↑

Properties

Label 24.0.965...256.1
Degree $24$
Signature $[0, 12]$
Discriminant $9.660\times 10^{34}$
Root discriminant \(28.69\)
Ramified primes $2,3,71$
Class number $8$ (GRH)
Class group [8] (GRH)
Galois group $C_2^2\times D_6$ (as 24T30)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 8*x^21 + 3*x^20 - 4*x^19 + 32*x^18 - 20*x^17 + 37*x^16 - 116*x^15 + 72*x^14 - 128*x^13 + 380*x^12 - 256*x^11 + 288*x^10 - 928*x^9 + 592*x^8 - 640*x^7 + 2048*x^6 - 512*x^5 + 768*x^4 - 4096*x^3 + 4096)
 
gp: K = bnfinit(y^24 - 8*y^21 + 3*y^20 - 4*y^19 + 32*y^18 - 20*y^17 + 37*y^16 - 116*y^15 + 72*y^14 - 128*y^13 + 380*y^12 - 256*y^11 + 288*y^10 - 928*y^9 + 592*y^8 - 640*y^7 + 2048*y^6 - 512*y^5 + 768*y^4 - 4096*y^3 + 4096, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 8*x^21 + 3*x^20 - 4*x^19 + 32*x^18 - 20*x^17 + 37*x^16 - 116*x^15 + 72*x^14 - 128*x^13 + 380*x^12 - 256*x^11 + 288*x^10 - 928*x^9 + 592*x^8 - 640*x^7 + 2048*x^6 - 512*x^5 + 768*x^4 - 4096*x^3 + 4096);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 8*x^21 + 3*x^20 - 4*x^19 + 32*x^18 - 20*x^17 + 37*x^16 - 116*x^15 + 72*x^14 - 128*x^13 + 380*x^12 - 256*x^11 + 288*x^10 - 928*x^9 + 592*x^8 - 640*x^7 + 2048*x^6 - 512*x^5 + 768*x^4 - 4096*x^3 + 4096)
 

