↑

Properties

Label 24.0.507...625.1
Degree $24$
Signature $[0, 12]$
Discriminant $5.072\times 10^{35}$
Root discriminant \(30.74\)
Ramified primes $3,5,7$
Class number $9$ (GRH)
Class group [3, 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 + 26*x^18 - 53*x^12 + 18954*x^6 + 531441)
 
gp: K = bnfinit(y^24 + 26*y^18 - 53*y^12 + 18954*y^6 + 531441, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 + 26*x^18 - 53*x^12 + 18954*x^6 + 531441);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 + 26*x^18 - 53*x^12 + 18954*x^6 + 531441)
 

\( x^{24} + 26x^{18} - 53x^{12} + 18954x^{6} + 531441 \) 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:   \(507202869744901554493542558837890625\) \(\medspace = 3^{36}\cdot 5^{12}\cdot 7^{12}\) 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:  \(30.74\)
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:  $3^{3/2}5^{1/2}7^{1/2}\approx 30.740852297878796$
Ramified primes:   \(3\), \(5\), \(7\) 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:  \(315=3^{2}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{315}(64,·)$, $\chi_{315}(1,·)$, $\chi_{315}(134,·)$, $\chi_{315}(71,·)$, $\chi_{315}(139,·)$, $\chi_{315}(76,·)$, $\chi_{315}(209,·)$, $\chi_{315}(146,·)$, $\chi_{315}(211,·)$, $\chi_{315}(281,·)$, $\chi_{315}(29,·)$, $\chi_{315}(286,·)$, $\chi_{315}(34,·)$, $\chi_{315}(104,·)$, $\chi_{315}(41,·)$, $\chi_{315}(106,·)$, $\chi_{315}(274,·)$, $\chi_{315}(239,·)$, $\chi_{315}(176,·)$, $\chi_{315}(244,·)$, $\chi_{315}(181,·)$, $\chi_{315}(169,·)$, $\chi_{315}(314,·)$, $\chi_{315}(251,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{2048}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{12}+\frac{1}{8}a^{6}+\frac{1}{8}$, $\frac{1}{24}a^{13}+\frac{5}{24}a^{7}-\frac{1}{2}a^{4}-\frac{11}{24}a$, $\frac{1}{72}a^{14}+\frac{17}{72}a^{8}+\frac{1}{72}a^{2}$, $\frac{1}{216}a^{15}+\frac{53}{216}a^{9}+\frac{1}{216}a^{3}-\frac{1}{2}$, $\frac{1}{648}a^{16}-\frac{55}{648}a^{10}-\frac{215}{648}a^{4}$, $\frac{1}{1944}a^{17}+\frac{269}{1944}a^{11}+\frac{433}{1944}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{618192}a^{18}-\frac{1}{432}a^{15}+\frac{7}{2916}a^{12}+\frac{55}{432}a^{9}-\frac{547}{2916}a^{6}+\frac{215}{432}a^{3}+\frac{291}{848}$, $\frac{1}{1854576}a^{19}-\frac{1}{1296}a^{16}+\frac{7}{8748}a^{13}+\frac{55}{1296}a^{10}+\frac{911}{8748}a^{7}-\frac{433}{1296}a^{4}+\frac{715}{2544}a$, $\frac{1}{5563728}a^{20}-\frac{1}{3888}a^{17}+\frac{7}{26244}a^{14}+\frac{703}{3888}a^{11}+\frac{5285}{26244}a^{8}+\frac{1511}{3888}a^{5}+\frac{3259}{7632}a^{2}$, $\frac{1}{16691184}a^{21}-\frac{701}{314928}a^{15}-\frac{1}{16}a^{12}+\frac{61235}{314928}a^{9}-\frac{1}{16}a^{6}+\frac{3511}{11448}a^{3}-\frac{1}{16}$, $\frac{1}{50073552}a^{22}-\frac{701}{944784}a^{16}-\frac{1}{48}a^{13}-\frac{96229}{944784}a^{10}+\frac{7}{48}a^{7}-\frac{13661}{34344}a^{4}-\frac{1}{48}a$, $\frac{1}{150220656}a^{23}-\frac{701}{2834352}a^{17}-\frac{1}{144}a^{14}+\frac{376163}{2834352}a^{11}-\frac{17}{144}a^{8}-\frac{30833}{103032}a^{5}-\frac{1}{144}a^{2}$ 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

