Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 + 7*x^22 - 7*x^21 + 35*x^20 - 34*x^19 + 153*x^18 - 146*x^17 + 629*x^16 - 588*x^15 + 1618*x^14 - 1394*x^13 + 3557*x^12 - 2395*x^11 + 6504*x^10 - 4802*x^9 + 10534*x^8 - 8224*x^7 + 6187*x^6 - 3548*x^5 + 2534*x^4 - 378*x^3 + 56*x^2 - 8*x + 1)
gp: K = bnfinit(y^24 - y^23 + 7*y^22 - 7*y^21 + 35*y^20 - 34*y^19 + 153*y^18 - 146*y^17 + 629*y^16 - 588*y^15 + 1618*y^14 - 1394*y^13 + 3557*y^12 - 2395*y^11 + 6504*y^10 - 4802*y^9 + 10534*y^8 - 8224*y^7 + 6187*y^6 - 3548*y^5 + 2534*y^4 - 378*y^3 + 56*y^2 - 8*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - x^23 + 7*x^22 - 7*x^21 + 35*x^20 - 34*x^19 + 153*x^18 - 146*x^17 + 629*x^16 - 588*x^15 + 1618*x^14 - 1394*x^13 + 3557*x^12 - 2395*x^11 + 6504*x^10 - 4802*x^9 + 10534*x^8 - 8224*x^7 + 6187*x^6 - 3548*x^5 + 2534*x^4 - 378*x^3 + 56*x^2 - 8*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - x^23 + 7*x^22 - 7*x^21 + 35*x^20 - 34*x^19 + 153*x^18 - 146*x^17 + 629*x^16 - 588*x^15 + 1618*x^14 - 1394*x^13 + 3557*x^12 - 2395*x^11 + 6504*x^10 - 4802*x^9 + 10534*x^8 - 8224*x^7 + 6187*x^6 - 3548*x^5 + 2534*x^4 - 378*x^3 + 56*x^2 - 8*x + 1)
\( x^{24} - x^{23} + 7 x^{22} - 7 x^{21} + 35 x^{20} - 34 x^{19} + 153 x^{18} - 146 x^{17} + 629 x^{16} + \cdots + 1 \)
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: |
\(161761786626698377317203521728515625\)
\(\medspace = 3^{12}\cdot 5^{18}\cdot 7^{20}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(29.31\) | 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: | $3^{1/2}5^{3/4}7^{5/6}\approx 29.3113956234904$ | ||
| Ramified primes: |
\(3\), \(5\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $24$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(105=3\cdot 5\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{105}(64,·)$, $\chi_{105}(1,·)$, $\chi_{105}(67,·)$, $\chi_{105}(4,·)$, $\chi_{105}(79,·)$, $\chi_{105}(16,·)$, $\chi_{105}(17,·)$, $\chi_{105}(83,·)$, $\chi_{105}(22,·)$, $\chi_{105}(89,·)$, $\chi_{105}(88,·)$, $\chi_{105}(68,·)$, $\chi_{105}(26,·)$, $\chi_{105}(101,·)$, $\chi_{105}(37,·)$, $\chi_{105}(38,·)$, $\chi_{105}(104,·)$, $\chi_{105}(41,·)$, $\chi_{105}(43,·)$, $\chi_{105}(46,·)$, $\chi_{105}(47,·)$, $\chi_{105}(58,·)$, $\chi_{105}(59,·)$, $\chi_{105}(62,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{2048}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{586382566462601}a^{21}-\frac{171538548989333}{586382566462601}a^{20}-\frac{70125785132858}{586382566462601}a^{19}+\frac{213659624122068}{586382566462601}a^{18}+\frac{95502070532581}{586382566462601}a^{17}-\frac{192023246337468}{586382566462601}a^{16}-\frac{90437479961259}{586382566462601}a^{15}-\frac{27070111217202}{586382566462601}a^{14}+\frac{33815946558713}{586382566462601}a^{13}-\frac{180351711312451}{586382566462601}a^{12}+\frac{82019629063228}{586382566462601}a^{11}+\frac{214886966053931}{586382566462601}a^{10}-\frac{109698136901594}{586382566462601}a^{9}+\frac{158581104538649}{586382566462601}a^{8}+\frac{52627448885242}{586382566462601}a^{