Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 2*x^23 + 5*x^22 - 12*x^21 + 16*x^20 - 30*x^19 + 35*x^18 - 36*x^17 + 47*x^16 - 36*x^15 + 9*x^14 - 16*x^13 + 39*x^12 + 80*x^11 + 53*x^10 - 46*x^8 - 20*x^7 + 34*x^6 + 50*x^5 + 38*x^4 + 18*x^3 + 9*x^2 + 4*x + 1)
gp: K = bnfinit(y^24 - 2*y^23 + 5*y^22 - 12*y^21 + 16*y^20 - 30*y^19 + 35*y^18 - 36*y^17 + 47*y^16 - 36*y^15 + 9*y^14 - 16*y^13 + 39*y^12 + 80*y^11 + 53*y^10 - 46*y^8 - 20*y^7 + 34*y^6 + 50*y^5 + 38*y^4 + 18*y^3 + 9*y^2 + 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 2*x^23 + 5*x^22 - 12*x^21 + 16*x^20 - 30*x^19 + 35*x^18 - 36*x^17 + 47*x^16 - 36*x^15 + 9*x^14 - 16*x^13 + 39*x^12 + 80*x^11 + 53*x^10 - 46*x^8 - 20*x^7 + 34*x^6 + 50*x^5 + 38*x^4 + 18*x^3 + 9*x^2 + 4*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 2*x^23 + 5*x^22 - 12*x^21 + 16*x^20 - 30*x^19 + 35*x^18 - 36*x^17 + 47*x^16 - 36*x^15 + 9*x^14 - 16*x^13 + 39*x^12 + 80*x^11 + 53*x^10 - 46*x^8 - 20*x^7 + 34*x^6 + 50*x^5 + 38*x^4 + 18*x^3 + 9*x^2 + 4*x + 1)
\( x^{24} - 2 x^{23} + 5 x^{22} - 12 x^{21} + 16 x^{20} - 30 x^{19} + 35 x^{18} - 36 x^{17} + 47 x^{16} + \cdots + 1 \)
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: |
\(2754990144000000000000000000\)
\(\medspace = 2^{24}\cdot 3^{16}\cdot 5^{18}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(13.91\) | 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: | $2\cdot 3^{4/3}5^{3/4}\approx 28.934712524984697$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $12$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{16\!\cdots\!41}a^{23}+\frac{125004402368765}{16\!\cdots\!41}a^{22}-\frac{181872483737775}{16\!\cdots\!41}a^{21}-\frac{692289305945831}{16\!\cdots\!41}a^{20}-\frac{344318626782306}{16\!\cdots\!41}a^{19}-\frac{666441772493729}{16\!\cdots\!41}a^{18}-\frac{216584206881134}{16\!\cdots\!41}a^{17}-\frac{715052720422932}{16\!\cdots\!41}a^{16}+\frac{333893682608393}{16\!\cdots\!41}a^{15}-\frac{282478792253345}{16\!\cdots\!41}a^{14}+\frac{209562288719590}{16\!\cdots\!41}a^{13}+\frac{14579344490490}{16\!\cdots\!41}a^{12}-\frac{647605619942813}{16\!\cdots\!41}a^{11}-\frac{298397729799972}{16\!\cdots\!41}a^{10}+\frac{50380689934097}{16\!\cdots\!41}a^{9}-\frac{163402279732272}{16\!\cdots\!41}a^{8}-\frac{154289618103356}{16\!\cdots\!41}a^{7}-\frac{515290496409285}{16\!\cdots\!41}a^{6}-\frac{610745727853033}{16\!\cdots\!41}a^{5}+\frac{713158735293030}{16\!\cdots\!41}a^{4}+\frac{83841989991487}{16\!\cdots\!41}a^{3}+\frac{231051949626161}{16\!\cdots\!41}a^{2}-\frac{718099794527682}{16\!\cdots\!41}a-\frac{754456509957561}{16\!\cdots\!41}$
