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Properties

Label 24.0.132...784.1
Degree $24$
Signature $[0, 12]$
Discriminant $1.324\times 10^{35}$
Root discriminant \(29.07\)
Ramified primes $2,7,17$
Class number $6$ (GRH)
Class group [6] (GRH)
Galois group $C_6\times D_4$ (as 24T38)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 2*x^23 - x^22 + 8*x^21 - 5*x^20 - 22*x^19 + 39*x^18 - 102*x^17 + 107*x^16 + 184*x^15 - 573*x^14 + 2*x^13 + 1911*x^12 - 1908*x^11 + 766*x^10 + 360*x^9 - 740*x^8 + 432*x^7 - 136*x^6 + 352*x^5 - 272*x^4 + 64*x^3 + 96*x^2 - 128*x + 64)
 
gp: K = bnfinit(y^24 - 2*y^23 - y^22 + 8*y^21 - 5*y^20 - 22*y^19 + 39*y^18 - 102*y^17 + 107*y^16 + 184*y^15 - 573*y^14 + 2*y^13 + 1911*y^12 - 1908*y^11 + 766*y^10 + 360*y^9 - 740*y^8 + 432*y^7 - 136*y^6 + 352*y^5 - 272*y^4 + 64*y^3 + 96*y^2 - 128*y + 64, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 2*x^23 - x^22 + 8*x^21 - 5*x^20 - 22*x^19 + 39*x^18 - 102*x^17 + 107*x^16 + 184*x^15 - 573*x^14 + 2*x^13 + 1911*x^12 - 1908*x^11 + 766*x^10 + 360*x^9 - 740*x^8 + 432*x^7 - 136*x^6 + 352*x^5 - 272*x^4 + 64*x^3 + 96*x^2 - 128*x + 64);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 2*x^23 - x^22 + 8*x^21 - 5*x^20 - 22*x^19 + 39*x^18 - 102*x^17 + 107*x^16 + 184*x^15 - 573*x^14 + 2*x^13 + 1911*x^12 - 1908*x^11 + 766*x^10 + 360*x^9 - 740*x^8 + 432*x^7 - 136*x^6 + 352*x^5 - 272*x^4 + 64*x^3 + 96*x^2 - 128*x + 64)
 

\( x^{24} - 2 x^{23} - x^{22} + 8 x^{21} - 5 x^{20} - 22 x^{19} + 39 x^{18} - 102 x^{17} + 107 x^{16} + \cdots + 64 \) Copy content Toggle raw display

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:   \(132353116631035821576664572214902784\) \(\medspace = 2^{36}\cdot 7^{20}\cdot 17^{6}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(29.07\)
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:  $2^{3/2}7^{5/6}17^{1/2}\approx 59.02252990123237$
Ramified primes:   \(2\), \(7\), \(17\) Copy content Toggle raw display
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 a CM field.