7}-\frac{205693131590540}{586382566462601}a^{6}-\frac{5661939767723}{586382566462601}a^{5}-\frac{285708907636718}{586382566462601}a^{4}-\frac{216861545151990}{586382566462601}a^{3}+\frac{120963384291680}{586382566462601}a^{2}+\frac{206237200571440}{586382566462601}a-\frac{186258710837106}{586382566462601}$, $\frac{1}{586382566462601}a^{22}-\frac{186215045205283}{586382566462601}a^{20}-\frac{15661384932351}{586382566462601}a^{19}+\frac{71136246625847}{586382566462601}a^{18}-\frac{94132030330856}{586382566462601}a^{17}+\frac{173051526941893}{586382566462601}a^{16}-\frac{276103737934819}{586382566462601}a^{15}+\frac{105342405320544}{586382566462601}a^{14}+\frac{62354549756924}{586382566462601}a^{13}+\frac{12272366125966}{586382566462601}a^{12}+\frac{195850814920183}{586382566462601}a^{11}+\frac{102355701839496}{586382566462601}a^{10}-\frac{279782116565472}{586382566462601}a^{9}+\frac{193898351814680}{586382566462601}a^{8}-\frac{64494144274257}{586382566462601}a^{7}+\frac{72569749588502}{586382566462601}a^{6}+\frac{92788436701071}{586382566462601}a^{5}-\frac{107780608046877}{586382566462601}a^{4}-\frac{241904384436198}{586382566462601}a^{3}-\frac{161289357740575}{586382566462601}a^{2}+\frac{268365935802107}{586382566462601}a-\frac{139808032505916}{586382566462601}$, $\frac{1}{586382566462601}a^{23}-\frac{179373992853381}{586382566462601}a^{20}-\frac{161398474765477}{586382566462601}a^{19}+\frac{273417314765986}{586382566462601}a^{18}+\frac{42975809566870}{586382566462601}a^{17}+\frac{211995488701823}{586382566462601}a^{16}+\frac{219490812382700}{586382566462601}a^{15}-\frac{264594525294754}{586382566462601}a^{14}-\frac{60077322863200}{586382566462601}a^{13}-\frac{135662957979370}{586382566462601}a^{12}-\frac{285027841217279}{586382566462601}a^{11}-\frac{23733207765291}{586382566462601}a^{10}-\frac{269918783075074}{586382566462601}a^{9}+\frac{237259122924358}{586382566462601}a^{8}+\frac{243057377254032}{586382566462601}a^{7}-\frac{115133424425142}{586382566462601}a^{6}-\frac{257645977686546}{586382566462601}a^{5}-\frac{34917752774359}{586382566462601}a^{4}+\frac{34716266604898}{586382566462601}a^{3}-\frac{173574082386098}{586382566462601}a^{2}-\frac{278327275127020}{586382566462601}a-\frac{57165082815132}{586382566462601}$
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
$C_{26}$, which has order $26$ (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: |
\( -\frac{87543327880547}{586382566462601} a^{23} + \frac{87683025935477}{586382566462601} a^{22} - \frac{612803295163829}{586382566462601} a^{21} + \frac{613455219420169}{586382566462601} a^{20} - \frac{3064007668486569}{586382566462601} a^{19} + \frac{2979243826028043}{586382566462601} a^{18} - \frac{13393966184659606}{586382566462601} a^{17} + \frac{12792571563981727}{586382566462601} a^{16} - \frac{55063798633488708}{586382566462601} a^{15} + \frac{51520059395520031}{586382566462601} a^{14} - \frac{141639889116674326}{586382566462601} a^{13} + \frac{122085760214284783}{586382566462601} a^{12} - \frac{311364562414