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Properties

Label 24.0.291...576.1
Degree $24$
Signature $[0, 12]$
Discriminant $2.914\times 10^{33}$
Root discriminant \(24.79\)
Ramified primes $2,3,7$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $C_2^2\times C_6$ (as 24T3)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 2*x^22 + 8*x^18 - 16*x^16 + 64*x^12 - 256*x^8 + 512*x^6 - 2048*x^2 + 4096)
 
gp: K = bnfinit(y^24 - 2*y^22 + 8*y^18 - 16*y^16 + 64*y^12 - 256*y^8 + 512*y^6 - 2048*y^2 + 4096, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 2*x^22 + 8*x^18 - 16*x^16 + 64*x^12 - 256*x^8 + 512*x^6 - 2048*x^2 + 4096);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 2*x^22 + 8*x^18 - 16*x^16 + 64*x^12 - 256*x^8 + 512*x^6 - 2048*x^2 + 4096)
 

\( x^{24} - 2x^{22} + 8x^{18} - 16x^{16} + 64x^{12} - 256x^{8} + 512x^{6} - 2048x^{2} + 4096 \) 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:   \(2914041287899137980901233132568576\) \(\medspace = 2^{36}\cdot 3^{12}\cdot 7^{20}\) 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:  \(24.79\)
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))
 
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$$\chi_{168}(1,·)$, $\chi_{168}(5,·)$, $\chi_{168}(65,·)$, $\chi_{168}(137,·)$, $\chi_{168}(13,·)$, $\chi_{168}(17,·)$, $\chi_{168}(149,·)$, $\chi_{168}(89,·)$, $\chi_{168}(25,·)$, $\chi_{168}(157,·)$, $\chi_{168}(101,·)$, $\chi_{168}(97,·)$, $\chi_{168}(37,·)$, $\chi_{168}(145,·)$, $\chi_{168}(41,·)$, $\chi_{168}(125,·)$, $\chi_{168}(109,·)$, $\chi_{168}(29,·)$, $\chi_{168}(113,·)$, $\chi_{168}(53,·)$, $\chi_{168}(73,·)$, $\chi_{168}(121,·)$, $\chi_{168}(61,·)$, $\chi_{168}(85,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{2048}$

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

$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{128}a^{14}$, $\frac{1}{128}a^{15}$, $\frac{1}{256}a^{16}$, $\frac{1}{256}a^{17}$, $\frac{1}{512}a^{18}$, $\frac{1}{512}a^{19}$, $\frac{1}{1024}a^{20}$, $\frac{1}{1024}a^{21}$, $\frac{1}{2048}a^{22}$, $\frac{1}{2048}a^{23}$ 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

$C_{6}$, 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:   \( -\frac{1}{2048} a^{22} \)  (order $42$) 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:   $\frac{1}{256}a^{16}+\frac{1}{2}a^{2}-1$, $\frac{1}{2048}a^{22}+\frac{1}{256}a^{16}+\frac{1}{16}a^{8}+\frac{1}{2}a^{2}$, $\frac{1}{32}a^{10}-1$, $\frac{1}{1024}a^{20}-1$, $\frac{1}{2048}a^{22}+\frac{1}{256}a^{16}+\frac{1}{32}a^{10}+\frac{1}{16}a^{8}+\frac{1}{4}a^{4}+\frac{1}{2}a^{2}-1$, $\frac{1}{128}a^{15}+\frac{1}{16}a^{8}+a$, $\frac{1}{2048}a^{22}-\frac{1}{512}a^{19}+\frac{1}{256}a^{16}$, $\frac{1}{2048}a^{23}+\frac{1}{2048}a^{22}+\frac{1}{32}a^{10}+\frac{1}{16}a^{9}$, $\frac{1}{2048}a^{23}-\frac{1}{1024}a^{21}+\frac{1}{256}a^{17}-\frac{1}{128}a^{15}-\frac{1}{64}a^{13}+\frac{1}{32}a^{11}+\frac{1}{32}a^{10}-\frac{1}{8}a^{7}+\frac{1}{4}a^{5}+\frac{1}{2}a^{2}-a$, $\frac{1}{2048}a^{22}+\frac{1}{1024}a^{21}+\frac{1}{8}a^{7}+\frac{1}{8}a^{6}$, $\frac{1}{2048}a^{23}-\frac{1}{2048}a^{22}-\frac{1}{1024}a^{20}+\frac{1}{256}a^{17}-\frac{1}{128}a^{14}+\frac{1}{16}a^{9}-\frac{1}{8}a^{8}+\frac{1}{8}a^{6}+\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-a+1$ 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:  \( 25576950.522236273 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 25576950.522236273 \cdot 6}{42\cdot\sqrt{2914041287899137980901233132568576}}\cr\approx \mathstrut & 0.256248477211597 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^24 - 2*x^22 + 8*x^18 - 16*x^16 + 64*x^12 - 256*x^8 + 512*x^6 - 2048*x^2 + 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 - 2*x^22 + 8*x^18 - 16*x^16 + 64*x^12 - 256*x^8 + 512*x^6 - 2048*x^2 + 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 - 2*x^22 + 8*x^18 - 16*x^16 + 64*x^12 - 256*x^8 + 512*x^6 - 2048*x^2 + 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 - 2*x^22 + 8*x^18 - 16*x^16 + 64*x^12 - 256*x^8 + 512*x^6 - 2048*x^2 + 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

$C_2^2\times C_6$ (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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{42}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{21}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-3}, \sqrt{-14})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{2}, \sqrt{-7})\), \(\Q(\sqrt{2}, \sqrt{21})\), \(\Q(\sqrt{-6}, \sqrt{-14})\), \(\Q(\sqrt{-6}, \sqrt{-7})\), 6.0.64827.1, 6.6.1229312.1, 6.0.33191424.1, 6.0.8605184.1, 6.6.232339968.1, \(\Q(\zeta_{7})\), \(\Q(\zeta_{21})^+\), 8.0.796594176.2, 12.0.1101670627147776.2, 12.0.53981860730241024.2, \(\Q(\zeta_{21})\), 12.0.74049191673856.2, 12.12.53981860730241024.1, 12.0.53981860730241024.1, 12.0.53981860730241024.5

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 2.12.18.23$x^{12} - 12 x^{11} + 48 x^{10} - 344 x^{9} + 8244 x^{8} - 31136 x^{7} + 54848 x^{6} - 23104 x^{5} + 18864 x^{4} - 7360 x^{3} + 5120 x^{2} + 5760 x + 1472$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
2.12.18.23$x^{12} - 12 x^{11} + 48 x^{10} - 344 x^{9} + 8244 x^{8} - 31136 x^{7} + 54848 x^{6} - 23104 x^{5} + 18864 x^{4} - 7360 x^{3} + 5120 x^{2} + 5760 x + 1472$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
\(3\) 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 7.6.5.5$x^{6} + 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 7$$6$$1$$5$$C_6$$[\ ]_{6}$