↑

Properties

Label 24.0.497...896.1
Degree $24$
Signature $[0, 12]$
Discriminant $4.971\times 10^{33}$
Root discriminant \(25.35\)
Ramified primes $2,3,7$
Class number $3$ (GRH)
Class group [3] (GRH)
Galois group $C_2^2\times C_6$ (as 24T3)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 13*x^20 + 143*x^16 - 336*x^12 + 663*x^8 - 26*x^4 + 1)
 
gp: K = bnfinit(y^24 - 13*y^20 + 143*y^16 - 336*y^12 + 663*y^8 - 26*y^4 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 13*x^20 + 143*x^16 - 336*x^12 + 663*x^8 - 26*x^4 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 13*x^20 + 143*x^16 - 336*x^12 + 663*x^8 - 26*x^4 + 1)
 

\( x^{24} - 13x^{20} + 143x^{16} - 336x^{12} + 663x^{8} - 26x^{4} + 1 \) 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:   \(4971225787269833056964369392336896\) \(\medspace = 2^{48}\cdot 3^{12}\cdot 7^{16}\) 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:  \(25.35\)
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:  $2^{2}3^{1/2}7^{2/3}\approx 25.352413640746768$
Ramified primes:   \(2\), \(3\), \(7\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $24$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(168=2^{3}\cdot 3\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{168}(1,·)$, $\chi_{168}(67,·)$, $\chi_{168}(65,·)$, $\chi_{168}(137,·)$, $\chi_{168}(11,·)$, $\chi_{168}(79,·)$, $\chi_{168}(107,·)$, $\chi_{168}(149,·)$, $\chi_{168}(23,·)$, $\chi_{168}(25,·)$, $\chi_{168}(71,·)$, $\chi_{168}(155,·)$, $\chi_{168}(29,·)$, $\chi_{168}(95,·)$, $\chi_{168}(163,·)$, $\chi_{168}(37,·)$, $\chi_{168}(43,·)$, $\chi_{168}(109,·)$, $\chi_{168}(113,·)$, $\chi_{168}(53,·)$, $\chi_{168}(121,·)$, $\chi_{168}(127,·)$, $\chi_{168}(151,·)$, $\chi_{168}(85,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{2048}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13}a^{12}+\frac{1}{13}$, $\frac{1}{13}a^{13}+\frac{1}{13}a$, $\frac{1}{13}a^{14}+\frac{1}{13}a^{2}$, $\frac{1}{13}a^{15}+\frac{1}{13}a^{3}$, $\frac{1}{377}a^{16}+\frac{10}{377}a^{12}-\frac{12}{29}a^{8}+\frac{66}{377}a^{4}-\frac{120}{377}$, $\frac{1}{377}a^{17}+\frac{10}{377}a^{13}-\frac{12}{29}a^{9}+\frac{66}{377}a^{5}-\frac{120}{377}a$, $\frac{1}{377}a^{18}+\frac{10}{377}a^{14}-\frac{12}{29}a^{10}+\frac{66}{377}a^{6}-\frac{120}{377}a^{2}$, $\frac{1}{377}a^{19}+\frac{10}{377}a^{15}-\frac{12}{29}a^{11}+\frac{66}{377}a^{7}-\frac{120}{377}a^{3}$, $\frac{1}{16211}a^{20}-\frac{11}{16211}a^{16}-\frac{395}{16211}a^{12}-\frac{51}{16211}a^{8}+\frac{2}{16211}a^{4}-\frac{1656}{16211}$, $\frac{1}{16211}a^{21}-\frac{11}{16211}a^{17}-\frac{395}{16211}a^{13}-\frac{51}{16211}a^{9}+\frac{2}{16211}a^{5}-\frac{1656}{16211}a$, $\frac{1}{16211}a^{22}-\frac{11}{16211}a^{18}-\frac{395}{16211}a^{14}-\frac{51}{16211}a^{10}+\frac{2}{16211}a^{6}-\frac{1656}{16211}a^{2}$, $\frac{1}{16211}a^{23}-\frac{11}{16211}a^{19}-\frac{395}{16211}a^{15}-\frac{51}{16211}a^{11}+\frac{2}{16211}a^{7}-\frac{1656}{16211}a^{3}$ 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

