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Properties

Label 24.0.510...000.3
Degree $24$
Signature $[0, 12]$
Discriminant $5.107\times 10^{36}$
Root discriminant \(33.85\)
Ramified primes $2,5,7$
Class number $26$ (GRH)
Class group [26] (GRH)
Galois group $C_2\times C_{12}$ (as 24T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 5*x^22 + 20*x^20 + 75*x^18 + 275*x^16 + 1000*x^14 + 3625*x^12 + 5000*x^10 + 6875*x^8 + 9375*x^6 + 12500*x^4 + 15625*x^2 + 15625)
 
gp: K = bnfinit(y^24 + 5*y^22 + 20*y^20 + 75*y^18 + 275*y^16 + 1000*y^14 + 3625*y^12 + 5000*y^10 + 6875*y^8 + 9375*y^6 + 12500*y^4 + 15625*y^2 + 15625, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 + 5*x^22 + 20*x^20 + 75*x^18 + 275*x^16 + 1000*x^14 + 3625*x^12 + 5000*x^10 + 6875*x^8 + 9375*x^6 + 12500*x^4 + 15625*x^2 + 15625);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 + 5*x^22 + 20*x^20 + 75*x^18 + 275*x^16 + 1000*x^14 + 3625*x^12 + 5000*x^10 + 6875*x^8 + 9375*x^6 + 12500*x^4 + 15625*x^2 + 15625)
 

\( x^{24} + 5 x^{22} + 20 x^{20} + 75 x^{18} + 275 x^{16} + 1000 x^{14} + 3625 x^{12} + 5000 x^{10} + \cdots + 15625 \) 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:   \(5106705043047168064000000000000000000\) \(\medspace = 2^{24}\cdot 5^{18}\cdot 7^{20}\) 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:  \(33.85\)
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:  $2\cdot 5^{3/4}7^{5/6}\approx 33.8458843070916$
Ramified primes:   \(2\), \(5\), \(7\) 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:  \(140=2^{2}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{140}(1,·)$, $\chi_{140}(3,·)$, $\chi_{140}(69,·)$, $\chi_{140}(129,·)$, $\chi_{140}(9,·)$, $\chi_{140}(87,·)$, $\chi_{140}(81,·)$, $\chi_{140}(67,·)$, $\chi_{140}(23,·)$, $\chi_{140}(89,·)$, $\chi_{140}(27,·)$, $\chi_{140}(29,·)$, $\chi_{140}(107,·)$, $\chi_{140}(101,·)$, $\chi_{140}(103,·)$, $\chi_{140}(41,·)$, $\chi_{140}(43,·)$, $\chi_{140}(109,·)$, $\chi_{140}(47,·)$, $\chi_{140}(83,·)$, $\chi_{140}(121,·)$, $\chi_{140}(123,·)$, $\chi_{140}(61,·)$, $\chi_{140}(127,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{2048}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5}a^{4}$, $\frac{1}{5}a^{5}$, $\frac{1}{5}a^{6}$, $\frac{1}{5}a^{7}$, $\frac{1}{25}a^{8}$, $\frac{1}{25}a^{9}$, $\frac{1}{25}a^{10}$, $\frac{1}{25}a^{11}$, $\frac{1}{125}a^{12}$, $\frac{1}{125}a^{13}$, $\frac{1}{3625}a^{14}+\frac{11}{29}$, $\frac{1}{3625}a^{15}+\frac{11}{29}a$, $\frac{1}{18125}a^{16}+\frac{8}{29}a^{2}$, $\frac{1}{18125}a^{17}+\frac{8}{29}a^{3}$, $\frac{1}{18125}a^{18}+\frac{11}{145}a^{4}$, $\frac{1}{18125}a^{19}+\frac{11}{145}a^{5}$, $\frac{1}{90625}a^{20}+\frac{8}{145}a^{6}$, $\frac{1}{90625}a^{21}+\frac{8}{145}a^{7}$, $\frac{1}{90625}a^{22}+\frac{11}{725}a^{8}$, $\frac{1}{90625}a^{23}+\frac{11}{725}a^{9}$ 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

