LMFDB - Number field 24.0.161761786626698377317203521728515625.2

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 + 5*x^22 - 5*x^21 + 20*x^20 - 19*x^19 + 74*x^18 - 99*x^17 + 299*x^16 - 380*x^15 + 1106*x^14 - 1331*x^13 + 3936*x^12 - 4456*x^11 + 5996*x^10 - 6745*x^9 + 8969*x^8 - 7509*x^7 + 10409*x^6 + 2176*x^5 + 455*x^4 + 95*x^3 + 20*x^2 + 4*x + 1)

gp: K = bnfinit(y^24 - y^23 + 5*y^22 - 5*y^21 + 20*y^20 - 19*y^19 + 74*y^18 - 99*y^17 + 299*y^16 - 380*y^15 + 1106*y^14 - 1331*y^13 + 3936*y^12 - 4456*y^11 + 5996*y^10 - 6745*y^9 + 8969*y^8 - 7509*y^7 + 10409*y^6 + 2176*y^5 + 455*y^4 + 95*y^3 + 20*y^2 + 4*y + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - x^23 + 5*x^22 - 5*x^21 + 20*x^20 - 19*x^19 + 74*x^18 - 99*x^17 + 299*x^16 - 380*x^15 + 1106*x^14 - 1331*x^13 + 3936*x^12 - 4456*x^11 + 5996*x^10 - 6745*x^9 + 8969*x^8 - 7509*x^7 + 10409*x^6 + 2176*x^5 + 455*x^4 + 95*x^3 + 20*x^2 + 4*x + 1);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - x^23 + 5*x^22 - 5*x^21 + 20*x^20 - 19*x^19 + 74*x^18 - 99*x^17 + 299*x^16 - 380*x^15 + 1106*x^14 - 1331*x^13 + 3936*x^12 - 4456*x^11 + 5996*x^10 - 6745*x^9 + 8969*x^8 - 7509*x^7 + 10409*x^6 + 2176*x^5 + 455*x^4 + 95*x^3 + 20*x^2 + 4*x + 1)

$ x^{24} - x^{23} + 5 x^{22} - 5 x^{21} + 20 x^{20} - 19 x^{19} + 74 x^{18} - 99 x^{17} + 299 x^{16} + \cdots + 1 $ x^24 - x^23 + 5*x^22 - 5*x^21 + 20*x^20 - 19*x^19 + 74*x^18 - 99*x^17 + 299*x^16 - 380*x^15 + 1106*x^14 - 1331*x^13 + 3936*x^12 - 4456*x^11 + 5996*x^10 - 6745*x^9 + 8969*x^8 - 7509*x^7 + 10409*x^6 + 2176*x^5 + 455*x^4 + 95*x^3 + 20*x^2 + 4*x + 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:	$161761786626698377317203521728515625$ 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:	$3$, $5$, $7$ 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$$\chi_{105}(64,·)$, $\chi_{105}(1,·)$, $\chi_{105}(2,·)$, $\chi_{105}(4,·)$, $\chi_{105}(8,·)$, $\chi_{105}(76,·)$, $\chi_{105}(79,·)$, $\chi_{105}(16,·)$, $\chi_{105}(17,·)$, $\chi_{105}(19,·)$, $\chi_{105}(23,·)$, $\chi_{105}(68,·)$, $\chi_{105}(92,·)$, $\chi_{105}(94,·)$, $\chi_{105}(31,·)$, $\chi_{105}(32,·)$, $\chi_{105}(34,·)$, $\chi_{105}(38,·)$, $\chi_{105}(46,·)$, $\chi_{105}(47,·)$, $\chi_{105}(83,·)$, $\chi_{105}(53,·)$, $\chi_{105}(61,·)$, $\chi_{105}(62,·)$$\rbrace$
This is a CM field.
