193*y^4 - 165600*y^2 + 92416, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 39*x^22 + 648*x^20 - 5829*x^18 + 30363*x^16 - 94896*x^14 + 196451*x^12 - 293637*x^10 + 327912*x^8 - 264279*x^6 + 203193*x^4 - 165600*x^2 + 92416);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 39*x^22 + 648*x^20 - 5829*x^18 + 30363*x^16 - 94896*x^14 + 196451*x^12 - 293637*x^10 + 327912*x^8 - 264279*x^6 + 203193*x^4 - 165600*x^2 + 92416)
\( x^{24} - 39 x^{22} + 648 x^{20} - 5829 x^{18} + 30363 x^{16} - 94896 x^{14} + 196451 x^{12} + \cdots + 92416 \)
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: |
\(7338001027960597407453351936000000000000\)
\(\medspace = 2^{24}\cdot 3^{28}\cdot 5^{12}\cdot 23^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(45.82\) | 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: | $2\cdot 3^{7/6}5^{1/2}23^{1/2}\approx 77.27168447156222$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(23\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{4}a^{7}+\frac{1}{4}a$, $\frac{1}{4}a^{8}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{9}+\frac{1}{4}a^{3}$, $\frac{1}{4}a^{10}+\frac{1}{4}a^{4}$, $\frac{1}{4}a^{11}+\frac{1}{4}a^{5}$, $\frac{1}{4}a^{12}+\frac{1}{4}a^{6}$, $\frac{1}{4}a^{13}-\frac{1}{4}a$, $\frac{1}{32}a^{14}-\frac{1}{16}a^{8}-\frac{1}{2}a^{6}-\frac{3}{32}a^{2}-\frac{1}{2}$, $\frac{1}{32}a^{15}-\frac{1}{16}a^{9}-\frac{3}{32}a^{3}$, $\frac{1}{32}a^{16}-\frac{1}{16}a^{10}-\frac{3}{32}a^{4}$, $\frac{1}{32}a^{17}-\frac{1}{16}a^{11}-\frac{3}{32}a^{5}$, $\frac{1}{32}a^{18}-\frac{1}{16}a^{12}-\frac{3}{32}a^{6}$, $\frac{1}{32}a^{19}-\frac{1}{16}a^{13}-\frac{3}{32}a^{7}$, $\frac{1}{1792}a^{20}+\frac{3}{224}a^{18}+\frac{11}{1792}a^{14}-\frac{3}{28}a^{12}-\frac{5}{56}a^{10}+\frac{99}{1792}a^{8}-\frac{89}{224}a^{6}+\frac{1}{8}a^{4}+\frac{89}{1792}a^{2}+\frac{9}{112}$, $\frac{1}{136192}a^{21}+\frac{3}{17024}a^{19}-\frac{1}{76}a^{17}-\frac{773}{136192}a^{15}+\frac{23}{1064}a^{13}-\frac{257}{4256}a^{11}+\frac{11523}{136192}a^{9}+\frac{79}{17024}a^{7}-\frac{219}{608}a^{5}-\frac{21751}{136192}a^{3}-\frac{327}{8512}a$, $\frac{1}{30\!\cdots\!44}a^{22}-\frac{10\!\cdots\!33}{38\!\cdots\!68}a^{20}-\frac{54\!\cdots\!71}{96\!\cdots\!92}a^{18}-\frac{13\!\cdots\!45}{30\!\cdots\!44}a^{16}-\frac{11\!\cdots\!85}{96\!\cdots\!92}a^{14}-\frac{12\!\cdots\!87}{96\!\cdots\!92}a^{12}-\frac{45\!\cdots\!93}{30\!\cdots\!44}a^{10}+\frac{20\!\cdots\!39}{38\!\cdots\!68}a^{8}+\frac{28\!\cdots\!62}{30\!\cdots\!31}a^{6}-\frac{15\!\cdots\!01}{49\!\cdots\!76}a^{4}-\frac{78\!\cdots\!25}{19\!\cdots\!84}a^{2}+\frac{50\!\cdots\!77}{31\!\cdots\!98}$, $\frac{1}{30\!