\( x^{24} - 8 x^{21} + 3 x^{20} - 4 x^{19} + 32 x^{18} - 20 x^{17} + 37 x^{16} - 116 x^{15} + 72 x^{14} + \cdots + 4096 \) 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:   \(96596555035261369442314470090080256\) \(\medspace = 2^{48}\cdot 3^{12}\cdot 71^{8}\) 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:  \(28.69\)
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:  $2^{2}3^{1/2}71^{1/2}\approx 58.378078077305695$
Ramified primes:   \(2\), \(3\), \(71\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $8$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{8}+\frac{1}{4}a^{4}$, $\frac{1}{8}a^{13}-\frac{1}{4}a^{11}+\frac{3}{8}a^{9}-\frac{1}{2}a^{8}+\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{3}{8}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{16}a^{14}-\frac{1}{8}a^{12}+\frac{3}{16}a^{10}+\frac{1}{4}a^{9}+\frac{1}{8}a^{8}-\frac{1}{4}a^{7}+\frac{5}{16}a^{6}-\frac{1}{4}a^{5}+\frac{3}{8}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{15}-\frac{1}{16}a^{13}+\frac{3}{32}a^{11}+\frac{1}{8}a^{10}-\frac{7}{16}a^{9}+\frac{3}{8}a^{8}-\frac{11}{32}a^{7}-\frac{1}{8}a^{6}+\frac{3}{16}a^{5}-\frac{1}{2}a^{4}-\frac{1}{8}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{320}a^{16}-\frac{1}{80}a^{15}-\frac{3}{160}a^{14}+\frac{1}{20}a^{13}-\frac{37}{320}a^{12}-\frac{3}{40}a^{11}-\frac{21}{160}a^{10}+\frac{19}{80}a^{9}-\frac{23}{64}a^{8}+\frac{1}{40}a^{7}+\frac{49}{160}a^{6}-\frac{3}{20}a^{5}-\frac{27}{80}a^{4}+\frac{11}{40}a^{3}-\frac{1}{20}a^{2}+\frac{1}{10}a+\frac{1}{5}$, $\frac{1}{320}a^{17}-\frac{1}{160}a^{15}-\frac{1}{40}a^{14}-\frac{13}{320}a^{13}-\frac{3}{80}a^{12}-\frac{39}{160}a^{11}-\frac{3}{80}a^{10}-\frac{91}{320}a^{9}-\frac{13}{80}a^{8}-\frac{9}{32}a^{7}-\frac{7}{40}a^{6}+\frac{7}{16}a^{5}+\frac{17}{40}a^{4}-\frac{1}{5}a^{3}+\frac{2}{5}a^{2}-\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{640}a^{18}+\frac{1}{160}a^{15}+\frac{3}{128}a^{14}-\frac{1}{32}a^{13}-\frac{9}{80}a^{12}+\frac{5}{128}a^{10}+\frac{15}{32}a^{9}-\frac{1}{4}a^{8}+\frac{11}{32}a^{7}+\frac{17}{80}a^{6}+\frac{3}{16}a^{4}+\frac{7}{20}a^{3}-\frac{1}{4}a^{2}+\frac{1}{5}$, $\frac{1}{640}a^{19}-\frac{9}{640}a^{15}+\frac{1}{160}a^{14}+\frac{3}{80}a^{13}-\frac{3}{160}a^{12}-\frac{159}{640}a^{11}-\frac{3}{160}a^{10}-\frac{19}{40}a^{9}+\frac{1}{16}a^{8}+\frac{1}{10}a^{7}-\frac{29}{80}a^{6}-\frac{21}{80}a^{5}+\frac{11}{40}a^{4}+\frac{1}{5}a^{3}+\frac{1}{10}a^{2}-\frac{1}{2}a-\frac{2}{5}$, $\frac{1}{1280}a^{20}-\frac{1}{1280}a^{16}+\frac{3}{320}a^{15}-\frac{3}{160}a^{14}+\frac{9}{320}a^{13}-\frac{27}{256}a^{12}-\frac{21}{320}a^{11}+\frac{1}{8}a^{10}+\frac{11}{160}a^{9}+\frac{73}{160}a^{8}-\frac{19}{40}a^{7}-\frac{23}{160}a^{6}+\frac{1}{40}a^{5}+\frac{7}{40}a^{4}-\frac{1}{40}a^{3}+\frac{2}{5}a^{2}+\frac{2}{5}$, $\frac{1}{2560}a^{21}+\frac{3}{2560}a^{17}-\frac{1}{640}a^{16}+\frac{1}{80}a^{15}-\frac{3}{128}a^{14}-\frac{123}{2560}a^{13}+\frac{7}{128}a^{12}-\frac{51}{320}a^{11}+\frac{29}{320}a^{10}-\frac{169}{640}a^{9}+\frac{1}{40}a^{8}-\frac{21}{80}a^{7}+\frac{77}{160}a^{5}+\frac{1}{4}a^{4}-\frac{1}{5}a^{3}-\frac{9}{20}a^{2}+\frac{3}{10}a$, $\frac{1}{189440}a^{22}-\frac{13}{94720}a^{21}+\frac{1}{23680}a^{20}-\frac{1}{11840}a^{19}+\frac{107}{189440}a^{18}+\frac{103}{94720}a^{17}+\frac{1}{1184}a^{16}-\frac{15}{9472}a^{15}+\frac{497}{37888}a^{14}-\frac{463}{18944}a^{13}-\frac{2853}{23680}a^{12}+\frac{201}{4736}a^{11}-\frac{1459}{47360}a^{10}+\frac{10709}{23680}a^{9}-\frac{183}{592}a^{8}+\frac{155}{1184}a^{7}+\frac{2429}{11840}a^{6}-\frac{2437}{5920}a^{5}+\frac{1139}{2960}a^{4}-\frac{513}{1480}a^{3}-\frac{103}{370}a^{2}-\frac{89}{370}a-\frac{7}{37}$, $\frac{1}{378880}a^{23}+\frac{9}{47360}a^{21}-\frac{13}{47360}a^{20}-\frac{13}{378880}a^{19}+\frac{7}{94720}a^{18}-\frac{69}{47360}a^{17}+\frac{3}{94720}a^{16}-\frac{2059}{378880}a^{15}+\frac{1379}{94720}a^{14}-\frac{1171}{23680}a^{13}-\frac{1301}{11840}a^{12}-\frac{257}{2560}a^{11}-\frac{381}{11840}a^{10}-\frac{5093}{11840}a^{9}+\frac{59}{160}a^{8}-\frac{4631}{23680}a^{7}+\frac{2397}{5920}a^{6}+\frac{343}{2960}a^{5}+\frac{1011}{2960}a^{4}-\frac{10}{37}a^{3}+\frac{76}{185}a^{2}-\frac{4}{185}a+\frac{26}{185}$ 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:  $2$, $5$