Reflex fields:  unavailable$^{2048}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{14}-\frac{1}{2}a^{13}-\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{17}-\frac{1}{4}a^{15}-\frac{1}{4}a^{13}-\frac{1}{2}a^{12}-\frac{1}{4}a^{11}-\frac{1}{2}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{18}-\frac{1}{8}a^{16}-\frac{1}{4}a^{15}-\frac{1}{8}a^{14}-\frac{1}{8}a^{12}+\frac{3}{8}a^{10}-\frac{1}{4}a^{9}-\frac{1}{8}a^{8}-\frac{1}{8}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{3683576}a^{19}-\frac{32751}{920894}a^{18}-\frac{261231}{3683576}a^{17}+\frac{32073}{460447}a^{16}-\frac{524787}{3683576}a^{15}-\frac{384101}{1841788}a^{14}-\frac{27111}{3683576}a^{13}+\frac{861181}{1841788}a^{12}+\frac{924809}{3683576}a^{11}-\frac{40988}{460447}a^{10}-\frac{1594523}{3683576}a^{9}+\frac{684091}{1841788}a^{8}+\frac{1014673}{3683576}a^{7}-\frac{459089}{920894}a^{6}+\frac{187839}{460447}a^{5}-\frac{600749}{1841788}a^{4}+\frac{308609}{920894}a^{3}-\frac{195275}{460447}a^{2}+\frac{74322}{460447}a-\frac{179783}{460447}$, $\frac{1}{7367152}a^{20}+\frac{392231}{7367152}a^{18}-\frac{353703}{3683576}a^{17}-\frac{546345}{7367152}a^{16}-\frac{155695}{1841788}a^{15}-\frac{1344705}{7367152}a^{14}+\frac{37713}{141676}a^{13}-\frac{1132517}{7367152}a^{12}+\frac{728293}{3683576}a^{11}+\frac{2295855}{7367152}a^{10}+\frac{866895}{1841788}a^{9}+\frac{125711}{7367152}a^{8}-\frac{108653}{3683576}a^{7}+\frac{632689}{3683576}a^{6}+\frac{186483}{920894}a^{5}+\frac{769125}{1841788}a^{4}-\frac{232751}{920894}a^{3}+\frac{243095}{920894}a^{2}+\frac{73845}{460447}a-\frac{214041}{460447}$, $\frac{1}{7367152}a^{21}-\frac{1}{7367152}a^{19}+\frac{32751}{1841788}a^{18}+\frac{261231}{7367152}a^{17}+\frac{332155}{3683576}a^{16}+\frac{524787}{7367152}a^{15}+\frac{806911}{3683576}a^{14}-\frac{1814677}{7367152}a^{13}-\frac{200367}{1841788}a^{12}-\frac{924809}{7367152}a^{11}+\frac{624399}{3683576}a^{10}-\frac{2089053}{7367152}a^{9}+\frac{202268}{460447}a^{8}-\frac{676495}{3683576}a^{7}+\frac{1378625}{3683576}a^{6}-\frac{187839}{920894}a^{5}-\frac{97574}{460447}a^{4}+\frac{75919}{920894}a^{3}-\frac{132586}{460447}a^{2}-\frac{37161}{460447}a+\frac{54250}{460447}$, $\frac{1}{14734304}a^{22}-\frac{1}{14734304}a^{20}-\frac{1}{7367152}a^{19}+\frac{790671}{14734304}a^{18}-\frac{38245}{920894}a^{17}+\frac{954071}{14734304}a^{16}-\frac{65203}{460447}a^{15}+\frac{118427}{14734304}a^{14}+\frac{3651043}{7367152}a^{13}+\frac{5975839}{14734304}a^{12}-\frac{643499}{1841788}a^{11}+\frac{2964423}{14734304}a^{10}+\frac{1702521}{7367152}a^{9}-\frac{16835}{566704}a^{8}-\frac{1834351}{3683576}a^{7}+\frac{832057}{3683576}a^{6}-\frac{562161}{1841788}a^{5}-\frac{1619}{35419}a^{4}-\frac{37929}{920894}a^{3}-\frac{78248}{460447}a^{2}-\frac{207016}{460447}a+\frac{196912}{460447}$, $\frac{1}{14734304}a^{23}-\frac{1}{14734304}a^{21}-\frac{1}{14734304}a^{19}-\frac{394945}{7367152}a^{18}-\frac{1580557}{14734304}a^{17}-\frac{165929}{7367152}a^{16}-\frac{3158789}{14734304