\cdots\!44}a^{23}-\frac{20\!\cdots\!03}{77\!\cdots\!36}a^{21}+\frac{35\!\cdots\!09}{48\!\cdots\!96}a^{19}-\frac{28\!\cdots\!49}{30\!\cdots\!44}a^{17}+\frac{15\!\cdots\!57}{77\!\cdots\!36}a^{15}+\frac{15\!\cdots\!53}{96\!\cdots\!92}a^{13}+\frac{38\!\cdots\!67}{30\!\cdots\!44}a^{11}-\frac{67\!\cdots\!69}{77\!\cdots\!36}a^{9}-\frac{71\!\cdots\!87}{96\!\cdots\!92}a^{7}-\frac{21\!\cdots\!01}{49\!\cdots\!76}a^{5}+\frac{33\!\cdots\!15}{77\!\cdots\!36}a^{3}+\frac{13\!\cdots\!47}{48\!\cdots\!96}a$
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
$C_{2}\times C_{90}$, which has order $180$ (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: |
\( \frac{864517039411035}{146267115883231837696} a^{23} - \frac{60159575870752349}{292534231766463675392} a^{21} + \frac{106290153800777601}{36566778970807959424} a^{19} - \frac{2936785930054899911}{146267115883231837696} a^{17} + \frac{953122452608806395}{15396538514024403968} a^{15} - \frac{154256040130002599}{4570847371350994928} a^{13} - \frac{1154417663020474821}{7698269257012201984} a^{11} + \frac{133772016713340543497}{292534231766463675392} a^{9} - \frac{33409727867553577971}{36566778970807959424} a^{7} + \frac{310100401791987721}{236295825336400384} a^{5} - \frac{11535685646696633047}{15396538514024403968} a^{3} + \frac{17386431391182374443}{18283389485403979712} a \)
(order $12$)
| 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: |
$\frac{204347744635511}{26\!\cdots\!04}a^{22}-\frac{18\!\cdots\!59}{65\!\cdots\!76}a^{20}+\frac{35\!\cdots\!95}{82\!\cdots\!72}a^{18}-\frac{86\!\cdots\!43}{26\!\cdots\!04}a^{16}+\frac{84\!\cdots\!57}{65\!\cdots\!76}a^{14}-\frac{25\!\cdots\!13}{11\!\cdots\!96}a^{12}+\frac{39\!\cdots\!01}{26\!\cdots\!04}a^{10}-\frac{15\!\cdots\!43}{93\!\cdots\!68}a^{8}+\frac{23\!\cdots\!89}{51\!\cdots\!42}a^{6}-\frac{18\!\cdots\!13}{26\!\cdots\!04}a^{4}+\frac{88\!\cdots\!79}{65\!\cdots\!76}a^{2}-\frac{29\!\cdots\!25}{41\!\cdots\!36}$, $\frac{15\!\cdots\!85}{14\!\cdots\!96}a^{23}-\frac{11\!\cdots\!83}{29\!\cdots\!92}a^{21}+\frac{24\!\cdots\!71}{36\!\cdots\!24}a^{19}-\frac{86\!\cdots\!57}{14\!\cdots\!96}a^{17}+\frac{85\!\cdots\!39}{29\!\cdots\!92}a^{15}-\frac{37\!\cdots\!07}{45\!\cdots\!28}a^{13}+\frac{20\!\cdots\!67}{14\!\cdots\!96}a^{11}-\frac{49\!\cdots\!61}{29\!\cdots\!92}a^{9}+\frac{62\!\cdots\!03}{36\!\cdots\!24}a^{7}-\frac{29\!\cdots\!45}{23\!\cdots\!84}a^{5}+\frac{42\!\cdots\!77}{29\!\cdots\!92}a^{3}-\frac{29\!\cdots\!35}{18\!\cdots\!12}a+1$, $\frac{362565293828901}{18\!\cdots\!48}a^{23}-\frac{76214801900863}{65\!\cdots\!76}a^{22}-\frac{19\!\cdots\!71}{25\!