Class group and class number

$C_{8}$, which has order $8$ (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:   \( -\frac{437}{189440} a^{23} - \frac{43}{94720} a^{22} + \frac{297}{94720} a^{21} + \frac{571}{47360} a^{20} - \frac{187}{37888} a^{19} - \frac{803}{94720} a^{18} - \frac{2761}{94720} a^{17} + \frac{159}{23680} a^{16} - \frac{9}{189440} a^{15} + \frac{10427}{94720} a^{14} + \frac{2349}{94720} a^{13} - \frac{5309}{47360} a^{12} - \frac{13649}{47360} a^{11} + \frac{819}{11840} a^{10} + \frac{9459}{23680} a^{9} + \frac{2091}{5920} a^{8} + \frac{2707}{11840} a^{7} - \frac{55}{74} a^{6} - \frac{1843}{1184} a^{5} - \frac{4287}{2960} a^{4} + \frac{2387}{740} a^{3} + \frac{465}{148} a^{2} + \frac{108}{185} a - \frac{881}{185} \)  (order $24$) 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:   $\frac{593}{189440}a^{23}-\frac{423}{189440}a^{22}-\frac{31}{5920}a^{21}-\frac{399}{23680}a^{20}+\frac{4091}{189440}a^{19}+\frac{1831}{189440}a^{18}+\frac{59}{1280}a^{17}-\frac{2561}{47360}a^{16}+\frac{637}{37888}a^{15}-\frac{38647}{189440}a^{14}+\frac{1265}{9472}a^{13}+\frac{2371}{23680}a^{12}+\frac{24931}{47360}a^{11}-\frac{28433}{47360}a^{10}-\frac{1679}{5920}a^{9}-\frac{439}{592}a^{8}+\frac{1795}{2368}a^{7}+\frac{8501}{11840}a^{6}+\frac{4023}{1480}a^{5}-\frac{517}{740}a^{4}-\frac{1811}{370}a^{3}-\frac{3533}{740}a^{2}+\frac{667}{185}a+\frac{1537}{185}$, $\frac{917}{189440}a^{23}-\frac{11}{37888}a^{22}-\frac{217}{23680}a^{21}-\frac{617}{23680}a^{20}+\frac{3463}{189440}a^{19}+\frac{4991}{189440}a^{18}+\frac{2841}{47360}a^{17}-\frac{1521}{47360}a^{16}-\frac{3547}{189440}a^{15}-\frac{46863}{189440}a^{14}-\frac{301}{47360}a^{13}+\frac{7553}{23680}a^{12}+\frac{29791}{47360}a^{11}-\frac{16139}{47360}a^{10}-\frac{5763}{5920}a^{9}-\frac{489}{740}a^{8}-\frac{235}{2368}a^{7}+\frac{24429}{11840}a^{6}+\frac{4911}{1480}a^{5}+\frac{6131}{2960}a^{4}-\frac{3251}{370}a^{3}-\frac{5189}{740}a^{2}+\frac{87}{74}a+\frac{435}{37}$, $\frac{437}{189440}a^{23}+\frac{43}{94720}a^{22}-\frac{297}{94720}a^{21}-\frac{571}{47360}a^{20}+\frac{187}{37888}a^{19}+\frac{803}{94720}a^{18}+\frac{2761}{94720}a^{17}-\frac{159}{23680}a^{16}+\frac{9}{189440}a^{15}-\frac{10427}{94720}a^{14}-\frac{2349}{94720}a^{13}+\frac{5309}{47360}a^{12}+\frac{13649}{47360}a^{11}-\frac{819}{11840}a^{10}-\frac{9459}{23680}a^{9}-\frac{2091}{5920}a^{8}-\frac{2707}{11840}a^{7}+\frac{55}{74}a^{6}+\frac{1843}{1184}a^{5}+\frac{4287}{2960}a^{4}-\frac{2387}{740}a^{3}-\frac{465}{148}a^{2}-\frac{108}{185}a+\frac{1066}{185}$, $\frac{387}{94720