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: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{182726643215655}{1627978215053041} a^{23} + \frac{962616722473751}{1627978215053041} a^{22} - \frac{2683818657405245}{1627978215053041} a^{21} + \frac{6455789481222930}{1627978215053041} a^{20} - \frac{12748207408726678}{1627978215053041} a^{19} + \frac{21221675082736354}{1627978215053041} a^{18} - \frac{32307569887879579}{1627978215053041} a^{17} + \frac{40882368026131925}{1627978215053041} a^{16} - \frac{45509506867823154}{1627978215053041} a^{15} + \frac{47167863732701051}{1627978215053041} a^{14} - \frac{39145484118482017}{1627978215053041} a^{13} + \frac{21803020531739926}{1627978215053041} a^{12} - \frac{12721052992563022}{1627978215053041} a^{11} + \frac{3465405237312246}{1627978215053041} a^{10} + \frac{17419680328333945}{1627978215053041} a^{9} - \frac{4770319527126960}{1627978215053041} a^{8} + \frac{6724773344866219}{1627978215053041} a^{7} - \frac{9567093455679535}{1627978215053041} a^{6} - \frac{2646793271721465}{1627978215053041} a^{5} + \frac{4931217970856872}{1627978215053041} a^{4} + \frac{2708519680365961}{1627978215053041} a^{3} + \frac{7387044184720634}{1627978215053041} a^{2} + \frac{986346911660636}{1627978215053041} a + \frac{536769987693331}{1627978215053041} \)
(order $20$)
| 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: |
$\frac{991403857763644}{16\!\cdots\!41}a^{23}-\frac{13\!\cdots\!88}{16\!\cdots\!41}a^{22}+\frac{30\!\cdots\!60}{16\!\cdots\!41}a^{21}-\frac{73\!\cdots\!42}{16\!\cdots\!41}a^{20}+\frac{47\!\cdots\!28}{16\!\cdots\!41}a^{19}-\frac{11\!\cdots\!16}{16\!\cdots\!41}a^{18}+\frac{34\!\cdots\!25}{16\!\cdots\!41}a^{17}+\frac{85\!\cdots\!64}{16\!\cdots\!41}a^{16}-\frac{44\!\cdots\!71}{16\!\cdots\!41}a^{15}+\frac{25\!\cdots\!18}{16\!\cdots\!41}a^{14}-\frac{54\!\cdots\!25}{16\!\cdots\!41}a^{13}+\frac{25\!\cdots\!45}{16\!\cdots\!41}a^{12}+\frac{11\!\cdots\!05}{16\!\cdots\!41}a^{11}+\frac{12\!\cdots\!87}{16\!\cdots\!41}a^{10}+\frac{71\!\cdots\!56}{16\!\cdots\!41}a^{9}-\frac{93\!\cdots\!33}{16\!\cdots\!41}a^{8}-\frac{65\!\cdots\!33}{16\!\cdots\!41}a^{7}-\frac{37\!\cdots\!94}{16\!\cdots\!41}a^{6}+\frac{51\!\cdots\!77}{16\!\cdots\!41}a^{5}+\frac{70\!\cdots\!16}{16\!\cdots\!41}a^{4}+\frac{44\!\cdots\!48}{16\!\cdots\!41}a^{3}+\frac{18\!\cdots\!35}{16\!\cdots\!41}a^{2}+\frac{76\!\cdots\!50}{16\!\cdots\!41}a+\frac{35\!\cdots\!17}{16\!\cdots\!41}$, $\frac{736257979987259}{16\!\cdots\!41}a^{23}-\frac{14\!\cdots\!06}{16\!\cdots\!41}a^{22}+\frac{32\!\cdots\!87}{16\!\cdots\!41}a^{21}-\frac{76\!\cdots\!90}{16\!\cdots\!41}a^{20}+\frac{92\!\cdots\!97}{16\!\cdots\!41}a^{19}-\frac{16\!\cdots\!85}{16\!\cdots\!41}a^{18}+\frac{18\!\cdots\!93}{16\!\cdots\!41}a^{17}-\frac{15\!\cdots\!55}{16\!\cdots\!41}a^{16}+\frac{21\!\cdots\!00}{16\!\cdots\!41}a^{15}-\frac{15\!\cdots\!12}{16\!\cdots\!41}a^{14}-\frac{41\!\cdots\!66}{16\!\cdots\!41}a^{13}-\frac{19\!\cdots\!64}{16\!\cdots\!41}a^{12}+\frac{28\!\cdots\!60}{16\!