$C_{3}$, which has order $3$ (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:   \( -\frac{7213}{16211} a^{23} + \frac{93791}{16211} a^{19} - \frac{1031701}{16211} a^{15} + \frac{2426101}{16211} a^{11} - \frac{4783341}{16211} a^{7} + \frac{187582}{16211} a^{3} \)  (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:   $\frac{1164}{16211}a^{20}-\frac{15083}{16211}a^{16}+\frac{165870}{16211}a^{12}-\frac{384702}{16211}a^{8}+\frac{759730}{16211}a^{4}-\frac{582}{16211}$, $\frac{280}{16211}a^{20}-\frac{3682}{16211}a^{16}+\frac{40502}{16211}a^{12}-\frac{98689}{16211}a^{8}+\frac{187782}{16211}a^{4}-\frac{7364}{16211}$, $\frac{76}{377}a^{22}-\frac{22}{559}a^{20}-\frac{985}{377}a^{18}+\frac{285}{559}a^{16}+\frac{10830}{377}a^{14}-\frac{3135}{559}a^{12}-\frac{25118}{377}a^{10}+\frac{7271}{559}a^{8}+\frac{49507}{377}a^{6}-\frac{14535}{559}a^{4}-\frac{38}{377}a^{2}+\frac{11}{559}$, $\frac{7364}{16211}a^{23}-\frac{95452}{16211}a^{19}+\frac{1049370}{16211}a^{15}-\frac{2433802}{16211}a^{11}+\frac{4783643}{16211}a^{7}-\frac{3682}{16211}a^{3}+1$, $\frac{76}{377}a^{22}+\frac{1466}{16211}a^{21}-\frac{985}{377}a^{18}-\frac{19007}{16211}a^{17}+\frac{10830}{377}a^{14}+\frac{208905}{16211}a^{13}-\frac{25118}{377}a^{10}-\frac{484513}{16211}a^{9}+\frac{49507}{377}a^{6}+\frac{947556}{16211}a^{5}-\frac{38}{377}a^{2}-\frac{733}{16211}a$, $\frac{13262}{16211}a^{23}+\frac{56}{16211}a^{22}-\frac{171897}{16211}a^{19}-\frac{616}{16211}a^{18}+\frac{1889835}{16211}a^{15}+\frac{6561}{16211}a^{14}-\frac{4383091}{16211}a^{11}-\frac{2856}{16211}a^{10}+\frac{8619730}{16211}a^{7}+\frac{112}{16211}a^{6}-\frac{6631}{16211}a^{3}+\frac{81844}{16211}a^{2}-1$, $\frac{56}{16211}a^{22}+\frac{1785}{16211}a^{21}-\frac{22}{559}a^{20}-\frac{616}{16211}a^{18}-\frac{23161}{16211}a^{17}+\frac{285}{559}a^{16}+\frac{6561}{16211}a^{14}+\frac{254771}{16211}a^{13}-\frac{3135}{559}a^{12}-\frac{2856}{16211}a^{10}-\frac{594694}{16211}a^{9}+\frac{7271}{559}a^{8}+\frac{112}{16211}a^{6}+\frac{1181211}{16211}a^{5}-\frac{14535}{559}a^{4}+\frac{81844}{16211}a^{2}-\frac{46322}{16211}a+\frac{11}{559}$, $\frac{7364}{16211}a^{23}-\frac{1466}{16211}a^{21}+\frac{22}{16211}a^{20}-\frac{95452}{16211}a^{19}+\frac{19007}{16211}a^{17}-\frac{242}{16211}a^{16}+\frac{1049370}{16211}a^{15}-\frac{208905}{16211}a^{13}+\frac{2533}{16211}a^{12}-\frac{2433802}{16211}a^{11}+\frac{484513}{16211}a^{9}-\frac{1122}{16211}a^{8}+\frac{4783643}{16211}a^{7}-\frac{947556}{16211}a^{5}+\frac{44}{16211}a^{4}-\frac{3682}{16211}a^{3}+\frac{733}{16211}a+\frac{7213}{16211}$, $\frac{13262}{16211}a^{23}-\frac{56}{16211}a^{22}-\frac{171897}{16211}a^{19}+\frac{616}{16211}a^{18}+\frac{1889835}{16211}a^{15}-\frac{6561}{16211}a^{14}-\frac{4383091}{16211}a^{11}+\frac{2856}{16211}a^{10}+\frac{8619730}{16211}a^{7}-\frac{112}{16211}a^{6}-\frac{6631}{16211}a^{3}-\frac{81844}{16211}a^{2}+1$, $\frac{7213}{16211}a^{23}-\frac{95}{16211}a^{22}-\frac{1466}{16211}a^{21}-\frac{93791}{16211}a^{19}+\frac{1045}{16211}a^{18}+\frac{19007}{16211}a^{17}+\frac{1031701}{16211}a^{15}-\frac{11108}{16211}a^{14}-\frac{208905}{16211}a^{13}-\frac{2426101}{16211}a^{11}+\frac{4845}{16211}a^{10}+\frac{484513}{16211}a^{9}+\frac{4783341}{16211}a^{7}-\frac{190}{16211}a^{6}-\frac{947556}{16211}a^{5}-\frac{187582}{16211}a^{3}-\frac{102056}{16211}a^{2}+\frac{733}{16211}a$, $\frac{7364}{16211}a^{23}+\frac{5786}{16211}a^{22}-\frac{3}{1247}a^{21}-\frac{95452}{16211}a^{19}-\frac{75213}{16211}a^{18}+\frac{33}{1247}a^{17}+\frac{1049370}{16211}a^{15}+\frac{827343}{16211}a^{14}-\frac{4547}{16211}a^{13}-\frac{2433802}{16211}a^{11}-\frac{1943577}{16211}a^{10}+\frac{153}{1247}a^{9}+\frac{4783643}{16211}a^{7}+\frac{3835863}{16211}a^{6}-\frac{6}{1247}a^{5}-\frac{3682}{16211}a^{3}-\frac{150426}{16211}a^{2}-\frac{36423}{16211}a$ 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:  \( 24013423.00725467 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 24013423.00725467 \cdot 3}{24\cdot\sqrt{4971225787269833056964369392336896}}\cr\approx \mathstrut & 0.161172432351781 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^24 - 13*x^20 + 143*x^16 - 336*x^12 + 663*x^8 - 26*x^4 + 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 - 13*x^20 + 143*x^16 - 336*x^12 + 663*x^8 - 26*x^4 + 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 - 13*x^20 + 143*x^16 - 336*x^12 + 663*x^8 - 26*x^4 + 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 - 13*x^20 + 143*x^16 - 336*x^12 + 663*x^8 - 26*x^4 + 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