467569}{586382566462601} a^{11} + \frac{209761940158527960}{586382566462601} a^{10} - \frac{569255340353525742}{586382566462601} a^{9} + \frac{420516518690863154}{586382566462601} a^{8} - \frac{922127632142534048}{586382566462601} a^{7} + \frac{719908900999969793}{586382566462601} a^{6} - \frac{541569125735784244}{586382566462601} a^{5} + \frac{308960363330819046}{586382566462601} a^{4} - \frac{221784594681567918}{586382566462601} a^{3} + \frac{33083880809898856}{586382566462601} a^{2} - \frac{4901308776871192}{586382566462601} a + \frac{700183641980291}{586382566462601} \)
(order $10$)
| 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: |
$\frac{87543327880547}{586382566462601}a^{23}-\frac{87683025935477}{586382566462601}a^{22}+\frac{612803295163829}{586382566462601}a^{21}-\frac{613455219420169}{586382566462601}a^{20}+\frac{30\!\cdots\!69}{586382566462601}a^{19}-\frac{29\!\cdots\!43}{586382566462601}a^{18}+\frac{13\!\cdots\!06}{586382566462601}a^{17}-\frac{12\!\cdots\!27}{586382566462601}a^{16}+\frac{55\!\cdots\!08}{586382566462601}a^{15}-\frac{51\!\cdots\!31}{586382566462601}a^{14}+\frac{14\!\cdots\!26}{586382566462601}a^{13}-\frac{12\!\cdots\!83}{586382566462601}a^{12}+\frac{31\!\cdots\!69}{586382566462601}a^{11}-\frac{20\!\cdots\!60}{586382566462601}a^{10}+\frac{56\!\cdots\!42}{586382566462601}a^{9}-\frac{42\!\cdots\!54}{586382566462601}a^{8}+\frac{92\!\cdots\!48}{586382566462601}a^{7}-\frac{71\!\cdots\!93}{586382566462601}a^{6}+\frac{54\!\cdots\!44}{586382566462601}a^{5}-\frac{30\!\cdots\!46}{586382566462601}a^{4}+\frac{22\!\cdots\!18}{586382566462601}a^{3}-\frac{33\!\cdots\!56}{586382566462601}a^{2}+\frac{49\!\cdots\!92}{586382566462601}a-\frac{113801075517690}{586382566462601}$, $\frac{29040181486792}{586382566462601}a^{23}-\frac{25468704605}{586382566462601}a^{22}+\frac{177871111606601}{586382566462601}a^{21}-\frac{3630022685849}{586382566462601}a^{20}+\frac{838535240431119}{586382566462601}a^{19}+\frac{3630022685849}{586382566462601}a^{18}+\frac{35\!\cdots\!73}{586382566462601}a^{17}+\frac{79860499088678}{586382566462601}a^{16}+\frac{14\!\cdots\!33}{586382566462601}a^{15}+\frac{660664128824518}{586382566462601}a^{14}+\frac{32\!\cdots\!81}{586382566462601}a^{13}+\frac{43\!\cdots\!67}{586382566462601}a^{12}+\frac{68\!\cdots\!78}{586382566462601}a^{11}+\frac{28\!\cdots\!98}{586382566462601}a^{10}+\frac{13\!\cdots\!21}{586382566462601}a^{9}+\frac{40\!\cdots\!29}{586382566462601}a^{8}+\frac{19\!\cdots\!10}{586382566462601}a^{7}+\frac{49\!\cdots\!22}{586382566462601}a^{6}-\frac{20\!\cdots\!38}{586382566462601}a^{5}+\frac{46\!\cdots\!12}{586382566462601}a^{4}-\frac{69\!\cdots\!25}{586382566462601}a^{3}+\frac{48\!\cdots\!59}{586382566462601}a^{2}-\frac{152460952805658}{586382566462601}a+\frac{21780136115094}{586382566462601}$, $\frac{70572190580254}{586382566462601}a^{23}-\frac{70115577806644}{586382566462601}a^{22}+\frac{494005334061778}{586382566462601}a^{21}-\frac{491874474451598}{586382566462601}a^{20}+\frac{24\!\cdots\!36}{586382566462601}a^{19}-\frac{23\!