\cdots\!41}a^{11}+\frac{54\!\cdots\!02}{16\!\cdots\!41}a^{10}+\frac{29\!\cdots\!83}{16\!\cdots\!41}a^{9}-\frac{98\!\cdots\!85}{16\!\cdots\!41}a^{8}-\frac{29\!\cdots\!14}{16\!\cdots\!41}a^{7}-\frac{58\!\cdots\!66}{16\!\cdots\!41}a^{6}+\frac{25\!\cdots\!21}{16\!\cdots\!41}a^{5}+\frac{33\!\cdots\!88}{16\!\cdots\!41}a^{4}+\frac{20\!\cdots\!55}{16\!\cdots\!41}a^{3}+\frac{10\!\cdots\!91}{16\!\cdots\!41}a^{2}+\frac{49\!\cdots\!07}{16\!\cdots\!41}a+\frac{20\!\cdots\!92}{16\!\cdots\!41}$, $\frac{18\!\cdots\!00}{16\!\cdots\!41}a^{23}-\frac{47\!\cdots\!11}{16\!\cdots\!41}a^{22}+\frac{12\!\cdots\!10}{16\!\cdots\!41}a^{21}-\frac{28\!\cdots\!09}{16\!\cdots\!41}a^{20}+\frac{45\!\cdots\!39}{16\!\cdots\!41}a^{19}-\frac{79\!\cdots\!99}{16\!\cdots\!41}a^{18}+\frac{10\!\cdots\!56}{16\!\cdots\!41}a^{17}-\frac{12\!\cdots\!69}{16\!\cdots\!41}a^{16}+\frac{14\!\cdots\!19}{16\!\cdots\!41}a^{15}-\frac{13\!\cdots\!12}{16\!\cdots\!41}a^{14}+\frac{78\!\cdots\!28}{16\!\cdots\!41}a^{13}-\frac{56\!\cdots\!18}{16\!\cdots\!41}a^{12}+\frac{91\!\cdots\!68}{16\!\cdots\!41}a^{11}+\frac{98\!\cdots\!66}{16\!\cdots\!41}a^{10}+\frac{32\!\cdots\!91}{16\!\cdots\!41}a^{9}-\frac{21\!\cdots\!77}{16\!\cdots\!41}a^{8}-\frac{61\!\cdots\!71}{16\!\cdots\!41}a^{7}+\frac{30\!\cdots\!80}{16\!\cdots\!41}a^{6}+\frac{60\!\cdots\!55}{16\!\cdots\!41}a^{5}+\frac{49\!\cdots\!90}{16\!\cdots\!41}a^{4}+\frac{37\!\cdots\!37}{16\!\cdots\!41}a^{3}+\frac{16\!\cdots\!17}{16\!\cdots\!41}a^{2}+\frac{77\!\cdots\!76}{16\!\cdots\!41}a+\frac{23\!\cdots\!71}{16\!\cdots\!41}$, $\frac{16\!\cdots\!80}{16\!\cdots\!41}a^{23}-\frac{38\!\cdots\!90}{16\!\cdots\!41}a^{22}+\frac{95\!\cdots\!78}{16\!\cdots\!41}a^{21}-\frac{22\!\cdots\!02}{16\!\cdots\!41}a^{20}+\frac{34\!\cdots\!79}{16\!\cdots\!41}a^{19}-\frac{61\!\cdots\!04}{16\!\cdots\!41}a^{18}+\frac{78\!\cdots\!58}{16\!\cdots\!41}a^{17}-\frac{86\!\cdots\!45}{16\!\cdots\!41}a^{16}+\frac{10\!\cdots\!54}{16\!\cdots\!41}a^{15}-\frac{93\!\cdots\!15}{16\!\cdots\!41}a^{14}+\frac{43\!\cdots\!28}{16\!\cdots\!41}a^{13}-\frac{36\!\cdots\!06}{16\!\cdots\!41}a^{12}+\frac{68\!\cdots\!14}{16\!\cdots\!41}a^{11}+\frac{11\!\cdots\!43}{16\!\cdots\!41}a^{10}+\frac{43\!\cdots\!61}{16\!\cdots\!41}a^{9}-\frac{10\!\cdots\!83}{16\!\cdots\!41}a^{8}-\frac{75\!\cdots\!30}{16\!\cdots\!41}a^{7}-\frac{11\!\cdots\!61}{16\!\cdots\!41}a^{6}+\frac{60\!\cdots\!62}{16\!\cdots\!41}a^{5}+\frac{61\!\cdots\!00}{16\!\cdots\!41}a^{4}+\frac{44\!\cdots\!41}{16\!\cdots\!41}a^{3}+\frac{13\!\cdots\!24}{16\!\cdots\!41}a^{2}+\frac{70\!\cdots\!69}{16\!\cdots\!41}a+\frac{24\!\cdots\!53}{16\!\cdots\!41}$, $\frac{23\!\cdots\!85}{16\!\cdots\!41}a^{23}-\frac{52\!\cdots\!48}{16\!\cdots\!41}a^{22}+\frac{12\!\cdots\!03}{16\!\cdots\!41}a^{21}-\frac{29\!\cdots\!77}{16\!\cdots\!41}a^{20}+\frac{41\!\cdots\!85}{16\!\cdots\!41}a^{19}-\frac{72\!