$C_{3}\times C_{3}$, which has order $9$ (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:   \( -\frac{1}{34344} a^{22} - \frac{16039}{34344} a^{4} \)  (order $18$) 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:   $\frac{1}{11448}a^{21}-\frac{13}{38637}a^{18}+\frac{1}{1458}a^{12}+\frac{13}{729}a^{6}+\frac{21763}{11448}a^{3}-\frac{338}{53}$, $\frac{221}{1390932}a^{20}+\frac{26}{38637}a^{18}-\frac{17}{52488}a^{14}-\frac{1}{729}a^{12}+\frac{287}{52488}a^{8}-\frac{26}{729}a^{6}+\frac{1377}{424}a^{2}+\frac{676}{53}$, $\frac{1}{34344}a^{22}-\frac{1}{3816}a^{20}+\frac{16039}{34344}a^{4}-\frac{19855}{3816}a^{2}$, $\frac{3589}{50073552}a^{22}+\frac{337}{1854576}a^{19}-\frac{55}{472392}a^{16}-\frac{41}{34992}a^{13}-\frac{2159}{472392}a^{10}-\frac{337}{34992}a^{7}+\frac{96229}{68688}a^{4}+\frac{4381}{1272}a$, $\frac{2159}{25036776}a^{22}-\frac{649}{8345592}a^{21}-\frac{287}{2781864}a^{20}+\frac{13}{38637}a^{18}-\frac{55}{472392}a^{16}+\frac{53}{157464}a^{15}-\frac{17}{52488}a^{14}-\frac{1}{1458}a^{12}-\frac{2159}{472392}a^{10}+\frac{649}{157464}a^{9}+\frac{287}{52488}a^{8}-\frac{13}{729}a^{6}+\frac{28067}{17172}a^{4}-\frac{8437}{5724}a^{3}-\frac{3731}{1908}a^{2}+\frac{729}{106}$, $\frac{2159}{25036776}a^{22}+\frac{1}{11448}a^{21}-\frac{221}{1390932}a^{20}-\frac{13}{38637}a^{18}-\frac{55}{472392}a^{16}+\frac{17}{52488}a^{14}+\frac{1}{1458}a^{12}-\frac{2159}{472392}a^{10}-\frac{287}{52488}a^{8}+\frac{13}{729}a^{6}+\frac{28067}{17172}a^{4}+\frac{21763}{11448}a^{3}-\frac{1377}{424}a^{2}-\frac{338}{53}$, $\frac{2159}{25036776}a^{22}+\frac{755}{4172796}a^{21}+\frac{287}{2781864}a^{20}+\frac{13}{38637}a^{18}-\frac{55}{472392}a^{16}-\frac{1}{78732}a^{15}+\frac{17}{52488}a^{14}-\frac{1}{1458}a^{12}-\frac{2159}{472392}a^{10}-\frac{755}{78732}a^{9}-\frac{287}{52488}a^{8}-\frac{13}{729}a^{6}+\frac{28067}{17172}a^{4}+\frac{9815}{2862}a^{3}+\frac{3731}{1908}a^{2}+\frac{623}{106}$, $\frac{5}{206064}a^{23}-\frac{715}{12518388}a^{22}+\frac{1171}{5563728}a^{20}+\frac{55}{472392}a^{16}-\frac{17}{104976}a^{14}+\frac{2159}{472392}a^{10}+\frac{287}{104976}a^{8}+\frac{114539}{206064}a^{5}-\frac{495}{424}a^{4}+\frac{4031}{954}a^{2}+1$, $\frac{2357}{25036776}a^{22}-\frac{3485}{16691184}a^{21}-\frac{451}{618192}a^{19}-\frac{55}{68688}a^{18}+\frac{43}{944784}a^{16}+\frac{53}{157464}a^{15}+\frac{1}{1458}a^{13}-\frac{1}{1296}a^{12}-\frac{6901}{944784}a^{10}+\frac{649}{157464}a^{9}+\frac{13}{729}a^{7}+\frac{55}{1296}a^{6}+\frac{131309}{68688}a^{4}-\frac{93313}{22896}a^{3}-\frac{36079}{2544}a-\frac{7071}{424}$, $\frac{2357}{75110328}a^{23}-\frac{3617}{16691184}a^{22}+\frac{3485}{5563728}a^{20}-\frac{221}{927288}a^{19}+\frac{43}{2834352}a^{17}+\frac{55}{314928}a^{16}-\frac{53}{52488}a^{14}+\frac{17}{34992}a^{13}-\frac{6901}{2834352}a^{11}+\frac{2159}{314928}a^{10}-\frac{649}{52488}a^{8}-\frac{287}{34992}a^{7}+\frac{131309}{206064}a^{5}-\frac{8305}{1908}a^{4}+\frac{93313}{7632}a^{2}-\frac{3707}{848}a$, $\frac{2357}{25036776}a^{22}-\frac{11}{1390932}a^{21}-\frac{451}{618192}a^{19}+\frac{451}{206064}a^{18}+\frac{43}{944784}a^{16}-\frac{17}{104976}a^{15}+\frac{1}{1458}a^{13}-\frac{1}{486}a^{12}-\frac{6901}{944784}a^{10}+\frac{287}{104976}a^{9}+\frac{13}{729}a^{7}-\frac{13}{243}a^{6}+\frac{131309}{68688}a^{4}-\frac{6347}{22896}a^{3}-\frac{36079}{2544}a+\frac{36927}{848}$ 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:  \( 70281481.48527941 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 70281481.48527941 \cdot 9}{18\cdot\sqrt{507202869744901554493542558837890625}}\cr\approx \mathstrut & 0.186800692168980 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^24 + 26*x^18 - 53*x^12 + 18954*x^6 + 531441)
 
