↑

Properties

Label 24.0.925...625.1
Degree $24$
Signature $[0, 12]$
Discriminant $9.253\times 10^{34}$
Root discriminant \(28.64\)
Ramified primes $5,7,41$
Class number $4$ (GRH)
Class group [2, 2] (GRH)
Galois group $C_6\times D_4$ (as 24T38)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - 2*x^22 + 3*x^21 + x^20 - 2*x^19 + x^18 + 34*x^17 - 37*x^16 - 59*x^15 + 98*x^14 + 17*x^13 - 45*x^12 + 34*x^11 + 392*x^10 - 472*x^9 - 592*x^8 + 1088*x^7 + 64*x^6 - 256*x^5 + 256*x^4 + 1536*x^3 - 2048*x^2 - 2048*x + 4096)
 
gp: K = bnfinit(y^24 - y^23 - 2*y^22 + 3*y^21 + y^20 - 2*y^19 + y^18 + 34*y^17 - 37*y^16 - 59*y^15 + 98*y^14 + 17*y^13 - 45*y^12 + 34*y^11 + 392*y^10 - 472*y^9 - 592*y^8 + 1088*y^7 + 64*y^6 - 256*y^5 + 256*y^4 + 1536*y^3 - 2048*y^2 - 2048*y + 4096, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - x^23 - 2*x^22 + 3*x^21 + x^20 - 2*x^19 + x^18 + 34*x^17 - 37*x^16 - 59*x^15 + 98*x^14 + 17*x^13 - 45*x^12 + 34*x^11 + 392*x^10 - 472*x^9 - 592*x^8 + 1088*x^7 + 64*x^6 - 256*x^5 + 256*x^4 + 1536*x^3 - 2048*x^2 - 2048*x + 4096);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - x^23 - 2*x^22 + 3*x^21 + x^20 - 2*x^19 + x^18 + 34*x^17 - 37*x^16 - 59*x^15 + 98*x^14 + 17*x^13 - 45*x^12 + 34*x^11 + 392*x^10 - 472*x^9 - 592*x^8 + 1088*x^7 + 64*x^6 - 256*x^5 + 256*x^4 + 1536*x^3 - 2048*x^2 - 2048*x + 4096)
 

\( x^{24} - x^{23} - 2 x^{22} + 3 x^{21} + x^{20} - 2 x^{19} + x^{18} + 34 x^{17} - 37 x^{16} - 59 x^{15} + \cdots + 4096 \) 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:   \(92534566049630892119774472900390625\) \(\medspace = 5^{12}\cdot 7^{20}\cdot 41^{6}\) 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:  \(28.64\)
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:  $5^{1/2}7^{5/6}41^{1/2}\approx 72.46449954207208$
Ramified primes:   \(5\), \(7\), \(41\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{13}-\frac{1}{2}a^{12}-\frac{1}{4}a^{11}+\frac{1}{4}a^{10}-\frac{1}{2}a^{9}+\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{15}-\frac{1}{8}a^{14}-\frac{1}{4}a^{13}+\frac{3}{8}a^{12}+\frac{1}{8}a^{11}-\frac{1}{4}a^{10}+\frac{1}{8}a^{9}+\frac{1}{4}a^{8}+\frac{3}{8}a^{7}-\frac{3}{8}a^{6}+\frac{1}{4}a^{5}+\frac{1}{8}a^{4}+\frac{3}{8}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{16}a^{16}-\frac{1}{16}a^{15}-\frac{1}{8}a^{14}+\frac{3}{16}a^{13}+\frac{1}{16}a^{12}-\frac{1}{8}a^{11}+\frac{1}{16}a^{10}+\frac{1}{8}a^{9}-\frac{5}{16}a^{8}+\frac{5}{16}a^{7}+\frac{1}{8}a^{6}+\frac{1}{16}a^{5}+\frac{3}{16}a^{4}+\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{17}-\frac{1}{32}a^{16}-\frac{1}{16}a^{15}+\frac{3}{32}a^{14}+\frac{1}{32}a^{13}-\frac{1}{16}a^{12}+\frac{1}{32}a^{11}+\frac{1}{16}a^{10}-\frac{5}{32}a^{9}+