}a^{23}-\frac{949}{94720}a^{22}+\frac{443}{47360}a^{21}-\frac{943}{47360}a^{20}+\frac{6157}{94720}a^{19}-\frac{1743}{18944}a^{18}+\frac{6267}{47360}a^{17}-\frac{11919}{47360}a^{16}+\frac{34431}{94720}a^{15}-\frac{58701}{94720}a^{14}+\frac{49817}{47360}a^{13}-\frac{56267}{47360}a^{12}+\frac{18689}{11840}a^{11}-\frac{68859}{23680}a^{10}+\frac{41903}{11840}a^{9}-\frac{5901}{1480}a^{8}+\frac{8157}{1480}a^{7}-\frac{17839}{2960}a^{6}+\frac{22427}{2960}a^{5}-\frac{4327}{370}a^{4}+\frac{5779}{740}a^{3}-\frac{796}{185}a^{2}+\frac{2493}{185}a-\frac{413}{37}$, $\frac{35}{37888}a^{23}+\frac{29}{189440}a^{22}-\frac{293}{94720}a^{21}-\frac{5}{1184}a^{20}-\frac{15}{37888}a^{19}+\frac{3267}{189440}a^{18}+\frac{567}{94720}a^{17}+\frac{649}{47360}a^{16}-\frac{10569}{189440}a^{15}+\frac{273}{37888}a^{14}-\frac{10751}{94720}a^{13}+\frac{5933}{23680}a^{12}-\frac{2871}{47360}a^{11}+\frac{2725}{9472}a^{10}-\frac{18519}{23680}a^{9}+\frac{2171}{5920}a^{8}-\frac{9121}{11840}a^{7}+\frac{19997}{11840}a^{6}-\frac{3437}{5920}a^{5}+\frac{6077}{2960}a^{4}-\frac{744}{185}a^{3}-\frac{251}{740}a^{2}-\frac{867}{370}a+\frac{1166}{185}$, $\frac{1353}{189440}a^{23}+\frac{159}{189440}a^{22}-\frac{1311}{94720}a^{21}-\frac{193}{4736}a^{20}+\frac{887}{37888}a^{19}+\frac{8129}{189440}a^{18}+\frac{8213}{94720}a^{17}-\frac{1791}{47360}a^{16}-\frac{931}{37888}a^{15}-\frac{69209}{189440}a^{14}-\frac{1193}{18944}a^{13}+\frac{1107}{2368}a^{12}+\frac{47159}{47360}a^{11}-\frac{22633}{47360}a^{10}-\frac{35473}{23680}a^{9}-\frac{607}{592}a^{8}-\frac{2131}{11840}a^{7}+\frac{33743}{11840}a^{6}+\frac{31901}{5920}a^{5}+\frac{10591}{2960}a^{4}-\frac{2366}{185}a^{3}-\frac{8849}{740}a^{2}+\frac{212}{185}a+\frac{3482}{185}$, $\frac{253}{378880}a^{23}-\frac{9}{11840}a^{22}-\frac{419}{94720}a^{21}-\frac{71}{11840}a^{20}+\frac{1319}{378880}a^{19}+\frac{2651}{94720}a^{18}+\frac{211}{18944}a^{17}-\frac{31}{94720}a^{16}-\frac{23679}{378880}a^{15}-\frac{381}{94720}a^{14}-\frac{8573}{94720}a^{13}+\frac{13691}{47360}a^{12}+\frac{3167}{94720}a^{11}+\frac{1863}{11840}a^{10}-\frac{22847}{23680}a^{9}+\frac{2041}{5920}a^{8}-\frac{7803}{23680}a^{7}+\frac{1201}{592}a^{6}-\frac{427}{1184}a^{5}+\frac{4373}{2960}a^{4}-\frac{4303}{740}a^{3}-\frac{2241}{740}a^{2}-\frac{419}{370}a+10$, $\frac{1183}{378880}a^{23}-\frac{343}{94720}a^{22}-\frac{313}{94720}a^{21}-\frac{119}{5920}a^{20}+\frac{2617}{75776}a^{19}-\frac{371}{23680}a^{18}+\frac{7217}{94720}a^{17}-\frac{2289}{18944}a^{16}+\frac{49579}{378880}a^{15}-\frac{18017}{47360}a^{14}+\frac{8073}{18944}a^{13}-\frac{14283}{47360}a^{12}+\frac{96201}{94720}a^{11}-\frac{35597}{23680}a^{10}+\frac{22261}{23680}a^{9}-\frac{23999}{11840}a^{8}+\frac{59847}{23680}a^{7}-\frac{963}{592}a^{6}+\frac{6081}{1184}a^{5}-\frac{33}{8}a^{4}-\frac{303}{148}a^{3}-\frac{231}{37}a^{2}+\frac{1338}{185}a+\frac{526}{185}$, $\frac{587}{75776}a^{23}-\frac{159}{23680}a^{22}-\frac{289}{23680}a^{21}-\frac{1997}{47360}a^{20}+\frac{22677}{378880}a^{19}+\frac{1333}{94720}a^{18}+\frac{3129}{23680}a^{17}-\frac{15551}{94720}a^{16}+\frac{6055}{75776}a^{15}-\frac{55607}{94720}a^{14}+\frac{21697}{47360}a^{13}+\frac{2391}{23680}a^{12}+\frac{139413}{94720}a^{11}-\frac{10953}{5920}a^{10}-\frac{1569}{5920}a^{9}-\frac{3437}{1480}a^{8}+\frac{12699}{4736}a^{7}+\frac{7543}{5920}a^{6}+\frac{5599}{740}a^{5}-\frac{1747}{592}a^{4}-\frac{8577}{740}a^{3}-\frac{923}{74}a^{2}+\frac{3703}{370}a+\frac{3392}{185}$, $\frac{955}{75776}a^{23}+\frac{23}{2368}a^{22}-\frac{1841}{47360}a^{21}-\frac{653}{9472}a^{20}-\frac{1371}{378880}a^{19}+\frac{16573}{94720}a^{18}+\frac{3269}{47360}a^{17}+\frac{13969}{94720}a^{16}-\frac{153037}{378880}a^{15}-\frac{22839}{94720}a^{14}-\frac{24249}{23680}a^{13}+\frac{51483}{23680}a^{12}+\frac{60181}{94720}a^{11}+\frac{36851}{23680}a^{10}-\frac{15499}{2368}a^{9}+\frac{262}{185}a^{8}-\frac{131541}{23680}a^{7}+\frac{14129}{1184}a^{6}+\frac{2589}{592}a^{5}+\frac{25693}{1480}a^{4}-\frac{26473}{740}a^{3}-\frac{14907}{740}a^{2}-\frac{3011}{370}a+\frac{9763}{185}$, $\frac{397}{23680}a^{23}-\frac{173}{94720}a^{22}-\frac{3007}{94720}a^{21}-\frac{137}{1480}a^{20}+\frac{223}{2960}a^{19}+\frac{7453}{94720}a^{18}+\frac{21111}{94720}a^{17}-\frac{779}{4736}a^{16}+\frac{339}{23680}a^{15}-\frac{93473}{94720}a^{14}+\frac{4133}{18944}a^{13}+\frac{3939}{4736}a^{12}+\frac{61601}{23680}a^{11}-\frac{12081}{5920}a^{10}-\frac{58573}{23680}a^{9}-\frac{39629}{11840}a^{8}+\frac{4351}{2960}a^{7}+\frac{14881}{2960}a^{6}+\frac{81613}{5920}a^{5}+\frac{1179}{370}a^{4}-\frac{20813}{740}a^{3}-\frac{4923}{185}a^{2}+\frac{1681}{185}a+\frac{7654}{185}$ 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:  \( 125335839.85669757 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 125335839.85669757 \cdot 8}{24\cdot\sqrt{96596555035261369442314470090080256}}\cr\approx \mathstrut & 0.508899028694790 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^24 - 8*x^21 + 3*x^20 - 4*x^19 + 32*x^18 - 20*x^17 + 37*x^16 - 116*x^15 + 72*x^14 - 128*x^13 + 380*x^12 - 256*x^11 + 288*x^10 - 928*x^9 + 592*x^8 - 640*x^7 + 2048*x^6 - 512*x^5 + 768*x^4 - 4096*x^3 + 4096)
 