Reflex fields:	unavailable$^{2048}$

Integral basis (with respect to field generator $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2146586119}a^{19}-\frac{80671238}{2146586119}a^{18}-\frac{363514156}{2146586119}a^{17}+\frac{40829208}{2146586119}a^{16}+\frac{329015334}{2146586119}a^{15}+\frac{607502231}{2146586119}a^{14}-\frac{911196026}{2146586119}a^{13}-\frac{744303332}{2146586119}a^{12}+\frac{227825652}{2146586119}a^{11}+\frac{10477406}{2146586119}a^{10}+\frac{981394019}{2146586119}a^{9}+\frac{716121545}{2146586119}a^{8}+\frac{876952384}{2146586119}a^{7}-\frac{537125891}{2146586119}a^{6}+\frac{479576926}{2146586119}a^{5}+\frac{489657714}{2146586119}a^{4}-\frac{256694009}{2146586119}a^{3}+\frac{377586222}{2146586119}a^{2}+\frac{74691672}{2146586119}a-\frac{80780534}{2146586119}$, $\frac{1}{2146586119}a^{20}-\frac{935272120}{2146586119}a^{18}+\frac{674003709}{2146586119}a^{17}-\frac{121919952}{2146586119}a^{16}-\frac{923153692}{2146586119}a^{15}-\frac{226411398}{2146586119}a^{14}+\frac{744303298}{2146586119}a^{13}+\frac{151546479}{2146586119}a^{12}-\frac{642200301}{2146586119}a^{11}-\frac{576258079}{2146586119}a^{10}+\frac{15925612}{2146586119}a^{9}-\frac{748133373}{2146586119}a^{8}-\frac{850996194}{2146586119}a^{7}-\frac{510269076}{2146586119}a^{6}-\frac{617694871}{2146586119}a^{5}+\frac{572504157}{2146586119}a^{4}-\frac{907932223}{2146586119}a^{3}-\frac{909147676}{2146586119}a^{2}+\frac{185130283}{2146586119}a+\frac{560390440}{2146586119}$, $\frac{1}{2146586119}a^{21}-\frac{1067568172}{2146586119}a^{14}+\frac{991307695}{2146586119}a^{7}+\frac{401941670}{2146586119}$, $\frac{1}{2146586119}a^{22}-\frac{1067568172}{2146586119}a^{15}+\frac{991307695}{2146586119}a^{8}+\frac{401941670}{2146586119}a$, $\frac{1}{2146586119}a^{23}-\frac{1067568172}{2146586119}a^{16}+\frac{991307695}{2146586119}a^{9}+\frac{401941670}{2146586119}a^{2}$ 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, 1/2146586119*a^19 - 80671238/2146586119*a^18 - 363514156/2146586119*a^17 + 40829208/2146586119*a^16 + 329015334/2146586119*a^15 + 607502231/2146586119*a^14 - 911196026/2146586119*a^13 - 744303332/2146586119*a^12 + 227825652/2146586119*a^11 + 10477406/2146586119*a^10 + 981394019/2146586119*a^9 + 716121545/2146586119*a^8 + 876952384/2146586119*a^7 - 537125891/2146586119*a^6 + 479576926/2146586119*a^5 + 489657714/2146586119*a^4 - 256694009/2146586119*a^3 + 377586222/2146586119*a^2 + 74691672/2146586119*a - 80780534/2146586119, 1/2146586119*a^20 - 935272120/2146586119*a^18 + 674003709/2146586119*a^17 - 121919952/2146586119*a^16 - 923153692/2146586119*a^15 - 226411398/2146586119*a^14 + 744303298/2146586119*a^13 + 151546479/2146586119*a^12 - 642200301/2146586119*a^11 - 576258079/2146586119*a^10 + 15925612/2146586119*a^9 - 748133373/2146586119*a^8 - 850996194/2146586119*a^7 - 510269076/2146586119*a^6 - 617694871/2146586119*a^5 + 572504157/2146586119*a^4 - 907932223/2146586119*a^3 - 909147676/2146586119*a^2 + 185130283/2146586119*a + 560390440/2146586119, 1/2146586119*a^21 - 1067568172/2146586119*a^14 + 991307695/2146586119*a^7 + 401941670/2146586119, 1/2146586119*a^22 - 1067568172/2146586119*a^15 + 991307695/2146586119*a^8 + 401941670/2146586119*a, 1/2146586119*a^23 - 1067568172/2146586119*a^16 + 991307695/2146586119*a^9 + 401941670/2146586119*a^2