\cdots\!71}{586382566462601}a^{18}+\frac{10\!\cdots\!07}{586382566462601}a^{17}-\frac{10\!\cdots\!79}{586382566462601}a^{16}+\frac{44\!\cdots\!01}{586382566462601}a^{15}-\frac{41\!\cdots\!32}{586382566462601}a^{14}+\frac{11\!\cdots\!12}{586382566462601}a^{13}-\frac{98\!\cdots\!71}{586382566462601}a^{12}+\frac{25\!\cdots\!48}{586382566462601}a^{11}-\frac{16\!\cdots\!15}{586382566462601}a^{10}+\frac{45\!\cdots\!67}{586382566462601}a^{9}-\frac{33\!\cdots\!88}{586382566462601}a^{8}+\frac{74\!\cdots\!86}{586382566462601}a^{7}-\frac{58\!\cdots\!91}{586382566462601}a^{6}+\frac{43\!\cdots\!63}{586382566462601}a^{5}-\frac{25\!\cdots\!48}{586382566462601}a^{4}+\frac{17\!\cdots\!96}{586382566462601}a^{3}-\frac{26\!\cdots\!82}{586382566462601}a^{2}+\frac{39\!\cdots\!04}{586382566462601}a-\frac{565110239544577}{586382566462601}$, $\frac{174539058017743}{586382566462601}a^{23}-\frac{161066427083344}{586382566462601}a^{22}+\frac{29478515848958}{14302013816161}a^{21}-\frac{11\!\cdots\!98}{586382566462601}a^{20}+\frac{60\!\cdots\!36}{586382566462601}a^{19}-\frac{54\!\cdots\!81}{586382566462601}a^{18}+\frac{26\!\cdots\!07}{586382566462601}a^{17}-\frac{23\!\cdots\!79}{586382566462601}a^{16}+\frac{10\!\cdots\!01}{586382566462601}a^{15}-\frac{94\!\cdots\!32}{586382566462601}a^{14}+\frac{27\!\cdots\!08}{586382566462601}a^{13}-\frac{22\!\cdots\!71}{586382566462601}a^{12}+\frac{60\!\cdots\!48}{586382566462601}a^{11}-\frac{37\!\cdots\!15}{586382566462601}a^{10}+\frac{11\!\cdots\!67}{586382566462601}a^{9}-\frac{75\!\cdots\!87}{586382566462601}a^{8}+\frac{17\!\cdots\!86}{586382566462601}a^{7}-\frac{12\!\cdots\!91}{586382566462601}a^{6}+\frac{97\!\cdots\!63}{586382566462601}a^{5}-\frac{54\!\cdots\!48}{586382566462601}a^{4}+\frac{39\!\cdots\!82}{586382566462601}a^{3}-\frac{33\!\cdots\!82}{586382566462601}a^{2}+\frac{48\!\cdots\!04}{586382566462601}a-\frac{12\!\cdots\!78}{586382566462601}$, $\frac{89292053731181}{586382566462601}a^{23}-\frac{90963568461899}{586382566462601}a^{22}+\frac{624230695674536}{586382566462601}a^{21}-\frac{634757892883786}{586382566462601}a^{20}+\frac{31\!\cdots\!26}{586382566462601}a^{19}-\frac{30\!\cdots\!36}{586382566462601}a^{18}+\frac{13\!\cdots\!25}{586382566462601}a^{17}-\frac{13\!\cdots\!30}{586382566462601}a^{16}+\frac{56\!\cdots\!59}{586382566462601}a^{15}-\frac{53\!\cdots\!32}{586382566462601}a^{14}+\frac{14\!\cdots\!82}{586382566462601}a^{13}-\frac{12\!\cdots\!00}{586382566462601}a^{12}+\frac{31\!\cdots\!58}{586382566462601}a^{11}-\frac{21\!\cdots\!23}{586382566462601}a^{10}+\frac{57\!\cdots\!60}{586382566462601}a^{9}-\frac{43\!\cdots\!87}{586382566462601}a^{8}+\frac{93\!\cdots\!65}{586382566462601}a^{7}-\frac{74\!\cdots\!90}{586382566462601}a^{6}+\frac{54\!\cdots\!07}{586382566462601}a^{5}-\frac{31\!\cdots\!48}{586382566462601}a^{4}+\frac{22\!\cdots\!34}{586382566462601}a^{3}-\frac{33\!\cdots\!27}{586382566462601}a^{2}+\frac{987950945860634}{586382566462601}a-\frac{440849523415875}{586382566462601}$, $\frac{23718648344081}{586382566462601}a^{23}-\frac{23530191183095}{586382566462601}a^{22}+\frac{166030538408567}{586382566462601}a^{21}-\frac{165151071657299}{586382566462601}a^{20}+\frac{830186573622155}{586382566462601}a^{19}-\frac{802696310005865}{586382566462601}a^{18}+\frac{36\!\cdots\!10}{586382566462601}a^{17}-\frac{34\!\cdots\!53}{586382566462601}a^{16}+\frac{14\!\cdots\!20}{586382566462601}a^{15}-\frac{13\!\cdots\!34}{586382566462601}a^{14}+\frac{38\!\cdots\!02}{586382566462601}a^{13}-\frac{32\!\cdots\!61}{586382566462601}a^{12}+\frac{84\!\cdots\!39}{586382566462601}a^{11}-\frac{56\!\cdots\!16}{586382566462601}a^{10}+\frac{15\!\cdots\!40}{586382566462601}a^{9}-\frac{11\!\cdots\!70}{586382566462601}a^{8}+\frac{24\!\cdots\!64}{586382566462601}a^{7}-\frac{19\!\cdots\!91}{586382566462601}a^{6}+\frac{14\!\cdots\!56}{586382566462601}a^{5}-\frac{86\!\cdots\!99}{586382566462601}a^{4}+\frac{60\!\cdots\!90}{586382566462601}a^{3}-\frac{89\!\cdots\!00}{586382566462601}a^{2}+\frac{13\!\cdots\!24}{586382566462601}a-\frac{189969053440465}{586382566462601}$, $\frac{134396870116720}{586382566462601}a^{23}-\frac{134268412559026}{586382566462601}a^{22}+\frac{940778090817040}{586382566462601}a^{21}-\frac{940178622214468}{586382566462601}a^{20}+\frac{47\!\cdots\!50}{586382566462601}a^{19}-\frac{45\!\cdots\!49}{586382566462601}a^{18}+\frac{20\!\cdots\!03}{586382566462601}a^{17}-\frac{19\!\cdots\!53}{586382566462601}a^{16}+\frac{84\!\cdots\!89}{586382566462601}a^{15}-\frac{78\!\cdots\!29}{586382566462601}a^{14}+\frac{21\!\cdots\!36}{586382566462601}a^{13}-\frac{18\!\cdots\!93}{586382566462601}a^{12}+\frac{47\!\cdots\!78}{586382566462601}a^{11}-\frac{32\!\cdots\!59}{586382566462601}a^{10}+\frac{87\!\cdots\!69}{586382566462601}a^{9}-\frac{64\!\cdots\!72}{586382566462601}a^{8}+\frac{14\!\cdots\!70}{586382566462601}a^{7}-\frac{11\!\cdots\!93}{586382566462601}a^{6}+\frac{83\!\cdots\!51}{586382566462601}a^{5}-\frac{47\!\cdots\!95}{586382566462601}a^{4}+\frac{34\!\cdots\!24}{586382566462601}a^{3}-\frac{50\!\cdots\!38}{586382566462601}a^{2}+\frac{75\!\cdots\!72}{586382566462601}a-\frac{10\!\cdots\!03}{586382566462601}$, $\frac{7332945769128}{586382566462601}a^{23}-\frac{33924090253}{586382566462601}a^{22}+\frac{44914292835909}{586382566462601}a^{21}-\frac{916618221141}{586382566462601}a^{20}+\frac{211738809083571}{586382566462601}a^{19}+\frac{916618221141}{586382566462601}a^{18}+\frac{904584182678114}{586382566462601}a^{17}+\frac{20165600865102}{586382566462601}a^{16}+\frac{36\!\cdots\!97}{586382566462601}a^{15}+\frac{166824516247662}{586382566462601}a^{14}+\frac{81\!\cdots\!29}{586382566462601}a^{13}+\frac{11\!\cdots\!94}{586382566462601}a^{12}+\frac{17\!\cdots\!02}{586382566462601}a^{11}+\frac{72\!\cdots\!82}{586382566462601}a^{10}+\frac{33\!\cdots\!89}{586382566462601}a^{9}+\frac{10\!\cdots\!61}{586382566462601}a^{8}+\frac{48\!\cdots\!57}{586382566462601}a^{7}+\frac{12\!\cdots\!98}{586382566462601}a^{6}-\frac{52\!\cdots\!42}{586382566462601}a^{5}+\frac{11\!\cdots\!08}{586382566462601}a^{4}-\frac{17\!\cdots\!25}{586382566462601}a^{3}+\frac{10\!