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 + 26*x^18 - 53*x^12 + 18954*x^6 + 531441, 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 + 26*x^18 - 53*x^12 + 18954*x^6 + 531441);
 
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 + 26*x^18 - 53*x^12 + 18954*x^6 + 531441);
 
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

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-15}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{-3}, \sqrt{-35})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{-15}, \sqrt{21})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-7}, \sqrt{-15})\), \(\Q(\sqrt{5}, \sqrt{21})\), 6.0.281302875.3, \(\Q(\zeta_{9})\), 6.6.843908625.1, 6.0.2250423.1, 6.6.820125.1, 6.6.6751269.1, 6.0.2460375.1, 8.0.121550625.1, 12.0.712181767349390625.1, 12.0.79131307483265625.1, 12.0.712181767349390625.3, 12.0.45579633110361.1, 12.0.6053445140625.1, 12.0.712181767349390625.2, 12.12.712181767349390625.1

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 ${\href{/padicField/2.6.0.1}{6} }^{4}$ R R R ${\href{/padicField/11.6.0.1}{6} }^{4}$ ${\href{/padicField/13.6.0.1}{6} }^{4}$ ${\href{/padicField/17.2.0.1}{2} }^{12}$ ${\href{/padicField/19.2.0.1}{2} }^{12}$ ${\href{/padicField/23.6.0.1}{6} }^{4}$ ${\href{/padicField/29.6.0.1}{6} }^{4}$ ${\href{/padicField/31.6.0.1}{6} }^{4}$ ${\href{/padicField/37.2.0.1}{2} }^{12}$ ${\href{/padicField/41.6.0.1}{6} }^{4}$ ${\href{/padicField/43.6.0.1}{6} }^{4}$ ${\href{/padicField/47.6.0.1}{6} }^{4}$ ${\href{/padicField/53.2.0.1}{2} }^{12}$ ${\href{/padicField/59.6.0.1}{6} }^{4}$

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
\(3\) Copy content Toggle raw display 3.12.18.82$x^{12} + 24 x^{11} + 252 x^{10} + 1558 x^{9} + 6450 x^{8} + 19068 x^{7} + 41627 x^{6} + 68094 x^{5} + 83298 x^{4} + 74306 x^{3} + 45618 x^{2} + 17400 x + 3277$$6$$2$$18$$C_6\times C_2$$[2]_{2}^{2}$
3.12.18.82$x^{12} + 24 x^{11} + 252 x^{10} + 1558 x^{9} + 6450 x^{8} + 19068 x^{7} + 41627 x^{6} + 68094 x^{5} + 83298 x^{4} + 74306 x^{3} + 45618 x^{2} + 17400 x + 3277$$6$$2$$18$$C_6\times C_2$$[2]_{2}^{2}$
\(5\) 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 7.12.6.1$x^{12} + 44 x^{10} + 10 x^{9} + 786 x^{8} + 22 x^{7} + 6899 x^{6} - 3434 x^{5} + 31050 x^{4} - 28440 x^{3} + 84557 x^{2} - 48082 x + 107648$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
7.12.6.1$x^{12} + 44 x^{10} + 10 x^{9} + 786 x^{8} + 22 x^{7} + 6899 x^{6} - 3434 x^{5} + 31050 x^{4} - 28440 x^{3} + 84557 x^{2} - 48082 x + 107648$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$