$C_2^2\times C_6$ (as 24T3):

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 $C_2^2\times C_6$
Character table for $C_2^2\times C_6$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-6}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{12})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{3})\), 6.0.153664.1, 6.0.64827.1, 6.6.4148928.1, 6.0.1229312.1, 6.6.1229312.1, 6.6.33191424.1, 6.0.33191424.1, \(\Q(\zeta_{24})\), 12.0.17213603549184.1, 12.0.96717311574016.1, 12.0.70506920137457664.2, 12.0.1101670627147776.1, 12.0.1101670627147776.2, 12.0.70506920137457664.1, 12.12.70506920137457664.1

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 R R ${\href{/padicField/5.6.0.1}{6} }^{4}$ R ${\href{/padicField/11.6.0.1}{6} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{12}$ ${\href{/padicField/17.6.0.1}{6} }^{4}$ ${\href{/padicField/19.6.0.1}{6} }^{4}$ ${\href{/padicField/23.6.0.1}{6} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{12}$ ${\href{/padicField/31.6.0.1}{6} }^{4}$ ${\href{/padicField/37.6.0.1}{6} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{12}$ ${\href{/padicField/43.2.0.1}{2} }^{12}$ ${\href{/padicField/47.6.0.1}{6} }^{4}$ ${\href{/padicField/53.6.0.1}{6} }^{4}$ ${\href{/padicField/59.6.0.1}{6} }^{4}$

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 Deg $24$$4$$6$$48$
\(3\) 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}$
\(7\) Copy content Toggle raw display 7.6.4.3$x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$