\cdots\!33}{16\!\cdots\!41}a^{18}+\frac{88\!\cdots\!97}{16\!\cdots\!41}a^{17}-\frac{87\!\cdots\!88}{16\!\cdots\!41}a^{16}+\frac{10\!\cdots\!22}{16\!\cdots\!41}a^{15}-\frac{86\!\cdots\!88}{16\!\cdots\!41}a^{14}+\frac{15\!\cdots\!88}{16\!\cdots\!41}a^{13}-\frac{11\!\cdots\!90}{16\!\cdots\!41}a^{12}+\frac{81\!\cdots\!22}{16\!\cdots\!41}a^{11}+\frac{16\!\cdots\!73}{16\!\cdots\!41}a^{10}+\frac{52\!\cdots\!31}{16\!\cdots\!41}a^{9}-\frac{29\!\cdots\!70}{16\!\cdots\!41}a^{8}-\frac{89\!\cdots\!94}{16\!\cdots\!41}a^{7}-\frac{38\!\cdots\!89}{16\!\cdots\!41}a^{6}+\frac{83\!\cdots\!27}{16\!\cdots\!41}a^{5}+\frac{80\!\cdots\!49}{16\!\cdots\!41}a^{4}+\frac{51\!\cdots\!45}{16\!\cdots\!41}a^{3}+\frac{24\!\cdots\!89}{16\!\cdots\!41}a^{2}+\frac{14\!\cdots\!63}{16\!\cdots\!41}a+\frac{49\!\cdots\!29}{16\!\cdots\!41}$, $\frac{155143502400767}{16\!\cdots\!41}a^{23}-\frac{93812486375543}{16\!\cdots\!41}a^{22}+\frac{302323471828985}{16\!\cdots\!41}a^{21}-\frac{11\!\cdots\!03}{16\!\cdots\!41}a^{20}+\frac{905562779113737}{16\!\cdots\!41}a^{19}-\frac{32\!\cdots\!25}{16\!\cdots\!41}a^{18}+\frac{40\!\cdots\!75}{16\!\cdots\!41}a^{17}-\frac{51\!\cdots\!64}{16\!\cdots\!41}a^{16}+\frac{11\!\cdots\!76}{16\!\cdots\!41}a^{15}-\frac{10\!\cdots\!76}{16\!\cdots\!41}a^{14}+\frac{60\!\cdots\!79}{16\!\cdots\!41}a^{13}-\frac{14\!\cdots\!60}{16\!\cdots\!41}a^{12}+\frac{17\!\cdots\!71}{16\!\cdots\!41}a^{11}+\frac{20\!\cdots\!39}{16\!\cdots\!41}a^{10}+\frac{18\!\cdots\!49}{16\!\cdots\!41}a^{9}-\frac{58\!\cdots\!24}{16\!\cdots\!41}a^{8}-\frac{26\!\cdots\!56}{16\!\cdots\!41}a^{7}-\frac{37\!\cdots\!74}{16\!\cdots\!41}a^{6}+\frac{15\!\cdots\!66}{16\!\cdots\!41}a^{5}+\frac{19\!\cdots\!41}{16\!\cdots\!41}a^{4}+\frac{74\!\cdots\!22}{16\!\cdots\!41}a^{3}-\frac{48\!\cdots\!85}{16\!\cdots\!41}a^{2}+\frac{52238610621824}{16\!\cdots\!41}a+\frac{491101743245076}{16\!\cdots\!41}$, $\frac{10\!\cdots\!04}{16\!\cdots\!41}a^{23}-\frac{21\!\cdots\!87}{16\!\cdots\!41}a^{22}+\frac{54\!\cdots\!49}{16\!\cdots\!41}a^{21}-\frac{12\!\cdots\!42}{16\!\cdots\!41}a^{20}+\frac{16\!\cdots\!76}{16\!\cdots\!41}a^{19}-\frac{30\!\cdots\!94}{16\!\cdots\!41}a^{18}+\frac{33\!\cdots\!66}{16\!\cdots\!41}a^{17}-\frac{30\!\cdots\!33}{16\!\cdots\!41}a^{16}+\frac{37\!\cdots\!18}{16\!\cdots\!41}a^{15}-\frac{21\!\cdots\!55}{16\!\cdots\!41}a^{14}-\frac{11\!\cdots\!26}{16\!\cdots\!41}a^{13}+\frac{39\!\cdots\!85}{16\!\cdots\!41}a^{12}+\frac{23\!\cdots\!72}{16\!\cdots\!41}a^{11}+\frac{99\!\cdots\!99}{16\!\cdots\!41}a^{10}+\frac{42\!\cdots\!67}{16\!\cdots\!41}a^{9}-\frac{32\!\cdots\!99}{16\!\cdots\!41}a^{8}-\frac{58\!\cdots\!11}{16\!\cdots\!41}a^{7}-\frac{15\!\cdots\!20}{16\!\cdots\!41}a^{6}+\frac{43\!\cdots\!77}{16\!\cdots\!41}a^{5}+\frac{54\!\cdots\!32}{16\!\cdots\!41}a^{4}+\frac{32\!\cdots\!00}{16\!\cdots\!41}a^{3}+\frac{14\!\cdots\!67}{16\!\cdots\!41}a^{2}+\frac{78\!\cdots\!39}{16\!\cdots\!41}a+\frac{42\!\cdots\!06}{16\!\cdots\!41}$, $\frac{25945484187578}{16\!\cdots\!41}a^{23}+\frac{36428594342262}{16\!\cdots\!41}a^{22}-\frac{50574237374880}{16\!\cdots\!41}a^{21}+\frac{56575414020325}{16\!\cdots\!41}a^{20}-\frac{171951387321319}{16\!\cdots\!41}a^{19}-\frac{742777979471293}{16\!\cdots\!41}a^{18}+\frac{12\!\cdots\!18}{16\!\cdots\!41}a^{17}-\frac{34\!\cdots\!22}{16\!\cdots\!41}a^{16}+\frac{67\!\cdots\!53}{16\!\cdots\!41}a^{15}-\frac{86\!\cdots\!29}{16\!\cdots\!41}a^{14}+\frac{10\!\cdots\!96}{16\!\cdots\!41}a^{13}-\frac{11\!\cdots\!23}{16\!\cdots\!41}a^{12}+\frac{95\!\cdots\!29}{16\!\cdots\!41}a^{11}-\frac{19\!\cdots\!63}{16\!\cdots\!41}a^{10}+\frac{81\!\cdots\!49}{16\!\cdots\!41}a^{9}+\frac{97\!\cdots\!91}{16\!\cdots\!41}a^{8}-\frac{23\!\cdots\!89}{16\!\cdots\!41}a^{7}+\frac{194684184294259}{16\!\cdots\!41}a^{6}-\frac{11\!\cdots\!51}{16\!\cdots\!41}a^{5}+\frac{42\!\cdots\!79}{16\!\cdots\!41}a^{4}+\frac{59\!\cdots\!10}{16\!\cdots\!41}a^{3}+\frac{76\!\cdots\!39}{16\!\cdots\!41}a^{2}+\frac{24\!\cdots\!08}{16\!\cdots\!41}a-\frac{459696461370420}{16\!\cdots\!41}$, $\frac{227366875605470}{16\!\cdots\!41}a^{23}-\frac{796147265047423}{16\!\cdots\!41}a^{22}+\frac{17\!\cdots\!22}{16\!\cdots\!41}a^{21}-\frac{38\!\cdots\!74}{16\!\cdots\!41}a^{20}+\frac{59\!\cdots\!08}{16\!\cdots\!41}a^{19}-\frac{80\!\cdots\!71}{16\!\cdots\!41}a^{18}+\frac{92\!\cdots\!58}{16\!\cdots\!41}a^{17}-\frac{57\!\cdots\!42}{16\!\cdots\!41}a^{16}+\frac{13\!\cdots\!04}{16\!\cdots\!41}a^{15}+\frac{36\!\cdots\!24}{16\!\cdots\!41}a^{14}-\frac{13\!\cdots\!86}{16\!\cdots\!41}a^{13}+\frac{20\!\cdots\!16}{16\!\cdots\!41}a^{12}-\frac{11\!\cdots\!48}{16\!\cdots\!41}a^{11}+\frac{18\!\cdots\!77}{16\!\cdots\!41}a^{10}-\frac{16\!\cdots\!80}{16\!\cdots\!41}a^{9}-\frac{37\!\cdots\!75}{16\!\cdots\!41}a^{8}-\frac{72\!\cdots\!80}{16\!\cdots\!41}a^{7}+\frac{51\!\cdots\!29}{16\!\cdots\!41}a^{6}+\frac{75\!\cdots\!35}{16\!\cdots\!41}a^{5}+\frac{35\!\cdots\!48}{16\!\cdots\!41}a^{4}-\frac{29\!\cdots\!06}{16\!\cdots\!41}a^{3}-\frac{24\!\cdots\!20}{16\!\cdots\!41}a^{2}-\frac{21\!\cdots\!22}{16\!\cdots\!41}a-\frac{14\!\cdots\!72}{16\!\cdots\!41}$, $\frac{797790212567187}{16\!\cdots\!41}a^{23}-\frac{27\!\cdots\!04}{16\!\cdots\!41}a^{22}+\frac{69\!\cdots\!15}{16\!\cdots\!41}a^{21}-\frac{16\!\cdots\!24}{16\!\cdots\!41}a^{20}+\frac{29\!\cdots\!11}{16\!\cdots\!41}a^{19}-\frac{48\!\cdots\!42}{16\!\cdots\!41}a^{18}+\frac{69\!\cdots\!88}{16\!\cdots\!41}a^{17}-\frac{81\!\cdots\!72}{16\!\cdots\!41}a^{16}+\frac{92\!\cdots\!44}{16\!\cdots\!41}a^{15}-\frac{93\!\cdots\!67}{16\!\cdots\!41}a^{14}+\frac{65\!\cdots\!44}{16\!\cdots\!41}a^{13}-\frac{36\!\cdots\!37}{16\!\cdots\!41}a^{12}+\frac{46\!\cdots\!47}{16\!\cdots\!41}a^{11}+\frac{13\!\cdots\!22}{16\!\cdots\!41}a^{10}-\frac{18\!\cdots\!42}{16\!\cdots\!41}a^{9}-\frac{19\!\cdots\!70}{16\!\cdots\!41}a^{8}-\frac{12\!\cdots\!17}{16\!\cdots\!41}a^{7}+\frac{20\!\cdots\!84}{16\!\cdots\!41}a^{6}+\frac{25\!\cdots\!43}{16\!\cdots\!41}a^{5}-\frac{492633687405077}{16\!\cdots\!41}a^{4}-\frac{23\!\cdots\!74}{16\!\cdots\!41}a^{3}-\frac{57\!\cdots\!20}{16\!\cdots\!41}a^{2}+\frac{26\!\cdots\!01}{16\!\cdots\!41}a-\frac{913371545079125}{16\!\cdots\!41}$, $\frac{18\!\cdots\!35}{16\!\cdots\!41}a^{23}-\frac{47\!\cdots\!78}{16\!\cdots\!41}a^{22}+\frac{12\!\cdots\!14}{16\!\cdots\!41}a^{21}-\frac{29\!\cdots\!56}{16\!\cdots\!41}a^{20}+\frac{46\!\cdots\!03}{16\!\cdots\!41}a^{19}-\frac{81\!\cdots\!33}{16\!\cdots\!41}a^{18}+\frac{10\!\cdots\!03}{16\!\cdots\!41}a^{17}-\frac{12\!\cdots\!15}{16\!\cdots\!41}a^{16}+\frac{15\!\cdots\!29}{16\!\cdots\!41}a^{15}-\frac{13\!\cdots\!45}{16\!\cdots\!41}a^{14}+\frac{81\!\cdots\!80}{16\!\cdots\!41}a^{13}-\frac{59\!\cdots\!59}{16\!\cdots\!41}a^{12}+\frac{86\!\cdots\!85}{16\!\cdots\!41}a^{11}+\frac{11\!\cdots\!79}{16\!\cdots\!41}a^{10}+\frac{34\!\cdots\!78}{16\!\cdots\!41}a^{9}-\frac{74\!\cdots\!86}{16\!\cdots\!41}a^{8}-\frac{76\!\cdots\!81}{16\!\cdots\!41}a^{7}-\frac{15\!\cdots\!82}{16\!\cdots\!41}a^{6}+\frac{64\!\cdots\!76}{16\!\cdots\!41}a^{5}+\frac{56\!\cdots\!31}{16\!\cdots\!41}a^{4}+\frac{44\!\cdots\!96}{16\!\cdots\!41}a^{3}+\frac{12\!\cdots\!50}{16\!\cdots\!41}a^{2}+\frac{10\!\cdots\!33}{16\!\cdots\!41}a+\frac{40\!\cdots\!32}{16\!\cdots\!41}$
| 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: | \( 48570.384925004604 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 48570.384925004604 \cdot 1}{20\cdot\sqrt{2754990144000000000000000000}}\cr\approx \mathstrut & 0.175162020115116 \end{aligned}\] (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 + 5*x^22 - 12*x^21 + 16*x^20 - 30*x^19 + 35*x^18 - 36*x^17 + 47*x^16 - 36*x^15 + 9*x^14 - 16*x^13 + 39*x^12 + 80*x^11 + 53*x^10 - 46*x^8 - 20*x^7 + 34*x^6 + 50*x^5 + 38*x^4 + 18*x^3 + 9*x^2 + 4*x + 1)
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 + 5*x^22 - 12*x^21 + 16*x^20 - 30*x^19 + 35*x^18 - 36*x^17 + 47*x^16 - 36*x^15 + 9*x^14 - 16*x^13 + 39*x^12 + 80*x^11 + 53*x^10 - 46*x^8 - 20*x^7 + 34*x^6 + 50*x^5 + 38*x^4 + 18*x^3 + 9*x^2 + 4*x + 1, 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 + 5*x^22 - 12*x^21 + 16*x^20 - 30*x^19 + 35*x^18 - 36*x^17 + 47*x^16 - 36*x^15 + 9*x^14 - 16*x^13 + 39*x^12 + 80*x^11 + 53*x^10 - 46*x^8 - 20*x^7 + 34*x^6 + 50*x^5 + 38*x^4 + 18*x^3 + 9*x^2 + 4*x + 1);
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 + 5*x^22 - 12*x^21 + 16*x^20 - 30*x^19 + 35*x^18 - 36*x^17 + 47*x^16 - 36*x^15 + 9*x^14 - 16*x^13 + 39*x^12 + 80*x^11 + 53*x^10 - 46*x^8 - 20*x^7 + 34*x^6 + 50*x^5 + 38*x^4 + 18*x^3 + 9*x^2 + 4*x + 1);
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
$S_3\times C_{12}$ (as 24T65):
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 72 |
| The 36 conjugacy class representatives for $S_3\times C_{12}$ |
| Character table for $S_3\times C_{12}$ |
Intermediate fields
| \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{5})\), \(\Q(\zeta_{20})^+\), \(\Q(i, \sqrt{5})\), 6.0.648000.1, \(\Q(\zeta_{20})\), 12.0.419904000000.1 |
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 36 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 | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }^{4}$ | ${\href{/padicField/13.12.0.1}{12} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{6}$ | ${\href{/padicField/19.2.0.1}{2} }^{12}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{6}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{12}$ | ${\href{/padicField/43.12.0.1}{12} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{3}$ | ${\href{/padicField/53.4.0.1}{4} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $24$ | $2$ | $12$ | $24$ | |||
|
\(3\)
| 3.12.16.25 | $x^{12} + 24 x^{11} + 216 x^{10} + 804 x^{9} + 216 x^{8} - 6480 x^{7} - 11610 x^{6} + 16200 x^{5} + 48600 x^{4} + 33156 x^{3} + 198936 x^{2} + 190593$ | $3$ | $4$ | $16$ | $C_{12}$ | $[2]^{4}$ |
| 3.12.0.1 | $x^{12} + x^{6} + x^{5} + x^{4} + x^{2} + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
|
\(5\)
| 5.12.9.2 | $x^{12} + 12 x^{10} + 12 x^{9} + 69 x^{8} + 108 x^{7} + 42 x^{6} - 396 x^{5} + 840 x^{4} + 252 x^{3} + 1476 x^{2} + 684 x + 1601$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
| 5.12.9.2 | $x^{12} + 12 x^{10} + 12 x^{9} + 69 x^{8} + 108 x^{7} + 42 x^{6} - 396 x^{5} + 840 x^{4} + 252 x^{3} + 1476 x^{2} + 684 x + 1601$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.20.2t1.a.a | $1$ | $ 2^{2} \cdot 5 $ | \(\Q(\sqrt{-5}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.36.6t1.b.a | $1$ | $ 2^{2} \cdot 3^{2}$ | 6.0.419904.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.45.6t1.a.a | $1$ | $ 3^{2} \cdot 5 $ | 6.6.820125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.45.6t1.a.b | $1$ | $ 3^{2} \cdot 5 $ | 6.6.820125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.180.6t1.b.a | $1$ | $ 2^{2} \cdot 3^{2} \cdot 5 $ | 6.0.52488000.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.36.6t1.b.b | $1$ | $ 2^{2} \cdot 3^{2}$ | 6.0.419904.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.180.6t1.b.b | $1$ | $ 2^{2} \cdot 3^{2} \cdot 5 $ | 6.0.52488000.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| * | 1.20.4t1.a.a | $1$ | $ 2^{2} \cdot 5 $ | \(\Q(\zeta_{20})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
| * | 1.5.4t1.a.a | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ |
| * | 1.5.4t1.a.b | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ |
| * | 1.20.4t1.a.b | $1$ | $ 2^{2} \cdot 5 $ | \(\Q(\zeta_{20})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
| 1.45.12t1.a.a | $1$ | $ 3^{2} \cdot 5 $ | 12.0.84075626953125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.180.12t1.b.a | $1$ | $ 2^{2} \cdot 3^{2} \cdot 5 $ | 12.12.344373768000000000.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.45.12t1.a.b | $1$ | $ 3^{2} \cdot 5 $ | 12.0.84075626953125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.180.12t1.b.b | $1$ | $ 2^{2} \cdot 3^{2} \cdot 5 $ | 12.12.344373768000000000.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.180.12t1.b.c | $1$ | $ 2^{2} \cdot 3^{2} \cdot 5 $ | 12.12.344373768000000000.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.180.12t1.b.d | $1$ | $ 2^{2} \cdot 3^{2} \cdot 5 $ | 12.12.344373768000000000.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.45.12t1.a.c | $1$ | $ 3^{2} \cdot 5 $ | 12.0.84075626953125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.45.12t1.a.d | $1$ | $ 3^{2} \cdot 5 $ | 12.0.84075626953125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 2.1620.3t2.b.a | $2$ | $ 2^{2} \cdot 3^{4} \cdot 5 $ | 3.1.1620.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.1620.6t3.g.a | $2$ | $ 2^{2} \cdot 3^{4} \cdot 5 $ | 6.2.13122000.3 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| * | 2.180.6t5.b.a | $2$ | $ 2^{2} \cdot 3^{2} \cdot 5 $ | 6.0.648000.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.180.12t18.a.a | $2$ | $ 2^{2} \cdot 3^{2} \cdot 5 $ | 12.0.419904000000.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.180.6t5.b.b | $2$ | $ 2^{2} \cdot 3^{2} \cdot 5 $ | 6.0.648000.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.180.12t18.a.b | $2$ | $ 2^{2} \cdot 3^{2} \cdot 5 $ | 12.0.419904000000.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| 2.8100.12t11.c.a | $2$ | $ 2^{2} \cdot 3^{4} \cdot 5^{2}$ | 12.0.21523360500000000.3 | $S_3 \times C_4$ (as 12T11) | $0$ | $0$ | |
| 2.8100.12t11.c.b | $2$ | $ 2^{2} \cdot 3^{4} \cdot 5^{2}$ | 12.0.21523360500000000.3 | $S_3 \times C_4$ (as 12T11) | $0$ | $0$ | |
| * | 2.900.24t65.b.a | $2$ | $ 2^{2} \cdot 3^{2} \cdot 5^{2}$ | 24.0.2754990144000000000000000000.1 | $S_3\times C_{12}$ (as 24T65) | $0$ | $0$ |
| * | 2.900.24t65.b.b | $2$ | $ 2^{2} \cdot 3^{2} \cdot 5^{2}$ | 24.0.2754990144000000000000000000.1 | $S_3\times C_{12}$ (as 24T65) | $0$ | $0$ |
| * | 2.900.24t65.b.c | $2$ | $ 2^{2} \cdot 3^{2} \cdot 5^{2}$ | 24.0.2754990144000000000000000000.1 | $S_3\times C_{12}$ (as 24T65) | $0$ | $0$ |
| * | 2.900.24t65.b.d | $2$ | $ 2^{2} \cdot 3^{2} \cdot 5^{2}$ | 24.0.2754990144000000000000000000.1 | $S_3\times C_{12}$ (as 24T65) | $0$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.