\cdots\!72}a^{21}+\frac{701103014166395}{16\!\cdots\!44}a^{20}+\frac{37\!\cdots\!55}{32\!\cdots\!84}a^{19}-\frac{26\!\cdots\!45}{41\!\cdots\!36}a^{18}-\frac{18\!\cdots\!65}{18\!\cdots\!48}a^{17}+\frac{33\!\cdots\!19}{65\!\cdots\!76}a^{16}+\frac{11\!\cdots\!59}{25\!\cdots\!72}a^{15}-\frac{35\!\cdots\!29}{16\!\cdots\!44}a^{14}-\frac{25\!\cdots\!75}{20\!\cdots\!24}a^{13}+\frac{15\!\cdots\!77}{36\!\cdots\!53}a^{12}+\frac{30\!\cdots\!93}{12\!\cdots\!36}a^{11}-\frac{29\!\cdots\!89}{65\!\cdots\!76}a^{10}-\frac{82\!\cdots\!33}{25\!\cdots\!72}a^{9}+\frac{45\!\cdots\!23}{23\!\cdots\!92}a^{8}+\frac{98\!\cdots\!59}{32\!\cdots\!84}a^{7}+\frac{21\!\cdots\!53}{41\!\cdots\!36}a^{6}-\frac{27\!\cdots\!55}{15\!\cdots\!68}a^{5}-\frac{83\!\cdots\!75}{65\!\cdots\!76}a^{4}+\frac{42\!\cdots\!77}{25\!\cdots\!72}a^{3}-\frac{27\!\cdots\!15}{16\!\cdots\!44}a^{2}-\frac{18\!\cdots\!67}{16\!\cdots\!92}a-\frac{13\!\cdots\!23}{10\!\cdots\!84}$, $\frac{18\!\cdots\!29}{44\!\cdots\!92}a^{23}+\frac{28\!\cdots\!57}{50\!\cdots\!96}a^{22}-\frac{24\!\cdots\!01}{15\!\cdots\!72}a^{21}-\frac{27\!\cdots\!01}{12\!\cdots\!24}a^{20}+\frac{48\!\cdots\!79}{19\!\cdots\!84}a^{19}+\frac{13\!\cdots\!81}{39\!\cdots\!32}a^{18}-\frac{93\!\cdots\!53}{44\!\cdots\!92}a^{17}-\frac{14\!\cdots\!37}{50\!\cdots\!96}a^{16}+\frac{15\!\cdots\!41}{15\!\cdots\!72}a^{15}+\frac{17\!\cdots\!31}{12\!\cdots\!24}a^{14}-\frac{24\!\cdots\!25}{96\!\cdots\!92}a^{13}-\frac{85\!\cdots\!21}{22\!\cdots\!04}a^{12}+\frac{13\!\cdots\!21}{30\!\cdots\!44}a^{11}+\frac{32\!\cdots\!55}{50\!\cdots\!96}a^{10}-\frac{78\!\cdots\!59}{15\!\cdots\!72}a^{9}-\frac{14\!\cdots\!01}{17\!\cdots\!32}a^{8}+\frac{84\!\cdots\!53}{19\!\cdots\!84}a^{7}+\frac{12\!\cdots\!83}{15\!\cdots\!28}a^{6}-\frac{18\!\cdots\!05}{71\!\cdots\!68}a^{5}-\frac{38\!\cdots\!73}{81\!\cdots\!84}a^{4}+\frac{56\!\cdots\!15}{15\!\cdots\!72}a^{3}+\frac{61\!\cdots\!49}{12\!\cdots\!24}a^{2}-\frac{16\!\cdots\!49}{96\!\cdots\!92}a-\frac{12\!\cdots\!69}{41\!\cdots\!56}$, $\frac{20\!\cdots\!37}{44\!\cdots\!92}a^{23}-\frac{10\!\cdots\!05}{25\!\cdots\!48}a^{22}-\frac{68\!\cdots\!29}{38\!\cdots\!68}a^{21}+\frac{41\!\cdots\!33}{25\!\cdots\!48}a^{20}+\frac{18\!\cdots\!71}{63\!\cdots\!96}a^{19}-\frac{85\!\cdots\!65}{31\!\cdots\!56}a^{18}-\frac{11\!\cdots\!61}{44\!\cdots\!92}a^{17}+\frac{60\!\cdots\!45}{25\!\cdots\!48}a^{16}+\frac{24\!\cdots\!87}{19\!\cdots\!84}a^{15}-\frac{30\!\cdots\!85}{25\!\cdots\!48}a^{14}-\frac{34\!\cdots\!15}{96\!\cdots\!92}a^{13}+\frac{40\!\cdots\!65}{11\!\cdots\!52}a^{12}+\frac{20\!\cdots\!93}{30\!\cdots\!44}a^{11}-\frac{16\!\cdots\!39}{25\!\cdots\!48}a^{10}-\frac{34\!\cdots\!69}{38\!