\frac{5}{32}a^{8}+\frac{1}{16}a^{7}-\frac{15}{32}a^{6}-\frac{13}{32}a^{5}+\frac{1}{16}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{64}a^{18}-\frac{1}{64}a^{17}-\frac{1}{32}a^{16}+\frac{3}{64}a^{15}+\frac{1}{64}a^{14}-\frac{1}{32}a^{13}+\frac{1}{64}a^{12}-\frac{15}{32}a^{11}+\frac{27}{64}a^{10}+\frac{5}{64}a^{9}-\frac{15}{32}a^{8}+\frac{17}{64}a^{7}+\frac{19}{64}a^{6}-\frac{15}{32}a^{5}+\frac{1}{8}a^{4}-\frac{3}{8}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{128}a^{19}-\frac{1}{128}a^{18}-\frac{1}{64}a^{17}+\frac{3}{128}a^{16}+\frac{1}{128}a^{15}-\frac{1}{64}a^{14}+\frac{1}{128}a^{13}+\frac{17}{64}a^{12}-\frac{37}{128}a^{11}-\frac{59}{128}a^{10}-\frac{15}{64}a^{9}+\frac{17}{128}a^{8}-\frac{45}{128}a^{7}+\frac{17}{64}a^{6}+\frac{1}{16}a^{5}+\frac{5}{16}a^{4}+\frac{3}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{256}a^{20}-\frac{1}{256}a^{19}-\frac{1}{128}a^{18}+\frac{3}{256}a^{17}+\frac{1}{256}a^{16}-\frac{1}{128}a^{15}+\frac{1}{256}a^{14}+\frac{17}{128}a^{13}-\frac{37}{256}a^{12}-\frac{59}{256}a^{11}+\frac{49}{128}a^{10}+\frac{17}{256}a^{9}-\frac{45}{256}a^{8}+\frac{17}{128}a^{7}-\frac{15}{32}a^{6}+\frac{5}{32}a^{5}-\frac{5}{16}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{512}a^{21}-\frac{1}{512}a^{20}-\frac{1}{256}a^{19}+\frac{3}{512}a^{18}+\frac{1}{512}a^{17}-\frac{1}{256}a^{16}+\frac{1}{512}a^{15}+\frac{17}{256}a^{14}-\frac{37}{512}a^{13}-\frac{59}{512}a^{12}+\frac{49}{256}a^{11}+\frac{17}{512}a^{10}-\frac{45}{512}a^{9}+\frac{17}{256}a^{8}-\frac{15}{64}a^{7}+\frac{5}{64}a^{6}-\frac{5}{32}a^{5}+\frac{1}{8}a^{4}+\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{1024}a^{22}-\frac{1}{1024}a^{21}-\frac{1}{512}a^{20}+\frac{3}{1024}a^{19}+\frac{1}{1024}a^{18}-\frac{1}{512}a^{17}+\frac{1}{1024}a^{16}+\frac{17}{512}a^{15}-\frac{37}{1024}a^{14}-\frac{59}{1024}a^{13}+\frac{49}{512}a^{12}+\frac{17}{1024}a^{11}-\frac{45}{1024}a^{10}+\frac{17}{512}a^{9}+\frac{49}{128}a^{8}-\frac{59}{128}a^{7}+\frac{27}{64}a^{6}+\frac{1}{16}a^{5}+\frac{1}{16}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{2048}a^{23}-\frac{1}{2048}a^{22}-\frac{1}{1024}a^{21}+\frac{3}{2048}a^{20}+\frac{1}{2048}a^{19}-\frac{1}{1024}a^{18}+\frac{1}{2048}a^{17}+\frac{17}{1024}a^{16}-\frac{37}{2048}a^{15}-\frac{59}{2048}a^{14}+\frac{49}{1024}a^{13}+\frac{17}{2048}a^{12}-\frac{45}{2048}a^{11}+\frac{17}{1024}a^{10}+\frac{49}{256}a^{9}-\frac{59}{256}a^{8}-\frac{37}{128}a^{7}-\frac{15}{32}a^{6}+\frac{1}{32}a^{5}-\frac{1}{8}a^{4}+\frac{1}{8}a^{3}-\frac{1}{4}a^{2}$ Copy content Toggle raw display

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