\cdots\!06}{586382566462601}a^{2}-\frac{38497965287922}{586382566462601}a+\frac{5499709326846}{586382566462601}$, $\frac{20283575986478}{586382566462601}a^{23}-\frac{8362706389456}{586382566462601}a^{22}+\frac{126452850694455}{586382566462601}a^{21}-\frac{56735692398896}{586382566462601}a^{20}+\frac{602444529527456}{586382566462601}a^{19}-\frac{259046975709508}{586382566462601}a^{18}+\frac{25\!\cdots\!03}{586382566462601}a^{17}-\frac{10\!\cdots\!01}{586382566462601}a^{16}+\frac{10\!\cdots\!79}{586382566462601}a^{15}-\frac{41\!\cdots\!81}{586382566462601}a^{14}+\frac{23\!\cdots\!55}{586382566462601}a^{13}-\frac{78\!\cdots\!90}{586382566462601}a^{12}+\frac{50\!\cdots\!08}{586382566462601}a^{11}-\frac{36\!\cdots\!36}{586382566462601}a^{10}+\frac{92\!\cdots\!41}{586382566462601}a^{9}-\frac{16\!\cdots\!52}{586382566462601}a^{8}+\frac{13\!\cdots\!68}{586382566462601}a^{7}-\frac{33\!\cdots\!05}{586382566462601}a^{6}-\frac{45\!\cdots\!57}{586382566462601}a^{5}+\frac{16\!\cdots\!74}{586382566462601}a^{4}-\frac{24\!\cdots\!35}{586382566462601}a^{3}+\frac{31\!\cdots\!12}{586382566462601}a^{2}-\frac{65\!\cdots\!94}{586382566462601}a+\frac{697405275536662}{586382566462601}$, $\frac{24237853443615}{586382566462601}a^{23}-\frac{13160455103162}{586382566462601}a^{22}+\frac{159642141902726}{586382566462601}a^{21}-\frac{93133698341726}{586382566462601}a^{20}+\frac{778343113701258}{586382566462601}a^{19}-\frac{443738092679245}{586382566462601}a^{18}+\frac{33\!\cdots\!95}{586382566462601}a^{17}-\frac{18\!\cdots\!12}{586382566462601}a^{16}+\frac{13\!\cdots\!06}{586382566462601}a^{15}-\frac{74\!\cdots\!52}{586382566462601}a^{14}+\frac{33\!\cdots\!34}{586382566462601}a^{13}-\frac{16\!\cdots\!27}{586382566462601}a^{12}+\frac{72\!\cdots\!64}{586382566462601}a^{11}-\frac{20\!\cdots\!36}{586382566462601}a^{10}+\frac{13\!\cdots\!62}{586382566462601}a^{9}-\frac{47\!\cdots\!49}{586382566462601}a^{8}+\frac{20\!\cdots\!76}{586382566462601}a^{7}-\frac{87\!\cdots\!12}{586382566462601}a^{6}+\frac{70\!\cdots\!68}{586382566462601}a^{5}-\frac{26\!\cdots\!04}{586382566462601}a^{4}+\frac{30\!\cdots\!57}{586382566462601}a^{3}+\frac{13\!\cdots\!10}{586382566462601}a^{2}+\frac{669446910244573}{586382566462601}a-\frac{8111936636020}{586382566462601}$, $\frac{57692459902251}{586382566462601}a^{23}-\frac{87693376777174}{586382566462601}a^{22}+\frac{429966728796766}{586382566462601}a^{21}-\frac{609723860922882}{586382566462601}a^{20}+\frac{22\!\cdots\!72}{586382566462601}a^{19}-\frac{29\!\cdots\!30}{586382566462601}a^{18}+\frac{97\!\cdots\!48}{586382566462601}a^{17}-\frac{12\!\cdots\!41}{586382566462601}a^{16}+\frac{40\!\cdots\!29}{586382566462601}a^{15}-\frac{52\!\cdots\!65}{586382566462601}a^{14}+\frac{10\!\cdots\!23}{586382566462601}a^{13}-\frac{12\!\cdots\!36}{586382566462601}a^{12}+\frac{24\!\cdots\!55}{586382566462601}a^{11}-\frac{23\!\cdots\!34}{586382566462601}a^{10}+\frac{43\!\cdots\!19}{586382566462601}a^{9}-\frac{46\!\cdots\!81}{586382566462601}a^{8}+\frac{72\!\cdots\!36}{586382566462601}a^{7}-\frac{77\!\cdots\!79}{586382566462601}a^{6}+\frac{56\!\cdots\!38}{586382566462601}a^{5}-\frac{35\!\cdots\!02}{586382566462601}a^{4}+\frac{22\!\cdots\!93}{586382566462601}a^{3}-\frac{84\!\cdots\!99}{586382566462601}a^{2}+\frac{50\!\cdots\!46}{586382566462601}a-\frac{136189226501412}{586382566462601}$