\cdots\!68}a^{9}+\frac{31\!\cdots\!05}{35\!\cdots\!64}a^{8}+\frac{83\!\cdots\!09}{96\!\cdots\!92}a^{7}-\frac{26\!\cdots\!25}{31\!\cdots\!56}a^{6}-\frac{40\!\cdots\!93}{71\!\cdots\!68}a^{5}+\frac{22\!\cdots\!85}{40\!\cdots\!92}a^{4}+\frac{12\!\cdots\!97}{24\!\cdots\!48}a^{3}-\frac{10\!\cdots\!95}{25\!\cdots\!48}a^{2}-\frac{53\!\cdots\!25}{12\!\cdots\!24}a+\frac{35\!\cdots\!67}{82\!\cdots\!12}$, $\frac{21\!\cdots\!29}{19\!\cdots\!84}a^{23}-\frac{25\!\cdots\!09}{25\!\cdots\!48}a^{22}-\frac{13\!\cdots\!83}{30\!\cdots\!44}a^{21}+\frac{26\!\cdots\!27}{62\!\cdots\!12}a^{20}+\frac{28\!\cdots\!39}{38\!\cdots\!68}a^{19}-\frac{23\!\cdots\!97}{31\!\cdots\!56}a^{18}-\frac{13\!\cdots\!07}{19\!\cdots\!84}a^{17}+\frac{18\!\cdots\!09}{25\!\cdots\!48}a^{16}+\frac{11\!\cdots\!87}{30\!\cdots\!44}a^{15}-\frac{25\!\cdots\!87}{62\!\cdots\!12}a^{14}-\frac{58\!\cdots\!41}{48\!\cdots\!96}a^{13}+\frac{19\!\cdots\!23}{15\!\cdots\!28}a^{12}+\frac{54\!\cdots\!45}{19\!\cdots\!84}a^{11}-\frac{56\!\cdots\!71}{25\!\cdots\!48}a^{10}-\frac{77\!\cdots\!99}{16\!\cdots\!76}a^{9}+\frac{14\!\cdots\!29}{62\!\cdots\!12}a^{8}+\frac{21\!\cdots\!31}{38\!\cdots\!68}a^{7}-\frac{45\!\cdots\!89}{31\!\cdots\!56}a^{6}-\frac{13\!\cdots\!65}{31\!\cdots\!36}a^{5}+\frac{15\!\cdots\!21}{40\!\cdots\!92}a^{4}+\frac{15\!\cdots\!27}{44\!\cdots\!92}a^{3}+\frac{58\!\cdots\!61}{89\!\cdots\!16}a^{2}-\frac{39\!\cdots\!07}{19\!\cdots\!84}a-\frac{813640986322091}{20\!\cdots\!28}$, $\frac{34\!\cdots\!99}{30\!\cdots\!44}a^{22}-\frac{33\!\cdots\!93}{77\!\cdots\!36}a^{20}+\frac{68\!\cdots\!81}{96\!\cdots\!92}a^{18}-\frac{19\!\cdots\!43}{30\!\cdots\!44}a^{16}+\frac{34\!\cdots\!93}{11\!\cdots\!48}a^{14}-\frac{88\!\cdots\!27}{96\!\cdots\!92}a^{12}+\frac{78\!\cdots\!51}{44\!\cdots\!92}a^{10}-\frac{19\!\cdots\!95}{77\!\cdots\!36}a^{8}+\frac{13\!\cdots\!21}{48\!\cdots\!96}a^{6}-\frac{10\!\cdots\!03}{49\!\cdots\!76}a^{4}+\frac{13\!\cdots\!09}{77\!\cdots\!36}a^{2}-\frac{32\!\cdots\!69}{25\!\cdots\!84}$, $\frac{18\!\cdots\!77}{15\!\cdots\!72}a^{23}+\frac{21\!\cdots\!03}{11\!\cdots\!48}a^{22}-\frac{13\!\cdots\!07}{30\!\cdots\!44}a^{21}-\frac{34\!\cdots\!51}{48\!\cdots\!96}a^{20}+\frac{37\!\cdots\!41}{55\!\cdots\!24}a^{19}+\frac{52\!\cdots\!03}{48\!\cdots\!96}a^{18}-\frac{86\!\cdots\!97}{15\!\cdots\!72}a^{17}-\frac{97\!\cdots\!11}{11\!\cdots\!48}a^{16}+\frac{81\!\cdots\!31}{30\!\cdots\!44}a^{15}+\frac{38\!\cdots\!89}{96\!\cdots\!92}a^{14}-\frac{18\!\cdots\!85}{24\!\cdots\!48}a^{13}-\frac{12\!\cdots\!49}{12\!\cdots\!24}a^{12}+\frac{23\!\cdots\!03}{15\!\cdots\!72}a^{11}+\frac{16\!\cdots\!63}{77\!\cdots\!36}a^{10}-\frac{58\!\cdots\!49}{30\!