}a^{15}+\frac{210869}{3683576}a^{14}+\frac{3710687}{14734304}a^{13}+\frac{980607}{7367152}a^{12}+\frac{916979}{14734304}a^{11}-\frac{439953}{920894}a^{10}+\frac{3390785}{7367152}a^{9}-\frac{2525879}{7367152}a^{8}-\frac{775215}{1841788}a^{7}+\frac{1379983}{3683576}a^{6}+\frac{733055}{1841788}a^{5}-\frac{655595}{1841788}a^{4}+\frac{268183}{920894}a^{3}-\frac{398451}{920894}a^{2}+\frac{211643}{460447}a-\frac{167457}{460447}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

$C_{6}$, which has order $6$ (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:   \( -\frac{32553}{7367152} a^{22} + \frac{2250029}{3683576} a^{15} - \frac{175628799}{7367152} a^{8} - \frac{1239808}{460447} a \)  (order $14$) Copy content Toggle raw display
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:   $\frac{217649}{3683576}a^{23}-\frac{727147}{7367152}a^{22}-\frac{300821}{3683576}a^{21}+\frac{1620785}{3683576}a^{20}-\frac{672385}{3683576}a^{19}-\frac{1211961}{920894}a^{18}+\frac{7129835}{3683576}a^{17}-\frac{10323827}{1841788}a^{16}+\frac{633229}{141676}a^{15}+\frac{5394063}{460447}a^{14}-\frac{110226707}{3683576}a^{13}-\frac{13117595}{1841788}a^{12}+\frac{400710231}{3683576}a^{11}-\frac{77314429}{920894}a^{10}+\frac{56952457}{1841788}a^{9}+\frac{23960587}{566704}a^{8}-\frac{29965599}{920894}a^{7}+\frac{11532775}{3683576}a^{6}-\frac{3270311}{460447}a^{5}+\frac{29872255}{1841788}a^{4}-\frac{5431662}{460447}a^{3}+\frac{1075816}{460447}a^{2}+\frac{3352564}{460447}a-\frac{199632}{35419}$, $\frac{148339}{3683576}a^{23}-\frac{560803}{7367152}a^{22}-\frac{148339}{3683576}a^{21}+\frac{1177201}{3683576}a^{20}-\frac{741695}{3683576}a^{19}-\frac{1631729}{1841788}a^{18}+\frac{445017}{283352}a^{17}-\frac{7565289}{1841788}a^{16}+\frac{3405561}{920894}a^{15}+\frac{3411797}{460447}a^{14}-\frac{83680977}{3683576}a^{13}+\frac{148339}{1841788}a^{12}+\frac{21805833}{283352}a^{11}-\frac{70757703}{920894}a^{10}+\frac{56813837}{1841788}a^{9}+\frac{282432879}{7367152}a^{8}-\frac{27442715}{920894}a^{7}+\frac{12641735}{3683576}a^{6}-\frac{2521763}{460447}a^{5}+\frac{6526916}{460447}a^{4}-\frac{5043526}{460447}a^{3}+\frac{1186712}{460447}a^{2}+\frac{3019876}{460447}a-\frac{1912977}{460447}$, $\frac{18905}{1133408}a^{23}-\frac{18905}{566704}a^{22}-\frac{18905}{1133408}a^{21}+\frac{37497}{283352}a^{20}-\frac{94525}{1133408}a^{19}-\frac{207955}{566704}a^{18}+\frac{737295}{1133408}a^{17}-\frac{964155}{566704}a^{16}+\frac{2022835}{1133408}a^{15}+\frac{434815}{141676}a^{14}-\frac{10662381}{1133408}a^{13}+\frac{18905}{566704}a^{12}+\frac{36127455}{1133408}a^{11}-\frac{9017685}{283352}a^{10}+\frac{7240615}{566704}a^{9}+\frac{850725}{141676}a^{8}-\frac{3497425}{283352}a^{7}+\frac{402877}{283352}a^{6}-\frac{321385}{141676}a^{5}+\frac{207955}{35419}a^{4}-\frac{321385}{70838}a^{3}+\frac{37810}{35419}a^{2}+\frac{56715}{35419}a-\frac{75620}{35419}$, $\frac{13911}{1841788}a^{23}+\frac{11521}{1841788}a^{22}-\frac{480481}{460447}a^{16}-\frac{795697}{920894