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{375299640}{2146586119} a^{23} + \frac{469124550}{2146586119} a^{22} - \frac{1970323110}{2146586119} a^{21} + \frac{2345622750}{2146586119} a^{20} - \frac{7975180155}{2146586119} a^{19} + \frac{9007191360}{2146586119} a^{18} - \frac{29554846650}{2146586119} a^{17} + \frac{44097707700}{2146586119} a^{16} - \frac{121503258450}{2146586119} a^{15} + \frac{170667511290}{2146586119} a^{14} - \frac{450734867640}{2146586119} a^{13} + \frac{603295786795}{2146586119} a^{12} - \frac{1602060338250}{2146586119} a^{11} + \frac{2041630041600}{2146586119} a^{10} - \frac{2668380440400}{2146586119} a^{9} + \frac{3093970232160}{2146586119} a^{8} - \frac{3998911489110}{2146586119} a^{7} + \frac{126194503950}{74020211} a^{6} - \frac{4610521190949}{2146586119} a^{5} + \frac{159971471550}{2146586119} a^{4} + \frac{33401667960}{2146586119} a^{3} + \frac{7036868250}{2146586119} a^{2} + \frac{1407373650}{2146586119} a + \frac{375299640}{2146586119} $ -(375299640)/(2146586119)a^(23) + (469124550)/(2146586119)a^(22) - (1970323110)/(2146586119)a^(21) + (2345622750)/(2146586119)a^(20) - (7975180155)/(2146586119)a^(19) + (9007191360)/(2146586119)a^(18) - (29554846650)/(2146586119)a^(17) + (44097707700)/(2146586119)a^(16) - (121503258450)/(2146586119)a^(15) + (170667511290)/(2146586119)a^(14) - (450734867640)/(2146586119)a^(13) + (603295786795)/(2146586119)a^(12) - (1602060338250)/(2146586119)a^(11) + (2041630041600)/(2146586119)a^(10) - (2668380440400)/(2146586119)a^(9) + (3093970232160)/(2146586119)a^(8) - (3998911489110)/(2146586119)a^(7) + (126194503950)/(74020211)a^(6) - (4610521190949)/(2146586119)a^(5) + (159971471550)/(2146586119)a^(4) + (33401667960)/(2146586119)a^(3) + (7036868250)/(2146586119)a^(2) + (1407373650)/(2146586119)*a + (375299640)/(2146586119) (order $14$)	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:	$\frac{231865976}{2146586119}a^{23}-\frac{289832470}{2146586119}a^{22}+\frac{1217296374}{2146586119}a^{21}-\frac{1449162350}{2146586119}a^{20}+\frac{4927024693}{2146586119}a^{19}-\frac{5564783424}{2146586119}a^{18}+\frac{18259445610}{2146586119}a^{17}-\frac{27244252180}{2146586119}a^{16}+\frac{75066609730}{2146586119}a^{15}-\frac{105441052586}{2146586119}a^{14}+\frac{278471037176}{2146586119}a^{13}-\frac{372719573107}{2146586119}a^{12}+\frac{989777885050}{2146586119}a^{11}-\frac{1261350909440}{2146586119}a^{10}+\frac{1648567089360}{2146586119}a^{9}-\frac{1911503106144}{2146586119}a^{8}+\frac{2470589940774}{2146586119}a^{7}-\frac{77964934430}{74020211}a^{6}+\frac{2849625874868}{2146586119}a^{5}-\frac{98832872270}{2146586119}a^{4}-\frac{20636071864}{2146586119}a^{3}-\frac{4347487050}{2146586119}a^{2}-\frac{869497410}{2146586119}a-\frac{231865976}{2146586119}$, $\frac{73451185}{2146586119}a^{23}+\frac{20373725}{2146586119}a^{22}+\frac{273431015}{2146586119}a^{21}+\frac{101922056}{2146586119}a^{20}+\frac{999836345}{2146586119}a^{19}+\frac{480925685}{2146586119}a^{18}+\frac{3652714400}{2146586119}a^{17}-\frac{328623975}{2146586119}a^{16}+\frac{12673238225}{2146586119}a^{15}+\frac{142197790}{2146586119}a^{14}+\frac{45581486865}{2146586119}a^{13}+\frac{6008438720}{2146586119}a^{12}+\frac{164222908950}{2146586119}a^{11}+\frac{41996365400}{2146586119}a^{10}+\frac{22329506300}{2146586119}a^{9}+\frac{67145917535}{2146586119}a^{8}+\frac{25934660315}{2146586119}a^{7}+\frac{289519634845}{2146586119}a^{6}+\frac{60526146476}{2146586119}a^{5}+\frac{1136453266750}{2146586119}a^{4}+\frac{237583293335}{2146586119}a^{3}+\frac{49668196625}{2146586119}a^{2}+\frac{10382390150}{2146586119}a+\frac{23716821}{2146586119}$, $\frac{453702664}{2146586119}a^{23}-\frac{469124550}{2146586119}a^{22}+\frac{2264334450}{2146586119}a^{21}-\frac{2326021994}{2146586119}a^{20}+\frac{9053221735}{2146586119}a^{19}-\frac{8830854485}{2146586119}a^{18}+\frac{33474997850}{2146586119}a