\cdots\!44}a^{9}-\frac{63\!\cdots\!89}{24\!\cdots\!48}a^{8}+\frac{99\!\cdots\!57}{55\!\cdots\!24}a^{7}+\frac{12\!\cdots\!61}{48\!\cdots\!96}a^{6}-\frac{34\!\cdots\!77}{24\!\cdots\!88}a^{5}-\frac{37\!\cdots\!43}{17\!\cdots\!92}a^{4}+\frac{27\!\cdots\!09}{30\!\cdots\!44}a^{3}+\frac{15\!\cdots\!97}{96\!\cdots\!92}a^{2}-\frac{55\!\cdots\!93}{27\!\cdots\!12}a-\frac{21\!\cdots\!11}{31\!\cdots\!98}$, $\frac{13\!\cdots\!15}{15\!\cdots\!72}a^{23}+\frac{21\!\cdots\!75}{30\!\cdots\!44}a^{22}-\frac{10\!\cdots\!43}{30\!\cdots\!44}a^{21}-\frac{14\!\cdots\!21}{55\!\cdots\!24}a^{20}+\frac{20\!\cdots\!47}{38\!\cdots\!68}a^{19}+\frac{21\!\cdots\!83}{48\!\cdots\!96}a^{18}-\frac{67\!\cdots\!79}{15\!\cdots\!72}a^{17}-\frac{12\!\cdots\!59}{30\!\cdots\!44}a^{16}+\frac{31\!\cdots\!13}{16\!\cdots\!76}a^{15}+\frac{45\!\cdots\!29}{24\!\cdots\!48}a^{14}-\frac{31\!\cdots\!73}{69\!\cdots\!28}a^{13}-\frac{47\!\cdots\!71}{96\!\cdots\!92}a^{12}+\frac{42\!\cdots\!43}{81\!\cdots\!88}a^{11}+\frac{20\!\cdots\!81}{30\!\cdots\!44}a^{10}-\frac{12\!\cdots\!39}{44\!\cdots\!92}a^{9}-\frac{16\!\cdots\!91}{38\!\cdots\!68}a^{8}-\frac{49\!\cdots\!05}{38\!\cdots\!68}a^{7}-\frac{15\!\cdots\!33}{96\!\cdots\!92}a^{6}+\frac{92\!\cdots\!97}{24\!\cdots\!88}a^{5}+\frac{38\!\cdots\!93}{49\!\cdots\!76}a^{4}+\frac{11\!\cdots\!47}{16\!\cdots\!76}a^{3}-\frac{21\!\cdots\!05}{19\!\cdots\!84}a^{2}-\frac{23\!\cdots\!63}{19\!\cdots\!84}a-\frac{15\!\cdots\!33}{15\!\cdots\!49}$, $\frac{15\!\cdots\!75}{15\!\cdots\!72}a^{23}-\frac{25\!\cdots\!57}{77\!\cdots\!36}a^{21}+\frac{42\!\cdots\!95}{96\!\cdots\!92}a^{19}-\frac{40\!\cdots\!71}{15\!\cdots\!72}a^{17}+\frac{32\!\cdots\!13}{77\!\cdots\!36}a^{15}+\frac{15\!\cdots\!17}{69\!\cdots\!28}a^{13}-\frac{14\!\cdots\!79}{15\!\cdots\!72}a^{11}+\frac{16\!\cdots\!75}{11\!\cdots\!48}a^{9}-\frac{18\!\cdots\!35}{96\!\cdots\!92}a^{7}+\frac{46\!\cdots\!97}{24\!\cdots\!88}a^{5}+\frac{74\!\cdots\!59}{77\!\cdots\!36}a^{3}+\frac{10\!\cdots\!55}{48\!\cdots\!96}a$, $\frac{98\!\cdots\!95}{30\!\cdots\!44}a^{23}+\frac{18\!\cdots\!01}{38\!\cdots\!68}a^{22}-\frac{22\!\cdots\!03}{19\!\cdots\!84}a^{21}-\frac{14\!\cdots\!15}{77\!\cdots\!36}a^{20}+\frac{17\!\cdots\!09}{96\!\cdots\!92}a^{19}+\frac{28\!\cdots\!29}{96\!\cdots\!92}a^{18}-\frac{46\!\cdots\!99}{30\!\cdots\!44}a^{17}-\frac{99\!\cdots\!53}{38\!\cdots\!68}a^{16}+\frac{24\!\cdots\!71}{38\!\cdots\!68}a^{15}+\frac{98\!\cdots\!91}{77\!\cdots\!36}a^{14}-\frac{12\!\cdots\!33}{96\!\cdots\!92}a^{13}-\frac{85\!\cdots\!83}{24\!\cdots\!48}a^{12}+\frac{15\!\cdots\!53}{30\!\cdots\!44}a^{11}+\frac{22\!\cdots\!63}{38\!\cdots\!68}a^{10}+\frac{14\!\cdots\!13}{63\!