}a^{15}+\frac{74953861}{1841788}a^{9}+\frac{62094043}{1841788}a^{8}+\frac{5805017}{920894}a^{2}+\frac{1753351}{460447}a-1$, $\frac{561881}{14734304}a^{23}-\frac{352665}{3683576}a^{22}-\frac{229193}{14734304}a^{21}+\frac{610539}{1841788}a^{20}-\frac{344065}{1133408}a^{19}-\frac{6061559}{7367152}a^{18}+\frac{27347263}{14734304}a^{17}-\frac{31761019}{7367152}a^{16}+\frac{87419223}{14734304}a^{15}+\frac{11370719}{1841788}a^{14}-\frac{28242457}{1133408}a^{13}+\frac{53902857}{7367152}a^{12}+\frac{1134622623}{14734304}a^{11}-\frac{374033813}{3683576}a^{10}+\frac{320828863}{7367152}a^{9}-\frac{53641771}{7367152}a^{8}-\frac{145145849}{3683576}a^{7}+\frac{2373512}{460447}a^{6}-\frac{11270865}{1841788}a^{5}+\frac{33156733}{1841788}a^{4}-\frac{13488785}{920894}a^{3}+\frac{137626}{35419}a^{2}+\frac{944867}{460447}a-\frac{3134692}{460447}$, $\frac{458291}{3683576}a^{23}-\frac{917121}{3683576}a^{22}-\frac{363157}{1841788}a^{21}+\frac{3912975}{3683576}a^{20}-\frac{1623235}{3683576}a^{19}-\frac{5851961}{1841788}a^{18}+\frac{17213459}{3683576}a^{17}-\frac{10142151}{920894}a^{16}+\frac{45185421}{3683576}a^{15}+\frac{26046549}{920894}a^{14}-\frac{266114051}{3683576}a^{13}-\frac{7918706}{460447}a^{12}+\frac{967414235}{3683576}a^{11}-\frac{373310513}{1841788}a^{10}-\frac{835769}{35419}a^{9}+\frac{164003289}{3683576}a^{8}-\frac{289376043}{3683576}a^{7}+\frac{27843091}{3683576}a^{6}-\frac{31582115}{1841788}a^{5}+\frac{72119051}{1841788}a^{4}-\frac{26226555}{920894}a^{3}-\frac{8820189}{920894}a^{2}+\frac{3989162}{460447}a-\frac{6265464}{460447}$, $\frac{65155}{1133408}a^{23}-\frac{569187}{7367152}a^{22}-\frac{1958719}{14734304}a^{21}+\frac{372527}{920894}a^{20}+\frac{433141}{14734304}a^{19}-\frac{10382245}{7367152}a^{18}+\frac{19235433}{14734304}a^{17}-\frac{32683201}{7367152}a^{16}+\frac{39947957}{14734304}a^{15}+\frac{6611317}{460447}a^{14}-\frac{363830067}{14734304}a^{13}-\frac{158940933}{7367152}a^{12}+\frac{1557443769}{14734304}a^{11}-\frac{116329133}{3683576}a^{10}-\frac{143692283}{7367152}a^{9}+\frac{38189743}{1841788}a^{8}-\frac{106930735}{3683576}a^{7}+\frac{188039}{920894}a^{6}-\frac{15685143}{1841788}a^{5}+\frac{2132447}{141676}a^{4}-\frac{1605343}{460447}a^{3}-\frac{4860589}{920894}a^{2}+\frac{1692854}{460447}a-\frac{2278316}{460447}$, $\frac{181247}{14734304}a^{23}-\frac{125799}{14734304}a^{22}-\frac{803577}{14734304}a^{21}+\frac{1473313}{14734304}a^{20}+\frac{81537}{1133408}a^{19}-\frac{6041605}{14734304}a^{18}+\frac{2415823}{14734304}a^{17}-\frac{6816377}{14734304}a^{16}-\frac{9074405}{14734304}a^{15}+\frac{5788945}{1133408}a^{14}-\frac{5958231}{1133408}a^{13}-\frac{150380125}{14734304}a^{12}+\frac{408148023}{14734304}a^{11}+\frac{106283077}{14734304}a^{10}-\frac{262350557}{7367152}a^{9}+\frac{17621033}{566704}a^{8}-\frac{42452721}{1841788}a^{7}+\frac{17244733}{3683576}a^{6}-\frac{5761425}{1841788}a^{5}+\frac{4307279}{920894}a^{4}+\frac{3068707}{920894}a^{3}-\frac{5738539}{920894}a^{2}+\frac{2420194}{460447}a-\frac{1423407}{460447}$, $\frac{938399}{7367152}a^{23}-\frac{545559}{14734304}a^{22}-\frac{642897}{1841788}a^{21}+\frac{7602111}{14734304}a^{20}+\frac{388139}{566704}a^{19}-\frac{36106185}{14734304}a^{18}+\frac{49646}{460447}a^{17}-\frac{137653317}{14734304}a^{16}-\frac{1615413}{460447}a^{15}+\frac{422317691}{14734304}a^{14}-\frac{12623161}{566704}a^{13}-\frac{1132166385}{14734304}a^{12}+\frac{282621961}{1841788}a^{11}+\frac{1613167287}{14734304}a^{10}+\frac{196858733}{3683576}a^{9}+\frac{439350445}{7367152}a^{8}+\frac{77154799}{3683576}a^{7}-\frac{24175763}{3683576}a^{6}-\frac{29662887}{1841788}a^{5}+\frac{30526617}{1841788}a^{4}+\frac{14161217}{920894}a^{3}+\frac{2727341}{460447}a^{2}+\frac{328909}{35419}a+\frac{2067500}{460447}$, $\frac{88111}{3683576}a^{23}-\frac{886675}{14734304}a^{22}+\frac{4587}{920894}a^{21}+\frac{2882109}{14734304}a^{20}-\frac{1723419}{7367152}a^{19}-\frac{6346539}{14734304}a^{18}+\frac{1131879}{920894}a^{17}-\frac{45106091}{14734304}a^{16}+\frac{14269131}{3683576}a^{15}+\frac{43061961}{14734304}a^{14}-\frac{113988327}{7367152}a^{13}+\frac{124450805}{14734304}a^{12}+\frac{80942363}{1841788}a^{11}-\frac{82421663}{1133408}a^{10}+\frac{29089817}{566704}a^{9}+\frac{6163741}{3683576}a^{8}-\frac{126634385}{3683576}a^{7}+\frac{8093825}{3683576}a^{6}-\frac{216439}{1841788}a^{5}+\frac{8561957}{920894}a^{4}-\frac{10977107}{920894}a^{3}+\frac{459377}{70838}a^{2}+\frac{762296}{460447}a-\frac{2479907}{460447}$, $\frac{476651}{7367152}a^{23}-\frac{1690205}{14734304}a^{22}-\frac{837283}{7367152}a^{21}+\frac{7763393}{14734304}a^{20}-\frac{292419}{1841788}a^{19}-\frac{24143911}{14734304}a^{18}+\frac{16042803}{7367152}a^{17}-\frac{81340463}{14734304}a^{16}+\frac{36796691}{7367152}a^{15}+\frac{219289449}{14734304}a^{14}-\frac{64313423}{1841788}a^{13}-\frac{193700799}{14734304}a^{12}+\frac{973624007}{7367152}a^{11}-\frac{98420623}{1133408}a^{10}-\frac{116994933}{7367152}a^{9}+\frac{316438833}{7367152}a^{8}-\frac{13216436}{460447}a^{7}-\frac{2859215}{1841788}a^{6}-\frac{13608809}{1841788}a^{5}+\frac{38911489}{1841788}a^{4}-\frac{5947510}{460447}a^{3}-\frac{4312287}{920894}a^{2}+\frac{3290351}{460447}a-\frac{2496479}{460447}$ Copy content Toggle raw display (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:  \( 81519832.15193881 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 81519832.15193881 \cdot 6}{14\cdot\sqrt{132353116631035821576664572214902784}}\cr\approx \mathstrut & 0.363561448917531 \end{aligned}\] (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 - x^22 + 8*x^21 - 5*x^20 - 22*x^19 + 39*x^18 - 102*x^17 + 107*x^16 + 184*x^15 - 573*x^14 + 2*x^13 + 1911*x^12 - 1908*x^11 + 766*x^10 + 360*x^9 - 740*x^8 + 432*x^7 - 136*x^6 + 352*x^5 - 272*x^4 + 64*x^3 + 96*x^2 - 128*x + 64)
 