$C_{2}\times C_{2}$, which has order $4$ (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:   \( -\frac{15}{256} a^{23} + \frac{15}{256} a^{22} + \frac{15}{128} a^{21} - \frac{1}{256} a^{20} - \frac{15}{256} a^{19} + \frac{15}{128} a^{18} - \frac{15}{256} a^{17} - \frac{255}{128} a^{16} + \frac{555}{256} a^{15} + \frac{885}{256} a^{14} - \frac{45}{128} a^{13} - \frac{255}{256} a^{12} + \frac{675}{256} a^{11} - \frac{255}{128} a^{10} - \frac{735}{32} a^{9} + \frac{885}{32} a^{8} + \frac{555}{16} a^{7} - \frac{225}{64} a^{6} - \frac{15}{4} a^{5} + 15 a^{4} - 15 a^{3} - 90 a^{2} + 120 a + 120 \)  (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:   $\frac{105}{2048}a^{23}-\frac{35}{2048}a^{22}-\frac{35}{512}a^{21}+\frac{35}{2048}a^{20}+\frac{35}{2048}a^{19}-\frac{13}{256}a^{18}+\frac{105}{2048}a^{17}+\frac{105}{64}a^{16}-\frac{1365}{2048}a^{15}-\frac{4025}{2048}a^{14}+\frac{245}{512}a^{13}+\frac{385}{2048}a^{12}-\frac{2231}{2048}a^{11}+\frac{175}{128}a^{10}+\frac{4655}{256}a^{9}-\frac{1085}{128}a^{8}-\frac{315}{16}a^{7}+\frac{105}{32}a^{6}-6a^{4}+\frac{35}{4}a^{3}+70a^{2}-35a-70$, $\frac{9}{256}a^{23}+\frac{9}{128}a^{22}-\frac{27}{256}a^{21}-\frac{9}{256}a^{20}+\frac{9}{128}a^{19}-\frac{9}{256}a^{18}-\frac{15}{256}a^{17}+\frac{333}{256}a^{16}+\frac{531}{256}a^{15}-\frac{441}{128}a^{14}-\frac{153}{256}a^{13}+\frac{405}{256}a^{12}-\frac{153}{128}a^{11}-\frac{179}{256}a^{10}+\frac{531}{32}a^{9}+\frac{333}{16}a^{8}-\frac{153}{4}a^{7}-\frac{9}{4}a^{6}+9a^{5}-9a^{4}-\frac{3}{8}a^{3}+72a^{2}+72a-145$, $\frac{11}{512}a^{23}+\frac{345}{512}a^{16}+\frac{3919}{512}a^{9}+31a^{2}+1$, $\frac{23}{512}a^{23}+\frac{1}{256}a^{22}-\frac{3}{256}a^{21}+\frac{5}{128}a^{20}-\frac{3}{256}a^{19}-\frac{21}{256}a^{18}+\frac{15}{128}a^{17}+\frac{699}{512}a^{16}-\frac{1}{32}a^{15}-\frac{3}{256}a^{14}+\frac{243}{256}a^{13}-\frac{75}{128}a^{12}-\frac{369}{256}a^{11}+\frac{689}{256}a^{10}+\frac{7183}{512}a^{9}-\frac{95}{128}a^{8}+\frac{57}{16}a^{7}+\frac{369}{64}a^{6}-\frac{21}{4}a^{5}-6a^{4}+\frac{123}{8}a^{3}+49a^{2}+\frac{3}{2}a+25$, $\frac{79}{2048}a^{23}+\frac{305}{2048}a^{22}-\frac{137}{1024}a^{21}-\frac{79}{2048}a^{20}+\frac{151}{2048}a^{19}-\frac{21}{1024}a^{18}-\frac{245}{2048}a^{17}+\frac{1443}{1024}a^{16}+\frac{9489}{2048}a^{15}-\frac{9037}{2048}a^{14}-\frac{727}{1024}a^{13}+\frac{3563}{2048}a^{12}-\frac{1835}{2048}a^{11}-\frac{2147}{1024}a^{10}+\frac{9211}{512}a^{9}+\frac{6425}{128}a^{8}-\frac{801}{16}a^{7}-\frac{195}{64}a^{6}+\frac{21}{2}a^{5}-\frac{121}{16}a^{4}-\frac{17}{2}a^{3}+79a^{2}+191a-195$, $\frac{19}{256}a^{23}-\frac{25}{512}a^{22}-\frac{61}{512}a^{21}-\frac{9}{256}a^{20}-\frac{1}{512}a^{19}+\frac{37}{512}a^{18}-\frac{37}{256}a^{17}+\frac{1223}{512}a^{16}-\frac{11}{8}a^{15}-\frac{2145}{512}a^{14}-\frac{479}{512}a^{13}+\frac{77}{256}a^{12}+\frac{629}{512}a^{11