^{17}-\frac{45685368936}{2146586119}a^{16}+\frac{135733407306}{2146586119}a^{15}-\frac{175077681390}{2146586119}a^{14}+\frac{501794837020}{2146586119}a^{13}-\frac{613488179915}{2146586119}a^{12}+\frac{1784624432840}{2146586119}a^{11}-\frac{2056311007844}{2146586119}a^{10}+\frac{2711972521744}{2146586119}a^{9}-\frac{3065353128400}{2146586119}a^{8}+\frac{4055753681510}{2146586119}a^{7}-\frac{3412965100290}{2146586119}a^{6}+\frac{4662090779985}{2146586119}a^{5}+\frac{974608985625}{2146586119}a^{4}-\frac{31147581020}{2146586119}a^{3}-\frac{226429314}{74020211}a^{2}-\frac{1309369870}{2146586119}a-\frac{355698884}{2146586119}$, $\frac{277925034}{2146586119}a^{23}-\frac{277502460}{2146586119}a^{22}+\frac{1386883689}{2146586119}a^{21}-\frac{1386874315}{2146586119}a^{20}+\frac{5547173300}{2146586119}a^{19}-\frac{5269814635}{2146586119}a^{18}+\frac{20524541210}{2146586119}a^{17}-\frac{27478032097}{2146586119}a^{16}+\frac{82935819409}{2146586119}a^{15}-\frac{105399813329}{2146586119}a^{14}+\frac{306762646569}{2146586119}a^{13}-\frac{369164383115}{2146586119}a^{12}+\frac{26626431840}{52355759}a^{11}-\frac{1235910211240}{2146586119}a^{10}+\frac{1658658109750}{2146586119}a^{9}-\frac{1869676379636}{2146586119}a^{8}+\frac{2486973085376}{2146586119}a^{7}-\frac{2082082410697}{2146586119}a^{6}+\frac{2887026343985}{2146586119}a^{5}+\frac{603532455040}{2146586119}a^{4}+\frac{126198192575}{2146586119}a^{3}-\frac{4005942330}{2146586119}a^{2}+\frac{9469161963}{2146586119}a+\frac{1929369060}{2146586119}$, $\frac{43883946}{2146586119}a^{23}+\frac{20373725}{2146586119}a^{22}+\frac{161453236}{2146586119}a^{21}+\frac{94462895}{2146586119}a^{20}+\frac{589582490}{2146586119}a^{19}+\frac{414015771}{2146586119}a^{18}+\frac{2160882200}{2146586119}a^{17}+\frac{266758502}{2146586119}a^{16}+\frac{7257887339}{2146586119}a^{15}+\frac{1824029644}{2146586119}a^{14}+\frac{26150372460}{2146586119}a^{13}+\frac{9887202440}{2146586119}a^{12}+\frac{94737261582}{2146586119}a^{11}+\frac{47583276989}{2146586119}a^{10}+\frac{3624494445}{2146586119}a^{9}+\frac{56255542475}{2146586119}a^{8}+\frac{4959874424}{2146586119}a^{7}+\frac{195646093660}{2146586119}a^{6}+\frac{40901093885}{2146586119}a^{5}+\frac{703097305375}{2146586119}a^{4}+\frac{236725489820}{2146586119}a^{3}+\frac{30728735085}{2146586119}a^{2}+\frac{10345094345}{2146586119}a+\frac{1342909379}{2146586119}$, $\frac{1402143}{2146586119}a^{23}-\frac{7010715}{2146586119}a^{22}+\frac{7010715}{2146586119}a^{21}-\frac{28042860}{2146586119}a^{20}+\frac{26640717}{2146586119}a^{19}-\frac{103758582}{2146586119}a^{18}+\frac{90954069}{2146586119}a^{17}-\frac{419240757}{2146586119}a^{16}+\frac{532814340}{2146586119}a^{15}-\frac{1550770158}{2146586119}a^{14}+\frac{1866252333}{2146586119}a^{13}-\frac{134605728}{52355759}a^{12}+\frac{6247949208}{2146586119}a^{11}-\frac{19291978399}{2146586119}a^{10}+\frac{9457454535}{2146586119}a^{9}-\frac{12575820567}{2146586119}a^{8}+\frac{10528691787}{2146586119}a^{7}-\frac{14594906487}{2146586119}a^{6}-\frac{3051063168}{2146586119}a^{5}-\frac{637975065}{2146586119}a^{4}-\frac{79603676271}{2146586119}a^{3}-\frac{28042860}{2146586119}a^{2}-\frac{5608572}{2146586119}a-\frac{1402143}{2146586119}$, $\frac{453702664}{2146586119}a^{23}-\frac{567128330}{2146586119}a^{22}+\frac{2381938986}{2146586119}a^{21}-\frac{2835641650}{2146586119}a^{20}+\frac{9641174486}{2146586119}a^{19}-\frac{10888863936}{2146586119}a^{18}+\frac{35729084790}{2146586119}a^{17}-\frac{53310063020}{2146586119}a^{16}+\frac{146886237470}{2146586119}a^{15}-\frac{206321286454}{2146586119}a^{14}+\frac{544896899464}{2146586