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 - x^22 + 8*x^21 - 5*x^20 - 22*x^19 + 39*x^18 - 102*x^17 + 107*x^16 + 184*x^15 - 573*x^14 + 2*x^13 + 1911*x^12 - 1908*x^11 + 766*x^10 + 360*x^9 - 740*x^8 + 432*x^7 - 136*x^6 + 352*x^5 - 272*x^4 + 64*x^3 + 96*x^2 - 128*x + 64, 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 - x^22 + 8*x^21 - 5*x^20 - 22*x^19 + 39*x^18 - 102*x^17 + 107*x^16 + 184*x^15 - 573*x^14 + 2*x^13 + 1911*x^12 - 1908*x^11 + 766*x^10 + 360*x^9 - 740*x^8 + 432*x^7 - 136*x^6 + 352*x^5 - 272*x^4 + 64*x^3 + 96*x^2 - 128*x + 64);
 
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 - x^22 + 8*x^21 - 5*x^20 - 22*x^19 + 39*x^18 - 102*x^17 + 107*x^16 + 184*x^15 - 573*x^14 + 2*x^13 + 1911*x^12 - 1908*x^11 + 766*x^10 + 360*x^9 - 740*x^8 + 432*x^7 - 136*x^6 + 352*x^5 - 272*x^4 + 64*x^3 + 96*x^2 - 128*x + 64);
 
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

$C_6\times D_4$ (as 24T38):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 48
The 30 conjugacy class representatives for $C_6\times D_4$
Character table for $C_6\times D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), 4.0.1088.2, 4.4.53312.1, \(\Q(\sqrt{2}, \sqrt{-7})\), 6.6.1229312.1, 6.0.8605184.1, \(\Q(\zeta_{7})\), 8.0.2842169344.2, 12.0.7424564871299072.1, 12.12.363803678693654528.1, 12.0.74049191673856.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Degree 24 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.12.0.1}{12} }^{2}$ ${\href{/padicField/5.12.0.1}{12} }^{2}$ R ${\href{/padicField/11.12.0.1}{12} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{12}$ R ${\href{/padicField/19.6.0.1}{6} }^{4}$ ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{6}$ ${\href{/padicField/31.6.0.1}{6} }^{4}$ ${\href{/padicField/37.12.0.1}{12} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{12}$ ${\href{/padicField/43.2.0.1}{2} }^{12}$ ${\href{/padicField/47.6.0.1}{6} }^{4}$ ${\href{/padicField/53.6.0.1}{6} }^{4}$ ${\href{/padicField/59.6.0.1}{6} }^{4}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.6.9.1$x^{6} + 44 x^{4} + 2 x^{3} + 589 x^{2} - 82 x + 2367$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.1$x^{6} + 44 x^{4} + 2 x^{3} + 589 x^{2} - 82 x + 2367$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.1$x^{6} + 44 x^{4} + 2 x^{3} + 589 x^{2} - 82 x + 2367$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.1$x^{6} + 44 x^{4} + 2 x^{3} + 589 x^{2} - 82 x + 2367$$2$$3$$9$$C_6$$[3]^{3}$
\(7\) Copy content Toggle raw display 7.6.5.5$x^{6} + 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.12.10.1$x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
\(17\) Copy content Toggle raw display 17.6.0.1$x^{6} + 2 x^{4} + 10 x^{2} + 3 x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
17.6.0.1$x^{6} + 2 x^{4} + 10 x^{2} + 3 x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
17.12.6.1$x^{12} + 918 x^{11} + 351241 x^{10} + 71712630 x^{9} + 8244584136 x^{8} + 506874732756 x^{7} + 13125344775560 x^{6} + 9625198256031 x^{5} + 28457943732288 x^{4} + 16844354225613 x^{3} + 132306217741765 x^{2} + 68598705820311 x + 44162739951115$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$