\cdots\!96}a^{9}-\frac{41\!\cdots\!65}{77\!\cdots\!36}a^{8}-\frac{25\!\cdots\!41}{48\!\cdots\!96}a^{7}+\frac{23\!\cdots\!53}{96\!\cdots\!92}a^{6}+\frac{31\!\cdots\!81}{49\!\cdots\!76}a^{5}+\frac{61\!\cdots\!99}{62\!\cdots\!72}a^{4}-\frac{21\!\cdots\!71}{55\!\cdots\!24}a^{3}-\frac{15\!\cdots\!79}{77\!\cdots\!36}a^{2}+\frac{14\!\cdots\!39}{24\!\cdots\!48}a+\frac{48\!\cdots\!15}{25\!\cdots\!84}$
| 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: | \( 1971189950.5562708 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 1971189950.5562708 \cdot 180}{12\cdot\sqrt{7338001027960597407453351936000000000000}}\cr\approx \mathstrut & 1.30674025973722 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^24 - 39*x^22 + 648*x^20 - 5829*x^18 + 30363*x^16 - 94896*x^14 + 196451*x^12 - 293637*x^10 + 327912*x^8 - 264279*x^6 + 203193*x^4 - 165600*x^2 + 92416)
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 - 39*x^22 + 648*x^20 - 5829*x^18 + 30363*x^16 - 94896*x^14 + 196451*x^12 - 293637*x^10 + 327912*x^8 - 264279*x^6 + 203193*x^4 - 165600*x^2 + 92416, 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 - 39*x^22 + 648*x^20 - 5829*x^18 + 30363*x^16 - 94896*x^14 + 196451*x^12 - 293637*x^10 + 327912*x^8 - 264279*x^6 + 203193*x^4 - 165600*x^2 + 92416);
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 - 39*x^22 + 648*x^20 - 5829*x^18 + 30363*x^16 - 94896*x^14 + 196451*x^12 - 293637*x^10 + 327912*x^8 - 264279*x^6 + 203193*x^4 - 165600*x^2 + 92416);
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 | R | R | ${\href{/padicField/7.2.0.1}{2} }^{12}$ | ${\href{/padicField/11.6.0.1}{6} }^{4}$ | ${\href{/padicField/13.6.0.1}{6} }^{4}$ | ${\href{/padicField/17.6.0.1}{6} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{12}$ | R | ${\href{/padicField/29.2.0.1}{2} }^{12}$ | ${\href{/padicField/31.6.0.1}{6} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{12}$ | ${\href{/padicField/41.2.0.1}{2} }^{12}$ | ${\href{/padicField/43.2.0.1}{2} }^{12}$ | ${\href{/padicField/47.2.0.1}{2} }^{12}$ | ${\href{/padicField/53.6.0.1}{6} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{12}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
|
\(3\)
| 3.12.14.6 | $x^{12} + 6 x^{8} + 15 x^{6} + 9 x^{4} + 18 x^{2} + 9$ | $6$ | $2$ | $14$ | $D_6$ | $[3/2]_{2}^{2}$ |
| 3.12.14.6 | $x^{12} + 6 x^{8} + 15 x^{6} + 9 x^{4} + 18 x^{2} + 9$ | $6$ | $2$ | $14$ | $D_6$ | $[3/2]_{2}^{2}$ | |
|
\(5\)
| 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 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
|
\(23\)
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |