Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 48*x^22 + 936*x^20 - 9584*x^18 + 55620*x^16 - 184752*x^14 + 341336*x^12 - 336864*x^10 + 183060*x^8 - 54976*x^6 + 8592*x^4 - 576*x^2 + 8)
gp: K = bnfinit(y^24 - 48*y^22 + 936*y^20 - 9584*y^18 + 55620*y^16 - 184752*y^14 + 341336*y^12 - 336864*y^10 + 183060*y^8 - 54976*y^6 + 8592*y^4 - 576*y^2 + 8, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 48*x^22 + 936*x^20 - 9584*x^18 + 55620*x^16 - 184752*x^14 + 341336*x^12 - 336864*x^10 + 183060*x^8 - 54976*x^6 + 8592*x^4 - 576*x^2 + 8);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 48*x^22 + 936*x^20 - 9584*x^18 + 55620*x^16 - 184752*x^14 + 341336*x^12 - 336864*x^10 + 183060*x^8 - 54976*x^6 + 8592*x^4 - 576*x^2 + 8)
\( x^{24} - 48 x^{22} + 936 x^{20} - 9584 x^{18} + 55620 x^{16} - 184752 x^{14} + 341336 x^{12} - 336864 x^{10} + \cdots + 8 \)
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: | $[24, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(18351423083070806589199715754737431920771072\)
\(\medspace = 2^{93}\cdot 3^{32}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(63.48\) | 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: | $2^{31/8}3^{4/3}\approx 63.48233698954048$ | ||
| Ramified primes: |
\(2\), \(3\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $24$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(288=2^{5}\cdot 3^{2}\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{288}(1,·)$, $\chi_{288}(133,·)$, $\chi_{288}(193,·)$, $\chi_{288}(265,·)$, $\chi_{288}(13,·)$, $\chi_{288}(205,·)$, $\chi_{288}(145,·)$, $\chi_{288}(277,·)$, $\chi_{288}(217,·)$, $\chi_{288}(25,·)$, $\chi_{288}(157,·)$, $\chi_{288}(229,·)$, $\chi_{288}(97,·)$, $\chi_{288}(37,·)$, $\chi_{288}(241,·)$, $\chi_{288}(169,·)$, $\chi_{288}(109,·)$, $\chi_{288}(253,·)$, $\chi_{288}(49,·)$, $\chi_{288}(181,·)$, $\chi_{288}(73,·)$, $\chi_{288}(121,·)$, $\chi_{288}(61,·)$, $\chi_{288}(85,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{2}a^{9}$, $\frac{1}{2}a^{10}$, $\frac{1}{2}a^{11}$, $\frac{1}{2}a^{12}$, $\frac{1}{2}a^{13}$, $\frac{1}{34}a^{14}-\frac{3}{17}a^{12}+\frac{1}{17}a^{10}+\frac{5}{34}a^{8}+\frac{2}{17}a^{6}+\frac{5}{17}a^{4}+\frac{4}{17}a^{2}-\frac{7}{17}$, $\frac{1}{34}a^{15}-\frac{3}{17}a^{13}+\frac{1}{17}a^{11}+\frac{5}{34}a^{9}+\frac{2}{17}a^{7}+\frac{5}{17}a^{5}+\frac{4}{17}a^{3}-\frac{7}{17}a$, $\frac{1}{68}a^{16}-\frac{4}{17}$, $\frac{1}{68}a^{17}-\frac{4}{17}a$, $\frac{1}{68}a^{18}-\frac{4}{17}a^{2}$, $\frac{1}{68}a^{19}-\frac{4}{17}a^{3}$, $\frac{1}{68}a^{20}-\frac{4}{17}a^{4}$, $\frac{1}{68}a^{21}-\frac{4}{17}a^{5}$, $\frac{1}{29305626052}a^{22}+\frac{12782034}{7326406513}a^{20}+\frac{90656021}{14652813026}a^{18}-\frac{38946948}{7326406513}a^{16}-\frac{31856446}{7326406513}a^{14}-\frac{357119348}{7326406513}a^{12}+\frac{1518502401}{14652813026}a^{10}-\frac{59039979}{7326406513}a^{8}+\frac{139399973}{7326406513}a^{6}-\frac{3436165571}{7326406513}a^{4}+\frac{3159080086}{7326406513}a^{2}-\frac{1618059094}{7326406513}$, $\frac{1}{29305626052}a^{23}+\frac{12782034}{7326406513}a^{21}+\frac{90656021}{14652813026}a^{19}-\frac{38946948}{7326406513}a^{17}-\frac{31856446}{7326406513}a^{15}-\frac{357119348}{7326406513}a^{13}+\frac{1518502401}{14652813026}a^{11}-\frac{59039979}{7326406513}a^{9}+\frac{139399973}{7326406513}a^{7}-\frac{3436165571}{7326406513}a^{5}+\frac{3159080086}{7326406513}a^{3}-\frac{1618059094}{7326406513}a$
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: | $23$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: |
$\frac{8302846608}{7326406513}a^{22}-\frac{395419860639}{7326406513}a^{20}+\frac{7623053411184}{7326406513}a^{18}-\frac{18050360826735}{1723860356}a^{16}+\frac{433028599807556}{7326406513}a^{14}-\frac{13\!\cdots\!62}{7326406513}a^{12}+\frac{23\!\cdots\!76}{7326406513}a^{10}-\frac{19\!\cdots\!82}{7326406513}a^{8}+\frac{800692045883496}{7326406513}a^{6}-\frac{158926474102572}{7326406513}a^{4}+\frac{12442950817744}{7326406513}a^{2}-\frac{193408723526}{7326406513}$, $\frac{3434221409}{7326406513}a^{22}-\frac{327426372543}{14652813026}a^{20}+\frac{3160568924946}{7326406513}a^{18}-\frac{1874892129477}{430965089}a^{16}+\frac{10618272345336}{430965089}a^{14}-\frac{33819588969777}{430965089}a^{12}+\frac{57774351960176}{430965089}a^{10}-\frac{97761788242869}{861930178}a^{8}+\frac{351154293931296}{7326406513}a^{6}-\frac{70755162305780}{7326406513}a^{4}+\frac{5594339938104}{7326406513}a^{2}-\frac{4612120182}{430965089}$, $\frac{17889732}{16998623}a^{22}-\frac{1705138101}{33997246}a^{20}+\frac{16452234165}{16998623}a^{18}-\frac{165808756815}{16998623}a^{16}+\frac{938106736380}{16998623}a^{14}-\frac{5965983544141}{33997246}a^{12}+\frac{5081236736670}{16998623}a^{10}-\frac{4277725426206}{16998623}a^{8}+\frac{1798511194184}{16998623}a^{6}-\frac{359708833089}{16998623}a^{4}+\frac{27892488708}{16998623}a^{2}-\frac{360510071}{16998623}$, $\frac{4342568691}{14652813026}a^{22}-\frac{103735355517}{7326406513}a^{20}+\frac{8036213039133}{29305626052}a^{18}-\frac{20357945097138}{7326406513}a^{16}+\frac{232350664863141}{14652813026}a^{14}-\frac{374814646799187}{7326406513}a^{12}+\frac{13\!\cdots\!05}{14652813026}a^{10}-\frac{579528046999512}{7326406513}a^{8}+\frac{260105805152608}{7326406513}a^{6}-\frac{3319135785030}{430965089}a^{4}+\frac{4863034828431}{7326406513}a^{2}-\frac{4985389923}{430965089}$, $\frac{33980311233}{14652813026}a^{23}-\frac{190557007663}{1723860356}a^{21}+\frac{62531668797969}{29305626052}a^{19}-\frac{630490151356821}{29305626052}a^{17}+\frac{892422344931671}{7326406513}a^{15}-\frac{28\!\cdots\!59}{7326406513}a^{13}+\frac{96\!\cdots\!69}{14652813026}a^{11}-\frac{40\!\cdots\!54}{7326406513}a^{9}+\frac{17\!\cdots\!76}{7326406513}a^{7}-\frac{346661960395356}{7326406513}a^{5}+\frac{26630928957824}{7326406513}a^{3}-\frac{292465967781}{7326406513}a-1$, $\frac{13061546325}{14652813026}a^{23}-\frac{311026484853}{7326406513}a^{21}+\frac{23984442046893}{29305626052}a^{19}-\frac{241367003669475}{29305626052}a^{17}+\frac{340614394724595}{7326406513}a^{15}-\frac{10\!\cdots\!18}{7326406513}a^{13}+\frac{36\!\cdots\!21}{14652813026}a^{11}-\frac{15\!\cdots\!42}{7326406513}a^{9}+\frac{629835033145248}{7326406513}a^{7}-\frac{125014014060084}{7326406513}a^{5}+\frac{9612931064272}{7326406513}a^{3}-\frac{78020879475}{7326406513}a-1$, $\frac{8766857205}{14652813026}a^{23}-\frac{834120670113}{29305626052}a^{21}+\frac{8027226822843}{14652813026}a^{19}-\frac{161157944641527}{29305626052}a^{17}+\frac{226481556971142}{7326406513}a^{15}-\frac{712036131987033}{7326406513}a^{13}+\frac{11\!\cdots\!59}{7326406513}a^{11}-\frac{952543953094906}{7326406513}a^{9}+\frac{367695181222209}{7326406513}a^{7}-\frac{62554568068476}{7326406513}a^{5}+\frac{3104946750000}{7326406513}a^{3}+\frac{67961387889}{7326406513}a-1$, $\frac{8766857205}{14652813026}a^{23}-\frac{834120670113}{29305626052}a^{21}+\frac{8027226822843}{14652813026}a^{19}-\frac{161157944641527}{29305626052}a^{17}+\frac{226481556971142}{7326406513}a^{15}-\frac{712036131987033}{7326406513}a^{13}+\frac{11\!\cdots\!59}{7326406513}a^{11}-\frac{952543953094906}{7326406513}a^{9}+\frac{367695181222209}{7326406513}a^{7}-\frac{62554568068476}{7326406513}a^{5}+\frac{3104946750000}{7326406513}a^{3}+\frac{67961387889}{7326406513}a+1$, $\frac{8302846608}{7326406513}a^{23}-\frac{395419860639}{7326406513}a^{21}+\frac{7623053411184}{7326406513}a^{19}-\frac{18050360826735}{1723860356}a^{17}+\frac{433028599807556}{7326406513}a^{15}-\frac{13\!\cdots\!62}{7326406513}a^{13}+\frac{23\!\cdots\!76}{7326406513}a^{11}-\frac{19\!\cdots\!82}{7326406513}a^{9}+\frac{800692045883496}{7326406513}a^{7}-\frac{158926474102572}{7326406513}a^{5}+\frac{12442950817744}{7326406513}a^{3}-\frac{200735130039}{7326406513}a+1$, $\frac{251544299}{430965089}a^{22}-\frac{203744074494}{7326406513}a^{20}+\frac{3930344000169}{7326406513}a^{18}-\frac{39590407986156}{7326406513}a^{16}+\frac{223813373509068}{7326406513}a^{14}-\frac{14\!\cdots\!53}{14652813026}a^{12}+\frac{12\!\cdots\!78}{7326406513}a^{10}-\frac{20\!\cdots\!99}{14652813026}a^{8}+\frac{424004030762008}{7326406513}a^{6}-\frac{84279344755579}{7326406513}a^{4}+\frac{6427322695044}{7326406513}a^{2}-\frac{84300204020}{7326406513}$, $\frac{19447542509}{7326406513}a^{22}-\frac{926590307676}{7326406513}a^{20}+\frac{17874535261245}{7326406513}a^{18}-\frac{720203095607991}{29305626052}a^{16}+\frac{59874307826944}{430965089}a^{14}-\frac{380262332478289}{861930178}a^{12}+\frac{323088358904114}{430965089}a^{10}-\frac{541646585089477}{861930178}a^{8}+\frac{19\!\cdots\!96}{7326406513}a^{6}-\frac{384716143469711}{7326406513}a^{4}+\frac{30058953388996}{7326406513}a^{2}-\frac{434521013734}{7326406513}$, $\frac{19131567343}{14652813026}a^{22}-\frac{910795382787}{14652813026}a^{20}+\frac{35098276305387}{29305626052}a^{18}-\frac{352917018470379}{29305626052}a^{16}+\frac{994727794493395}{14652813026}a^{14}-\frac{15\!\cdots\!84}{7326406513}a^{12}+\frac{52\!\cdots\!31}{14652813026}a^{10}-\frac{43\!\cdots\!13}{14652813026}a^{8}+\frac{891740534662184}{7326406513}a^{6}-\frac{173256328062842}{7326406513}a^{4}+\frac{13174255927417}{7326406513}a^{2}-\frac{194389544442}{7326406513}$, $\frac{11211011509}{14652813026}a^{22}-\frac{534897083577}{14652813026}a^{20}+\frac{20678488738917}{29305626052}a^{18}-\frac{52231111298247}{7326406513}a^{16}+\frac{593371924604565}{14652813026}a^{14}-\frac{949747659285396}{7326406513}a^{12}+\frac{32\!\cdots\!89}{14652813026}a^{10}-\frac{28\!\cdots\!97}{14652813026}a^{8}+\frac{611260099083904}{7326406513}a^{6}-\frac{127180470651290}{7326406513}a^{4}+\frac{10457374766535}{7326406513}a^{2}-\frac{9166545016}{430965089}$, $\frac{27827984809}{14652813026}a^{22}-\frac{1325617083723}{14652813026}a^{20}+\frac{51129303360151}{29305626052}a^{18}-\frac{128701116687189}{7326406513}a^{16}+\frac{727077361253133}{7326406513}a^{14}-\frac{23\!\cdots\!77}{7326406513}a^{12}+\frac{78\!\cdots\!43}{14652813026}a^{10}-\frac{65\!\cdots\!65}{14652813026}a^{8}+\frac{13\!\cdots\!08}{7326406513}a^{6}-\frac{271352974428380}{7326406513}a^{4}+\frac{1249362292653}{430965089}a^{2}-\frac{17087959996}{430965089}$, $\frac{4353848775}{14652813026}a^{22}-\frac{103675494951}{7326406513}a^{20}+\frac{7994814015631}{29305626052}a^{18}-\frac{80455667889825}{29305626052}a^{16}+\frac{113538131574865}{7326406513}a^{14}-\frac{359638151460906}{7326406513}a^{12}+\frac{12\!\cdots\!07}{14652813026}a^{10}-\frac{505874325229814}{7326406513}a^{8}+\frac{209945011048416}{7326406513}a^{6}-\frac{41671338020028}{7326406513}a^{4}+\frac{3201868219253}{7326406513}a^{2}-\frac{18680553312}{7326406513}$, $\frac{13061546325}{14652813026}a^{23}+\frac{3434221409}{7326406513}a^{22}-\frac{311026484853}{7326406513}a^{21}-\frac{327426372543}{14652813026}a^{20}+\frac{23984442046893}{29305626052}a^{19}+\frac{3160568924946}{7326406513}a^{18}-\frac{241367003669475}{29305626052}a^{17}-\frac{1874892129477}{430965089}a^{16}+\frac{340614394724595}{7326406513}a^{15}+\frac{10618272345336}{430965089}a^{14}-\frac{10\!\cdots\!18}{7326406513}a^{13}-\frac{33819588969777}{430965089}a^{12}+\frac{36\!\cdots\!21}{14652813026}a^{11}+\frac{57774351960176}{430965089}a^{10}-\frac{15\!\cdots\!42}{7326406513}a^{9}-\frac{97761788242869}{861930178}a^{8}+\frac{629835033145248}{7326406513}a^{7}+\frac{351154293931296}{7326406513}a^{6}-\frac{125014014060084}{7326406513}a^{5}-\frac{70755162305780}{7326406513}a^{4}+\frac{9612931064272}{7326406513}a^{3}+\frac{5594339938104}{7326406513}a^{2}-\frac{78020879475}{7326406513}a-\frac{4612120182}{430965089}$, $\frac{33980311233}{14652813026}a^{23}+\frac{11737068017}{7326406513}a^{22}-\frac{190557007663}{1723860356}a^{21}-\frac{1118266093821}{14652813026}a^{20}+\frac{62531668797969}{29305626052}a^{19}+\frac{10783622336130}{7326406513}a^{18}-\frac{630490151356821}{29305626052}a^{17}-\frac{25549929344643}{1723860356}a^{16}+\frac{892422344931671}{7326406513}a^{15}+\frac{613539229678268}{7326406513}a^{14}-\frac{28\!\cdots\!59}{7326406513}a^{13}-\frac{19\!\cdots\!71}{7326406513}a^{12}+\frac{96\!\cdots\!69}{14652813026}a^{11}+\frac{33\!\cdots\!68}{7326406513}a^{10}-\frac{40\!\cdots\!54}{7326406513}a^{9}-\frac{55\!\cdots\!37}{14652813026}a^{8}+\frac{17\!\cdots\!76}{7326406513}a^{7}+\frac{11\!\cdots\!92}{7326406513}a^{6}-\frac{346661960395356}{7326406513}a^{5}-\frac{229681636408352}{7326406513}a^{4}+\frac{26630928957824}{7326406513}a^{3}+\frac{18037290755848}{7326406513}a^{2}-\frac{292465967781}{7326406513}a-\frac{271814766620}{7326406513}$, $\frac{33980311233}{14652813026}a^{23}-\frac{8302846608}{7326406513}a^{22}-\frac{190557007663}{1723860356}a^{21}+\frac{395419860639}{7326406513}a^{20}+\frac{62531668797969}{29305626052}a^{19}-\frac{7623053411184}{7326406513}a^{18}-\frac{630490151356821}{29305626052}a^{17}+\frac{18050360826735}{1723860356}a^{16}+\frac{892422344931671}{7326406513}a^{15}-\frac{433028599807556}{7326406513}a^{14}-\frac{28\!\cdots\!59}{7326406513}a^{13}+\frac{13\!\cdots\!62}{7326406513}a^{12}+\frac{96\!\cdots\!69}{14652813026}a^{11}-\frac{23\!\cdots\!76}{7326406513}a^{10}-\frac{40\!\cdots\!54}{7326406513}a^{9}+\frac{19\!\cdots\!82}{7326406513}a^{8}+\frac{17\!\cdots\!76}{7326406513}a^{7}-\frac{800692045883496}{7326406513}a^{6}-\frac{346661960395356}{7326406513}a^{5}+\frac{158926474102572}{7326406513}a^{4}+\frac{26630928957824}{7326406513}a^{3}-\frac{12442950817744}{7326406513}a^{2}-\frac{292465967781}{7326406513}a+\frac{193408723526}{7326406513}$, $\frac{8766857205}{14652813026}a^{23}+\frac{8302846608}{7326406513}a^{22}-\frac{834120670113}{29305626052}a^{21}-\frac{395419860639}{7326406513}a^{20}+\frac{8027226822843}{14652813026}a^{19}+\frac{7623053411184}{7326406513}a^{18}-\frac{161157944641527}{29305626052}a^{17}-\frac{18050360826735}{1723860356}a^{16}+\frac{226481556971142}{7326406513}a^{15}+\frac{433028599807556}{7326406513}a^{14}-\frac{712036131987033}{7326406513}a^{13}-\frac{13\!\cdots\!62}{7326406513}a^{12}+\frac{11\!\cdots\!59}{7326406513}a^{11}+\frac{23\!\cdots\!76}{7326406513}a^{10}-\frac{952543953094906}{7326406513}a^{9}-\frac{19\!\cdots\!82}{7326406513}a^{8}+\frac{367695181222209}{7326406513}a^{7}+\frac{800692045883496}{7326406513}a^{6}-\frac{62554568068476}{7326406513}a^{5}-\frac{158926474102572}{7326406513}a^{4}+\frac{3104946750000}{7326406513}a^{3}+\frac{12442950817744}{7326406513}a^{2}+\frac{67961387889}{7326406513}a-\frac{193408723526}{7326406513}$, $\frac{8302846608}{7326406513}a^{23}-\frac{11737068017}{7326406513}a^{22}-\frac{395419860639}{7326406513}a^{21}+\frac{1118266093821}{14652813026}a^{20}+\frac{7623053411184}{7326406513}a^{19}-\frac{10783622336130}{7326406513}a^{18}-\frac{18050360826735}{1723860356}a^{17}+\frac{25549929344643}{1723860356}a^{16}+\frac{433028599807556}{7326406513}a^{15}-\frac{613539229678268}{7326406513}a^{14}-\frac{13\!\cdots\!62}{7326406513}a^{13}+\frac{19\!\cdots\!71}{7326406513}a^{12}+\frac{23\!\cdots\!76}{7326406513}a^{11}-\frac{33\!\cdots\!68}{7326406513}a^{10}-\frac{19\!\cdots\!82}{7326406513}a^{9}+\frac{55\!\cdots\!37}{14652813026}a^{8}+\frac{800692045883496}{7326406513}a^{7}-\frac{11\!\cdots\!92}{7326406513}a^{6}-\frac{158926474102572}{7326406513}a^{5}+\frac{229681636408352}{7326406513}a^{4}+\frac{12442950817744}{7326406513}a^{3}-\frac{18037290755848}{7326406513}a^{2}-\frac{200735130039}{7326406513}a+\frac{271814766620}{7326406513}$, $\frac{8302846608}{7326406513}a^{23}-\frac{3434221409}{7326406513}a^{22}-\frac{395419860639}{7326406513}a^{21}+\frac{327426372543}{14652813026}a^{20}+\frac{7623053411184}{7326406513}a^{19}-\frac{3160568924946}{7326406513}a^{18}-\frac{18050360826735}{1723860356}a^{17}+\frac{1874892129477}{430965089}a^{16}+\frac{433028599807556}{7326406513}a^{15}-\frac{10618272345336}{430965089}a^{14}-\frac{13\!\cdots\!62}{7326406513}a^{13}+\frac{33819588969777}{430965089}a^{12}+\frac{23\!\cdots\!76}{7326406513}a^{11}-\frac{57774351960176}{430965089}a^{10}-\frac{19\!\cdots\!82}{7326406513}a^{9}+\frac{97761788242869}{861930178}a^{8}+\frac{800692045883496}{7326406513}a^{7}-\frac{351154293931296}{7326406513}a^{6}-\frac{158926474102572}{7326406513}a^{5}+\frac{70755162305780}{7326406513}a^{4}+\frac{12442950817744}{7326406513}a^{3}-\frac{5594339938104}{7326406513}a^{2}-\frac{200735130039}{7326406513}a+\frac{4612120182}{430965089}$, $\frac{8766857205}{14652813026}a^{23}-\frac{11737068017}{7326406513}a^{22}-\frac{834120670113}{29305626052}a^{21}+\frac{1118266093821}{14652813026}a^{20}+\frac{8027226822843}{14652813026}a^{19}-\frac{10783622336130}{7326406513}a^{18}-\frac{161157944641527}{29305626052}a^{17}+\frac{25549929344643}{1723860356}a^{16}+\frac{226481556971142}{7326406513}a^{15}-\frac{613539229678268}{7326406513}a^{14}-\frac{712036131987033}{7326406513}a^{13}+\frac{19\!\cdots\!71}{7326406513}a^{12}+\frac{11\!\cdots\!59}{7326406513}a^{11}-\frac{33\!\cdots\!68}{7326406513}a^{10}-\frac{952543953094906}{7326406513}a^{9}+\frac{55\!\cdots\!37}{14652813026}a^{8}+\frac{367695181222209}{7326406513}a^{7}-\frac{11\!\cdots\!92}{7326406513}a^{6}-\frac{62554568068476}{7326406513}a^{5}+\frac{229681636408352}{7326406513}a^{4}+\frac{3104946750000}{7326406513}a^{3}-\frac{18037290755848}{7326406513}a^{2}+\frac{67961387889}{7326406513}a+\frac{271814766620}{7326406513}$, $\frac{13061546325}{14652813026}a^{23}+\frac{11737068017}{7326406513}a^{22}-\frac{311026484853}{7326406513}a^{21}-\frac{1118266093821}{14652813026}a^{20}+\frac{23984442046893}{29305626052}a^{19}+\frac{10783622336130}{7326406513}a^{18}-\frac{241367003669475}{29305626052}a^{17}-\frac{25549929344643}{1723860356}a^{16}+\frac{340614394724595}{7326406513}a^{15}+\frac{613539229678268}{7326406513}a^{14}-\frac{10\!\cdots\!18}{7326406513}a^{13}-\frac{19\!\cdots\!71}{7326406513}a^{12}+\frac{36\!\cdots\!21}{14652813026}a^{11}+\frac{33\!\cdots\!68}{7326406513}a^{10}-\frac{15\!\cdots\!42}{7326406513}a^{9}-\frac{55\!\cdots\!37}{14652813026}a^{8}+\frac{629835033145248}{7326406513}a^{7}+\frac{11\!\cdots\!92}{7326406513}a^{6}-\frac{125014014060084}{7326406513}a^{5}-\frac{229681636408352}{7326406513}a^{4}+\frac{9612931064272}{7326406513}a^{3}+\frac{18037290755848}{7326406513}a^{2}-\frac{78020879475}{7326406513}a-\frac{271814766620}{7326406513}$
| 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: | \( 71277883593115.94 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{24}\cdot(2\pi)^{0}\cdot 71277883593115.94 \cdot 1}{2\cdot\sqrt{18351423083070806589199715754737431920771072}}\cr\approx \mathstrut & 0.139575702501298 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^24 - 48*x^22 + 936*x^20 - 9584*x^18 + 55620*x^16 - 184752*x^14 + 341336*x^12 - 336864*x^10 + 183060*x^8 - 54976*x^6 + 8592*x^4 - 576*x^2 + 8)
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 - 48*x^22 + 936*x^20 - 9584*x^18 + 55620*x^16 - 184752*x^14 + 341336*x^12 - 336864*x^10 + 183060*x^8 - 54976*x^6 + 8592*x^4 - 576*x^2 + 8, 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 - 48*x^22 + 936*x^20 - 9584*x^18 + 55620*x^16 - 184752*x^14 + 341336*x^12 - 336864*x^10 + 183060*x^8 - 54976*x^6 + 8592*x^4 - 576*x^2 + 8);
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 - 48*x^22 + 936*x^20 - 9584*x^18 + 55620*x^16 - 184752*x^14 + 341336*x^12 - 336864*x^10 + 183060*x^8 - 54976*x^6 + 8592*x^4 - 576*x^2 + 8);
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
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 cyclic group of order 24 |
| The 24 conjugacy class representatives for $C_{24}$ |
| Character table for $C_{24}$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{16})^+\), 6.6.3359232.1, \(\Q(\zeta_{32})^+\), 12.12.369768517790072832.1 |
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]
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 | $24$ | ${\href{/padicField/7.12.0.1}{12} }^{2}$ | $24$ | $24$ | ${\href{/padicField/17.2.0.1}{2} }^{12}$ | ${\href{/padicField/19.8.0.1}{8} }^{3}$ | ${\href{/padicField/23.12.0.1}{12} }^{2}$ | $24$ | ${\href{/padicField/31.3.0.1}{3} }^{8}$ | ${\href{/padicField/37.8.0.1}{8} }^{3}$ | ${\href{/padicField/41.12.0.1}{12} }^{2}$ | $24$ | ${\href{/padicField/47.6.0.1}{6} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }^{3}$ | $24$ |
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 $24$ | $8$ | $3$ | $93$ | |||
|
\(3\)
| Deg $24$ | $3$ | $8$ | $32$ |