119}a^{13}-\frac{729325994669}{2146586119}a^{12}+\frac{1936743246950}{2146586119}a^{11}-\frac{2468142492160}{2146586119}a^{10}+\frac{3225825941040}{2146586119}a^{9}-\frac{3740324762016}{2146586119}a^{8}+\frac{4834315310586}{2146586119}a^{7}-\frac{152557520770}{74020211}a^{6}+\frac{5574496667900}{2146586119}a^{5}-\frac{193390760530}{2146586119}a^{4}-\frac{40379537096}{2146586119}a^{3}-\frac{8506924950}{2146586119}a^{2}-\frac{1701384990}{2146586119}a-\frac{453702664}{2146586119}$, $\frac{622021133}{2146586119}a^{23}-\frac{542728904}{2146586119}a^{22}+\frac{3006880641}{2146586119}a^{21}-\frac{2693287293}{2146586119}a^{20}+\frac{11929556280}{2146586119}a^{19}-\frac{10132602090}{2146586119}a^{18}+\frac{44070755590}{2146586119}a^{17}-\frac{55338186896}{2146586119}a^{16}+\frac{176465429745}{2146586119}a^{15}-\frac{210589391850}{2146586119}a^{14}+\frac{651148289840}{2146586119}a^{13}-\frac{732360696405}{2146586119}a^{12}+\frac{2318142187550}{2146586119}a^{11}-\frac{2432398441404}{2146586119}a^{10}+\frac{3288764078233}{2146586119}a^{9}-\frac{3629895436805}{2146586119}a^{8}+\frac{4923256935836}{2146586119}a^{7}-\frac{3827472703634}{2146586119}a^{6}+\frac{5699240414651}{2146586119}a^{5}+\frac{2315225256535}{2146586119}a^{4}+\frac{249126046365}{2146586119}a^{3}-\frac{7368266322}{2146586119}a^{2}+\frac{18622103646}{2146586119}a+\frac{3793914080}{2146586119}$, $\frac{622318097}{2146586119}a^{23}-\frac{600542229}{2146586119}a^{22}+\frac{3085456706}{2146586119}a^{21}-\frac{2997075014}{2146586119}a^{20}+\frac{12314346020}{2146586119}a^{19}-\frac{11360345828}{2146586119}a^{18}+\frac{45543766546}{2146586119}a^{17}-\frac{59889940821}{2146586119}a^{16}+\frac{4477276719}{52355759}a^{15}-\frac{229445738686}{2146586119}a^{14}+\frac{678569211351}{2146586119}a^{13}-\frac{802191174959}{2146586119}a^{12}+\frac{2415114388539}{2146586119}a^{11}-\frac{2680149310216}{2146586119}a^{10}+\frac{3615392307475}{2146586119}a^{9}-\frac{4042600750925}{2146586119}a^{8}+\frac{5402418825109}{2146586119}a^{7}-\frac{4440834500898}{2146586119}a^{6}+\frac{6265562466909}{2146586119}a^{5}+\frac{1624223557610}{2146586119}a^{4}+\frac{273881308115}{2146586119}a^{3}+\frac{70925108695}{2146586119}a^{2}+\frac{12038435620}{2146586119}a-\frac{465008485}{2146586119}$, $\frac{42122034}{2146586119}a^{23}-\frac{93915274}{2146586119}a^{22}+\frac{274491574}{2146586119}a^{21}-\frac{482164724}{2146586119}a^{20}+\frac{1162742905}{2146586119}a^{19}-\frac{1899539727}{2146586119}a^{18}+\frac{4347569540}{2146586119}a^{17}-\frac{8244693270}{2146586119}a^{16}+\frac{18635598825}{2146586119}a^{15}-\frac{32735462909}{2146586119}a^{14}+\frac{69965919710}{2146586119}a^{13}-\frac{118132723330}{2146586119}a^{12}+\frac{248429023197}{2146586119}a^{11}-\frac{408363749579}{2146586119}a^{10}+\frac{532100664914}{2146586119}a^{9}-\frac{651139195681}{2146586119}a^{8}+\frac{19603133602}{52355759}a^{7}-\frac{21154815294}{52355759}a^{6}+\frac{942480030351}{2146586119}a^{5}-\frac{546636624521}{2146586119}a^{4}+\frac{41197936615}{2146586119}a^{3}-\frac{1666012516}{2146586119}a^{2}+\frac{5284849193}{2146586119}a+\frac{1088431761}{2146586119}$, $\frac{1059259537}{2146586119}a^{23}-\frac{1147125480}{2146586119}a^{22}+\frac{5354407974}{2146586119}a^{21}-\frac{5698107934}{2146586119}a^{20}+\frac{21467436752}{2146586119}a^{19}-\frac{21695514938}{2146586119}a^{18}+\frac{79419659210}{2146586119}a^{17}-\frac{110648046919}{2146586119}a^{16}+\frac{323063092005}{2146586119}a^{15}-\frac{425271535226}{2146586119}a^{14}+\frac{1195250147496}{2146586119}a^{13}-\frac{1493411517203}{2146586119}a^{12}+\frac{4250277704866}{2146586119}a^{11}-\frac{122382982064}{52355759}a^{10}+\frac{6615205792705}{2146586119}a^{9}-\frac{7508357838152}{2146586119}a^{8}+\frac{9879484598534}{2146586119}a^{7}-\frac{8494856133910}{2146586119}a^{6}+\frac{11368091910132}{2146586119}a^{5}+\frac{1681950463875}{2146586119}a^{4}-\frac{77527962984}{2146586119}a^{3}+\frac{73397097730}{2146586119}a^{2}+\frac{21845287393}{2146586119}a-\frac{881576200}{2146586119}$ 