}-\frac{1729}{512}a^{10}+\frac{1819}{64}a^{9}-\frac{3807}{256}a^{8}-\frac{1651}{32}a^{7}-\frac{375}{64}a^{6}+\frac{117}{32}a^{5}+\frac{9}{2}a^{4}-\frac{39}{2}a^{3}+\frac{493}{4}a^{2}-\frac{125}{2}a-219$, $\frac{19}{2048}a^{23}+\frac{153}{2048}a^{22}+\frac{37}{1024}a^{21}-\frac{67}{2048}a^{20}+\frac{39}{2048}a^{19}+\frac{21}{1024}a^{18}-\frac{129}{2048}a^{17}+\frac{377}{1024}a^{16}+\frac{4997}{2048}a^{15}+\frac{1971}{2048}a^{14}-\frac{701}{1024}a^{13}+\frac{1103}{2048}a^{12}+\frac{549}{2048}a^{11}-\frac{1365}{1024}a^{10}+\frac{2799}{512}a^{9}+\frac{7023}{256}a^{8}+\frac{155}{16}a^{7}-\frac{233}{64}a^{6}+\frac{7}{2}a^{5}+\frac{3}{8}a^{4}-\frac{57}{8}a^{3}+28a^{2}+\frac{213}{2}a+40$, $\frac{163}{1024}a^{23}+\frac{93}{512}a^{22}-\frac{425}{1024}a^{21}-\frac{79}{1024}a^{20}+\frac{89}{512}a^{19}-\frac{139}{1024}a^{18}-\frac{201}{1024}a^{17}+\frac{5667}{1024}a^{16}+\frac{5541}{1024}a^{15}-\frac{6825}{512}a^{14}-\frac{1259}{1024}a^{13}+\frac{3971}{1024}a^{12}-\frac{2017}{512}a^{11}-\frac{2857}{1024}a^{10}+\frac{17223}{256}a^{9}+\frac{3533}{64}a^{8}-\frac{18931}{128}a^{7}-\frac{129}{32}a^{6}+\frac{697}{32}a^{5}-\frac{441}{16}a^{4}-\frac{29}{4}a^{3}+\frac{1133}{4}a^{2}+197a-566$, $\frac{155}{1024}a^{23}+\frac{61}{1024}a^{22}-\frac{137}{512}a^{21}-\frac{39}{1024}a^{20}+\frac{119}{1024}a^{19}-\frac{33}{512}a^{18}-\frac{109}{1024}a^{17}+\frac{2581}{512}a^{16}+\frac{1677}{1024}a^{15}-\frac{8661}{1024}a^{14}-\frac{283}{512}a^{13}+\frac{2547}{1024}a^{12}-\frac{1963}{1024}a^{11}-\frac{647}{512}a^{10}+\frac{1895}{32}a^{9}+\frac{3731}{256}a^{8}-\frac{11883}{128}a^{7}-\frac{79}{64}a^{6}+\frac{439}{32}a^{5}-\frac{213}{16}a^{4}-\frac{13}{8}a^{3}+\frac{973}{4}a^{2}+\frac{85}{2}a-354$, $\frac{249}{2048}a^{23}-\frac{179}{2048}a^{22}-\frac{57}{128}a^{21}+\frac{7}{2048}a^{20}+\frac{251}{2048}a^{19}-\frac{113}{512}a^{18}+\frac{133}{2048}a^{17}+\frac{1057}{256}a^{16}-\frac{6969}{2048}a^{15}-\frac{28065}{2048}a^{14}+\frac{147}{256}a^{13}+\frac{4493}{2048}a^{12}-\frac{10255}{2048}a^{11}+\frac{1487}{512}a^{10}+\frac{24471}{512}a^{9}-\frac{11587}{256}a^{8}-\frac{18827}{128}a^{7}+\frac{395}{64}a^{6}+\frac{37}{4}a^{5}-\frac{461}{16}a^{4}+\frac{95}{4}a^{3}+188a^{2}-203a-555$, $\frac{5}{512}a^{23}-\frac{3}{512}a^{22}-\frac{5}{256}a^{21}-\frac{7}{512}a^{20}+\frac{31}{512}a^{19}-\frac{25}{256}a^{18}+\frac{27}{512}a^{17}+\frac{15}{32}a^{16}-\frac{259}{512}a^{15}-\frac{229}{512}a^{14}-\frac{7}{256}a^{13}+\frac{603}{512}a^{12}-\frac{1139}{512}a^{11}+\frac{481}{256}a^{10}+\frac{1581}{256}a^{9}-\frac{1187}{128}a^{8}-\frac{21}{16}a^{7}+\frac{87}{64}a^{6}+\frac{45}{8}a^{5}-\frac{205}{16}a^{4}+\frac{119}{8}a^{3}+24a^{2}-49a+14$ 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:  \( 