| 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: | \( 9216624.607005743 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 9216624.607005743 \cdot 26}{10\cdot\sqrt{161761786626698377317203521728515625}}\cr\approx \mathstrut & 0.225561876967671 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 + 7*x^22 - 7*x^21 + 35*x^20 - 34*x^19 + 153*x^18 - 146*x^17 + 629*x^16 - 588*x^15 + 1618*x^14 - 1394*x^13 + 3557*x^12 - 2395*x^11 + 6504*x^10 - 4802*x^9 + 10534*x^8 - 8224*x^7 + 6187*x^6 - 3548*x^5 + 2534*x^4 - 378*x^3 + 56*x^2 - 8*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 - x^23 + 7*x^22 - 7*x^21 + 35*x^20 - 34*x^19 + 153*x^18 - 146*x^17 + 629*x^16 - 588*x^15 + 1618*x^14 - 1394*x^13 + 3557*x^12 - 2395*x^11 + 6504*x^10 - 4802*x^9 + 10534*x^8 - 8224*x^7 + 6187*x^6 - 3548*x^5 + 2534*x^4 - 378*x^3 + 56*x^2 - 8*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 - x^23 + 7*x^22 - 7*x^21 + 35*x^20 - 34*x^19 + 153*x^18 - 146*x^17 + 629*x^16 - 588*x^15 + 1618*x^14 - 1394*x^13 + 3557*x^12 - 2395*x^11 + 6504*x^10 - 4802*x^9 + 10534*x^8 - 8224*x^7 + 6187*x^6 - 3548*x^5 + 2534*x^4 - 378*x^3 + 56*x^2 - 8*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 - x^23 + 7*x^22 - 7*x^21 + 35*x^20 - 34*x^19 + 153*x^18 - 146*x^17 + 629*x^16 - 588*x^15 + 1618*x^14 - 1394*x^13 + 3557*x^12 - 2395*x^11 + 6504*x^10 - 4802*x^9 + 10534*x^8 - 8224*x^7 + 6187*x^6 - 3548*x^5 + 2534*x^4 - 378*x^3 + 56*x^2 - 8*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
$C_2\times C_{12}$ (as 24T2):
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)
| An abelian group of order 24 |
| The 24 conjugacy class representatives for $C_2\times C_{12}$ |
| Character table for $C_2\times C_{12}$ 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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.12.0.1}{12} }^{2}$ | R | R | R | ${\href{/padicField/11.6.0.1}{6} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{6}$ | ${\href{/padicField/17.12.0.1}{12} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{4}$ | ${\href{/padicField/23.12.0.1}{12} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{12}$ | ${\href{/padicField/31.6.0.1}{6} }^{4}$ | ${\href{/padicField/37.12.0.1}{12} }^{2}$ | ${\href{/padicField/41.1.0.1}{1} }^{24}$ | ${\href{/padicField/43.4.0.1}{4} }^{6}$ | ${\href{/padicField/47.12.0.1}{12} }^{2}$ | ${\href{/padicField/53.12.0.1}{12} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $24$ | $2$ | $12$ | $12$ | |||
|
\(5\)
| 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 | $x^{12} + 12 x^{10} + 12 x^{9} + 69 x^{8} + 108 x^{7} + 42 x^{6} - 396 x^{5} + 840 x^{4} + 252 x^{3} + 1476 x^{2} + 684 x + 1601$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ | |
|
\(7\)
| Deg $24$ | $6$ | $4$ | $20$ |