231865976/2146586119a^23 - 289832470/2146586119a^22 + 1217296374/2146586119a^21 - 1449162350/2146586119a^20 + 4927024693/2146586119a^19 - 5564783424/2146586119a^18 + 18259445610/2146586119a^17 - 27244252180/2146586119a^16 + 75066609730/2146586119a^15 - 105441052586/2146586119a^14 + 278471037176/2146586119a^13 - 372719573107/2146586119a^12 + 989777885050/2146586119a^11 - 1261350909440/2146586119a^10 + 1648567089360/2146586119a^9 - 1911503106144/2146586119a^8 + 2470589940774/2146586119a^7 - 77964934430/74020211a^6 + 2849625874868/2146586119a^5 - 98832872270/2146586119a^4 - 20636071864/2146586119a^3 - 4347487050/2146586119a^2 - 869497410/2146586119a - 231865976/2146586119, 73451185/2146586119a^23 + 20373725/2146586119a^22 + 273431015/2146586119a^21 + 101922056/2146586119a^20 + 999836345/2146586119a^19 + 480925685/2146586119a^18 + 3652714400/2146586119a^17 - 328623975/2146586119a^16 + 12673238225/2146586119a^15 + 142197790/2146586119a^14 + 45581486865/2146586119a^13 + 6008438720/2146586119a^12 + 164222908950/2146586119a^11 + 41996365400/2146586119a^10 + 22329506300/2146586119a^9 + 67145917535/2146586119a^8 + 25934660315/2146586119a^7 + 289519634845/2146586119a^6 + 60526146476/2146586119a^5 + 1136453266750/2146586119a^4 + 237583293335/2146586119a^3 + 49668196625/2146586119a^2 + 10382390150/2146586119a + 23716821/2146586119, 453702664/2146586119a^23 - 469124550/2146586119a^22 + 2264334450/2146586119a^21 - 2326021994/2146586119a^20 + 9053221735/2146586119a^19 - 8830854485/2146586119a^18 + 33474997850/2146586119a^17 - 45685368936/2146586119a^16 + 135733407306/2146586119a^15 - 175077681390/2146586119a^14 + 501794837020/2146586119a^13 - 613488179915/2146586119a^12 + 1784624432840/2146586119a^11 - 2056311007844/2146586119a^10 + 2711972521744/2146586119a^9 - 3065353128400/2146586119a^8 + 4055753681510/2146586119a^7 - 3412965100290/2146586119a^6 + 4662090779985/2146586119a^5 + 974608985625/2146586119a^4 - 31147581020/2146586119a^3 - 226429314/74020211a^2 - 1309369870/2146586119a - 355698884/2146586119, 277925034/2146586119a^23 - 277502460/2146586119a^22 + 1386883689/2146586119a^21 - 1386874315/2146586119a^20 + 5547173300/2146586119a^19 - 5269814635/2146586119a^18 + 20524541210/2146586119a^17 - 27478032097/2146586119a^16 + 82935819409/2146586119a^15 - 105399813329/2146586119a^14 + 306762646569/2146586119a^13 - 369164383115/2146586119a^12 + 26626431840/52355759a^11 - 1235910211240/2146586119a^10 + 1658658109750/2146586119a^9 - 1869676379636/2146586119a^8 + 2486973085376/2146586119a^7 - 2082082410697/2146586119a^6 + 2887026343985/2146586119a^5 + 603532455040/2146586119a^4 + 126198192575/2146586119a^3 - 4005942330/2146586119a^2 + 9469161963/2146586119a + 1929369060/2146586119, 43883946/2146586119a^23 + 20373725/2146586119a^22 + 161453236/2146586119a^21 + 94462895/2146586119a^20 + 589582490/2146586119a^19 + 414015771/2146586119a^18 + 2160882200/2146586119a^17 + 266758502/2146586119a^16 + 7257887339/2146586119a^15 + 1824029644/2146586119a^14 + 26150372460/2146586119a^13 + 9887202440/2146586119a^12 + 94737261582/2146586119a^11 + 47583276989/2146586119a^10 + 3624494445/2146586119a^9 + 56255542475/2146586119a^8 + 4959874424/2146586119a^7 + 195646093660/2146586119a^6 + 40901093885/2146586119a^5 + 703097305375/2146586119a^4 + 236725489820/2146586119a^3 + 30728735085/2146586119a^2 + 10345094345/2146586119a + 1342909379/2146586119, 1402143/2146586119a^23 - 7010715/2146586119a^22 + 7010715/2146586119a^21 - 28042860/2146586119a^20 + 26640717/2146586119a^19 - 103758582/2146586119a^18 + 90954069/2146586119a^17 - 419240757/2146586119a^16 + 532814340/2146586119a^15 - 1550770158/2146586119a^14 + 1866252333/2146586119a^13 - 134605728/52355759a^12 + 6247949208/2146586119a^11 - 19291978399/2146586119a^10 + 9457454535/2146586119a^9 - 12575820567/2146586119a^8 + 10528691787/2146586119a^7 - 14594906487/2146586119a^6 - 3051063168/2146586119a^5 - 637975065/2146586119a^4 - 79603676271/2146586119a^3 - 28042860/2146586119a^2 - 5608572/2146586119a - 1402143/2146586119, 453702664/2146586119a^23 - 567128330/2146586119a^22 + 2381938986/2146586119a^21 - 2835641650/2146586119a^20 + 9641174486/2146586119a^19 - 10888863936/2146586119a^18 + 35729084790/2146586119a^17 - 53310063020/2146586119a^16 + 146886237470/2146586119a^15 - 206321286454/2146586119a^14 + 544896899464/2146586119a^13 - 729325994669/2146586119a^12 + 1936743246950/2146586119a^11 - 2468142492160/2146586119a^10 + 3225825941040/2146586119a^9 - 3740324762016/2146586119a^8 + 4834315310586/2146586119a^7 - 152557520770/74020211a^6 + 5574496667900/2146586119a^5 - 193390760530/2146586119a^4 - 40379537096/2146586119a^3 - 8506924950/2146586119a^2 - 1701384990/2146586119a - 453702664/2146586119, 622021133/2146586119a^23 - 542728904/2146586119a^22 + 3006880641/2146586119a^21 - 2693287293/2146586119a^20 + 11929556280/2146586119a^19 - 10132602090/2146586119a^18 + 44070755590/2146586119a^17 - 55338186896/2146586119a^16 + 176465429745/2146586119a^15 - 210589391850/2146586119a^14 + 651148289840/2146586119a^13 - 732360696405/2146586119a^12 + 2318142187550/2146586119a^11 - 2432398441404/2146586119a^10 + 3288764078233/2146586119a^9 - 3629895436805/2146586119a^8 + 4923256935836/2146586119a^7 - 3827472703634/2146586119a^6 + 5699240414651/2146586119a^5 + 2315225256535/2146586119a^4 + 249126046365/2146586119a^3 - 7368266322/2146586119a^2 + 18622103646/2146586119a + 3793914080/2146586119, 622318097/2146586119a^23 - 600542229/2146586119a^22 + 3085456706/2146586119a^21 - 2997075014/2146586119a^20 + 12314346020/2146586119a^19 - 11360345828/2146586119a^18 + 45543766546/2146586119a^17 - 59889940821/2146586119a^16 + 4477276719/52355759a^15 - 229445738686/2146586119a^14 + 678569211351/2146586119a^13 - 802191174959/2146586119a^12 + 2415114388539/2146586119a^11 - 2680149310216/2146586119a^10 + 3615392307475/2146586119a^9 - 4042600750925/2146586119a^8 + 5402418825109/2146586119a^7 - 4440834500898/2146586119a^6 + 6265562466909/2146586119a^5 + 1624223557610/2146586119a^4 + 273881308115/2146586119a^3 + 70925108695/2146586119a^2 + 12038435620/2146586119a - 465008485/2146586119, 42122034/2146586119a^23 - 93915274/2146586119a^22 + 274491574/2146586119a^21 - 482164724/2146586119a^20 + 1162742905/2146586119a^19 - 1899539727/2146586119a^18 + 4347569540/2146586119a^17 - 8244693270/2146586119a^16 + 18635598825/2146586119a^15 - 32735462909/2146586119a^14 + 69965919710/2146586119a^13 - 118132723330/2146586119a^12 + 248429023197/2146586119a^11 - 408363749579/2146586119a^10 + 532100664914/2146586119a^9 - 651139195681/2146586119a^8 + 19603133602/52355759a^7 - 21154815294/52355759a^6 + 942480030351/2146586119a^5 - 546636624521/2146586119a^4 + 41197936615/2146586119a^3 - 1666012516/2146586119a^2 + 5284849193/2146586119a + 