54448598.00948402 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 54448598.00948402 \cdot 4}{14\cdot\sqrt{92534566049630892119774472900390625}}\cr\approx \mathstrut & 0.193608794688587 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - 2*x^22 + 3*x^21 + x^20 - 2*x^19 + x^18 + 34*x^17 - 37*x^16 - 59*x^15 + 98*x^14 + 17*x^13 - 45*x^12 + 34*x^11 + 392*x^10 - 472*x^9 - 592*x^8 + 1088*x^7 + 64*x^6 - 256*x^5 + 256*x^4 + 1536*x^3 - 2048*x^2 - 2048*x + 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 - x^23 - 2*x^22 + 3*x^21 + x^20 - 2*x^19 + x^18 + 34*x^17 - 37*x^16 - 59*x^15 + 98*x^14 + 17*x^13 - 45*x^12 + 34*x^11 + 392*x^10 - 472*x^9 - 592*x^8 + 1088*x^7 + 64*x^6 - 256*x^5 + 256*x^4 + 1536*x^3 - 2048*x^2 - 2048*x + 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 - x^23 - 2*x^22 + 3*x^21 + x^20 - 2*x^19 + x^18 + 34*x^17 - 37*x^16 - 59*x^15 + 98*x^14 + 17*x^13 - 45*x^12 + 34*x^11 + 392*x^10 - 472*x^9 - 592*x^8 + 1088*x^7 + 64*x^6 - 256*x^5 + 256*x^4 + 1536*x^3 - 2048*x^2 - 2048*x + 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 - x^23 - 2*x^22 + 3*x^21 + x^20 - 2*x^19 + x^18 + 34*x^17 - 37*x^16 - 59*x^15 + 98*x^14 + 17*x^13 - 45*x^12 + 34*x^11 + 392*x^10 - 472*x^9 - 592*x^8 + 1088*x^7 + 64*x^6 - 256*x^5 + 256*x^4 + 1536*x^3 - 2048*x^2 - 2048*x + 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

$C_6\times D_4$ (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-35}) \), \(\Q(\zeta_{7})^+\), 4.4.50225.1, 4.0.1025.1, \(\Q(\sqrt{5}, \sqrt{-7})\), 6.6.300125.1, \(\Q(\zeta_{7})\), 6.0.2100875.1, 8.0.2522550625.1, 12.12.304194947442640625.1, 12.0.6208060151890625.1, 12.0.4413675765625.1

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 ${\href{/padicField/2.6.0.1}{6} }^{4}$ ${\href{/padicField/3.12.0.1}{12} }^{2}$ R R ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{6}$ ${\href{/padicField/17.12.0.1}{12} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{4}$ ${\href{/padicField/23.6.0.1}{6} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }^{12}$ ${\href{/padicField/31.6.0.1}{6} }^{4}$ ${\href{/padicField/37.6.0.1}{6} }^{4}$ R ${\href{/padicField/43.2.0.1}{2} }^{12}$ ${\href{/padicField/47.12.0.1}{12} }^{2}$ ${\href{/padicField/53.12.0.1}{12} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }^{4}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(5\) Copy content Toggle raw display 5.12.6.1$x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
5.12.6.1$x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
\(7\) Copy content Toggle raw display Deg $24$$6$$4$$20$
\(41\) Copy content Toggle raw display 41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.2.1$x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$