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 - 8*x^21 + 3*x^20 - 4*x^19 + 32*x^18 - 20*x^17 + 37*x^16 - 116*x^15 + 72*x^14 - 128*x^13 + 380*x^12 - 256*x^11 + 288*x^10 - 928*x^9 + 592*x^8 - 640*x^7 + 2048*x^6 - 512*x^5 + 768*x^4 - 4096*x^3 + 4096, 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 - 8*x^21 + 3*x^20 - 4*x^19 + 32*x^18 - 20*x^17 + 37*x^16 - 116*x^15 + 72*x^14 - 128*x^13 + 380*x^12 - 256*x^11 + 288*x^10 - 928*x^9 + 592*x^8 - 640*x^7 + 2048*x^6 - 512*x^5 + 768*x^4 - 4096*x^3 + 4096);
 
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 - 8*x^21 + 3*x^20 - 4*x^19 + 32*x^18 - 20*x^17 + 37*x^16 - 116*x^15 + 72*x^14 - 128*x^13 + 380*x^12 - 256*x^11 + 288*x^10 - 928*x^9 + 592*x^8 - 640*x^7 + 2048*x^6 - 512*x^5 + 768*x^4 - 4096*x^3 + 4096);
 
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

$C_2^2\times D_6$ (as 24T30):

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 24 conjugacy class representatives for $C_2^2\times D_6$
Character table for $C_2^2\times D_6$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-1}) \), 3.3.568.1, \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\zeta_{12})\), \(\Q(\zeta_{8})\), 6.6.139373568.1, 6.6.10323968.1, 6.0.2580992.1, 6.0.8710848.1, 6.6.69686784.1, 6.0.278747136.2, 6.0.5161984.1, \(\Q(\zeta_{24})\), 12.0.310799863312809984.2, 12.12.310799863312809984.1, 12.0.4856247864262656.1, 12.0.310799863312809984.1, 12.0.77699965828202496.1, 12.0.426337261060096.1, 12.0.19424991457050624.1

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

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

Sibling fields

Degree 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 R ${\href{/padicField/5.2.0.1}{2} }^{12}$ ${\href{/padicField/7.6.0.1}{6} }^{4}$ ${\href{/padicField/11.2.0.1}{2} }^{12}$ ${\href{/padicField/13.6.0.1}{6} }^{4}$ ${\href{/padicField/17.2.0.1}{2} }^{12}$ ${\href{/padicField/19.6.0.1}{6} }^{4}$ ${\href{/padicField/23.6.0.1}{6} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{12}$ ${\href{/padicField/31.6.0.1}{6} }^{4}$ ${\href{/padicField/37.2.0.1}{2} }^{12}$ ${\href{/padicField/41.2.0.1}{2} }^{12}$ ${\href{/padicField/43.6.0.1}{6} }^{4}$ ${\href{/padicField/47.6.0.1}{6} }^{4}$ ${\href{/padicField/53.2.0.1}{2} }^{12}$ ${\href{/padicField/59.2.0.1}{2} }^{12}$

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 2.8.16.6$x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
\(3\) Copy content Toggle raw display 3.12.6.2$x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
3.12.6.2$x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
\(71\) Copy content Toggle raw display 71.2.0.1$x^{2} + 69 x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} + 69 x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} + 69 x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} + 69 x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
71.4.2.1$x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$