1088431761/2146586119, 1059259537/2146586119a^23 - 1147125480/2146586119a^22 + 5354407974/2146586119a^21 - 5698107934/2146586119a^20 + 21467436752/2146586119a^19 - 21695514938/2146586119a^18 + 79419659210/2146586119a^17 - 110648046919/2146586119a^16 + 323063092005/2146586119a^15 - 425271535226/2146586119a^14 + 1195250147496/2146586119a^13 - 1493411517203/2146586119a^12 + 4250277704866/2146586119a^11 - 122382982064/52355759a^10 + 6615205792705/2146586119a^9 - 7508357838152/2146586119a^8 + 9879484598534/2146586119a^7 - 8494856133910/2146586119a^6 + 11368091910132/2146586119a^5 + 1681950463875/2146586119a^4 - 77527962984/2146586119a^3 + 73397097730/2146586119a^2 + 21845287393/2146586119*a - 881576200/2146586119 (assuming GRH)	sage: UK.fundamental_units() gp: K.fu magma: [K\|fUK(g): g in Generators(UK)]; oscar: [K(fUK(a)) for a in gens(UK)]
Regulator:	$ 16114643.667937363 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 16114643.667937363 \cdot 26}{14\cdot\sqrt{161761786626698377317203521728515625}}\cr\approx \mathstrut & 0.281699756642720 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 + 5*x^22 - 5*x^21 + 20*x^20 - 19*x^19 + 74*x^18 - 99*x^17 + 299*x^16 - 380*x^15 + 1106*x^14 - 1331*x^13 + 3936*x^12 - 4456*x^11 + 5996*x^10 - 6745*x^9 + 8969*x^8 - 7509*x^7 + 10409*x^6 + 2176*x^5 + 455*x^4 + 95*x^3 + 20*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 - x^23 + 5*x^22 - 5*x^21 + 20*x^20 - 19*x^19 + 74*x^18 - 99*x^17 + 299*x^16 - 380*x^15 + 1106*x^14 - 1331*x^13 + 3936*x^12 - 4456*x^11 + 5996*x^10 - 6745*x^9 + 8969*x^8 - 7509*x^7 + 10409*x^6 + 2176*x^5 + 455*x^4 + 95*x^3 + 20*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 - x^23 + 5*x^22 - 5*x^21 + 20*x^20 - 19*x^19 + 74*x^18 - 99*x^17 + 299*x^16 - 380*x^15 + 1106*x^14 - 1331*x^13 + 3936*x^12 - 4456*x^11 + 5996*x^10 - 6745*x^9 + 8969*x^8 - 7509*x^7 + 10409*x^6 + 2176*x^5 + 455*x^4 + 95*x^3 + 20*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 - x^23 + 5*x^22 - 5*x^21 + 20*x^20 - 19*x^19 + 74*x^18 - 99*x^17 + 299*x^16 - 380*x^15 + 1106*x^14 - 1331*x^13 + 3936*x^12 - 4456*x^11 + 5996*x^10 - 6745*x^9 + 8969*x^8 - 7509*x^7 + 10409*x^6 + 2176*x^5 + 455*x^4 + 95*x^3 + 20*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

$C_2\times C_{12}$ (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

$\Q(\sqrt{-35}) $, $\Q(\sqrt{5}) $, $\Q(\sqrt{-7}) $, $\Q(\zeta_{7})^+$, $\Q(\sqrt{5}, \sqrt{-7})$, $\Q(\zeta_{15})^+$, 4.0.55125.1, 6.0.2100875.1, 6.6.300125.1, $\Q(\zeta_{7})$, 8.0.3038765625.2, 12.0.4413675765625.1, 12.12.8208085798828125.1, 12.0.402196204142578125.1

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	${\href{/padicField/2.12.0.1}{12} }^{2}$	R	R	R	${\href{/padicField/11.6.0.1}{6} }^{4}$	${\href{/padicField/13.4.0.1}{4} }^{6}$	${\href{/padicField/17.12.0.1}{12} }^{2}$	${\href{/padicField/19.6.0.1}{6} }^{4}$	${\href{/padicField/23.12.0.1}{12} }^{2}$	${\href{/padicField/29.1.0.1}{1} }^{24}$	${\href{/padicField/31.6.0.1}{6} }^{4}$	${\href{/padicField/37.12.0.1}{12} }^{2}$	${\href{/padicField/41.2.0.1}{2} }^{12}$	${\href{/padicField/43.4.0.1}{4} }^{6}$	${\href{/padicField/47.12.0.1}{12} }^{2}$	${\href{/padicField/53.12.0.1}{12} }^{2}$	${\href{/padicField/59.6.0.1}{6} }^{4}$

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
$3$ 3	3.12.6.1	$x^{12} + 18 x^{8} + 81 x^{4} - 486 x^{2} + 1458$	$2$	$6$	$6$	$C_{12}$	$[\ ]_{2}^{6}$
$3$ 3	3.12.6.1	$x^{12} + 18 x^{8} + 81 x^{4} - 486 x^{2} + 1458$	$2$	$6$	$6$	$C_{12}$	$[\ ]_{2}^{6}$
$5$ 5		Deg $24$	$4$	$6$	$18$
$7$ 7		Deg $24$	$6$	$4$	$20$

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