$C_{26}$, which has order $26$ (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:   \( -\frac{11}{90625} a^{22} - \frac{44}{90625} a^{20} - \frac{33}{18125} a^{18} - \frac{121}{18125} a^{16} - \frac{88}{3625} a^{14} - \frac{11}{125} a^{12} - \frac{8}{25} a^{10} - \frac{121}{725} a^{8} - \frac{33}{145} a^{6} - \frac{44}{145} a^{4} - \frac{11}{29} a^{2} - \frac{11}{29} \)  (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:   $\frac{3}{18125}a^{16}-\frac{34}{29}a^{2}$, $\frac{1}{18125}a^{18}-\frac{76}{145}a^{4}-1$, $\frac{3}{90625}a^{22}-\frac{199}{725}a^{8}-1$, $\frac{7}{90625}a^{22}+\frac{28}{90625}a^{20}+\frac{22}{18125}a^{18}+\frac{78}{18125}a^{16}+\frac{56}{3625}a^{14}+\frac{7}{125}a^{12}+\frac{1}{5}a^{10}+\frac{77}{725}a^{8}+\frac{21}{145}a^{6}-\frac{48}{145}a^{4}-\frac{14}{29}a^{2}+\frac{7}{29}$, $\frac{11}{90625}a^{22}+\frac{44}{90625}a^{20}+\frac{32}{18125}a^{18}+\frac{121}{18125}a^{16}+\frac{3}{125}a^{14}+\frac{11}{125}a^{12}+\frac{8}{25}a^{10}+\frac{121}{725}a^{8}+\frac{33}{145}a^{6}+\frac{24}{29}a^{4}+\frac{11}{29}a^{2}+2$, $\frac{13}{90625}a^{23}+\frac{13}{18125}a^{21}+\frac{7}{90625}a^{20}+\frac{52}{18125}a^{19}+\frac{39}{3625}a^{17}+\frac{143}{3625}a^{15}+\frac{18}{125}a^{13}+\frac{13}{25}a^{11}+\frac{104}{145}a^{9}+\frac{143}{145}a^{7}-\frac{89}{145}a^{6}+\frac{39}{29}a^{5}+\frac{52}{29}a^{3}+\frac{65}{29}a$, $\frac{13}{90625}a^{23}-\frac{11}{90625}a^{22}+\frac{13}{18125}a^{21}-\frac{44}{90625}a^{20}+\frac{52}{18125}a^{19}-\frac{33}{18125}a^{18}+\frac{39}{3625}a^{17}-\frac{24}{3625}a^{16}+\frac{143}{3625}a^{15}-\frac{88}{3625}a^{14}+\frac{18}{125}a^{13}-\frac{11}{125}a^{12}+\frac{13}{25}a^{11}-\frac{8}{25}a^{10}+\frac{104}{145}a^{9}-\frac{121}{725}a^{8}+\frac{143}{145}a^{7}-\frac{33}{145}a^{6}+\frac{39}{29}a^{5}-\frac{44}{145}a^{4}+\frac{52}{29}a^{3}-\frac{32}{29}a^{2}+\frac{65}{29}a-\frac{11}{29}$, $\frac{18}{90625}a^{23}+\frac{3}{90625}a^{22}+\frac{72}{90625}a^{21}+\frac{4}{90625}a^{20}+\frac{54}{18125}a^{19}+\frac{1}{18125}a^{18}+\frac{198}{18125}a^{17}+\frac{1}{18125}a^{16}+\frac{144}{3625}a^{15}+\frac{18}{125}a^{13}+\frac{13}{25}a^{11}+\frac{198}{725}a^{9}-\frac{199}{725}a^{8}+\frac{54}{145}a^{7}-\frac{11}{29}a^{6}+\frac{72}{145}a^{5}-\frac{76}{145}a^{4}+\frac{18}{29}a^{3}-\frac{21}{29}a^{2}+\frac{18}{29}a-1$, $\frac{8}{90625}a^{23}+\frac{8}{18125}a^{21}+\frac{3}{90625}a^{20}+\frac{32}{18125}a^{19}+\frac{24}{3625}a^{17}+\frac{88}{3625}a^{15}+\frac{11}{125}a^{13}+\frac{8}{25}a^{11}+\frac{64}{145}a^{9}+\frac{88}{145}a^{7}-\frac{34}{145}a^{6}+\frac{24}{29}a^{5}+\frac{32}{29}a^{3}+\frac{40}{29}a$, $\frac{11}{90625}a^{23}+\frac{44}{90625}a^{21}-\frac{4}{90625}a^{20}+\frac{33}{18125}a^{19}+\frac{121}{18125}a^{17}-\frac{1}{18125}a^{16}+\frac{88}{3625}a^{15}+\frac{11}{125}a^{13}+\frac{8}{25}a^{11}+\frac{121}{725}a^{9}+\frac{33}{145}a^{7}+\frac{11}{29}a^{6}+\frac{44}{145}a^{5}+\frac{11}{29}a^{3}+\frac{21}{29}a^{2}+\frac{11}{29}a$, $\frac{1}{90625}a^{20}+\frac{1}{18125}a^{18}+\frac{2}{18125}a^{17}+\frac{1}{18125}a^{16}-\frac{1}{3625}a^{14}-\frac{21}{145}a^{6}-\frac{76}{145}a^{4}-\frac{13}{29}a^{3}-\frac{21}{29}a^{2}-\frac{11}{29}$ 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:  \( 36584901.886295445 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 36584901.886295445 \cdot 26}{14\cdot\sqrt{5106705043047168064000000000000000000}}\cr\approx \mathstrut & 0.113824366041538 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^24 + 5*x^22 + 20*x^20 + 75*x^18 + 275*x^16 + 1000*x^14 + 3625*x^12 + 5000*x^10 + 6875*x^8 + 9375*x^6 + 12500*x^4 + 15625*x^2 + 15625)
 
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 + 5*x^22 + 20*x^20 + 75*x^18 + 275*x^16 + 1000*x^14 + 3625*x^12 + 5000*x^10 + 6875*x^8 + 9375*x^6 + 12500*x^4 + 15625*x^2 + 15625, 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 + 5*x^22 + 20*x^20 + 75*x^18 + 275*x^16 + 1000*x^14 + 3625*x^12 + 5000*x^10 + 6875*x^8 + 9375*x^6 + 12500*x^4 + 15625*x^2 + 15625);
 
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 + 5*x^22 + 20*x^20 + 75*x^18 + 275*x^16 + 1000*x^14 + 3625*x^12 + 5000*x^10 + 6875*x^8 + 9375*x^6 + 12500*x^4 + 15625*x^2 + 15625);
 
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

$C_2\times C_{12}$ (as 24T2):

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 $C_2\times C_{12}$
Character table for $C_2\times C_{12}$ is not computed

Intermediate fields

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\zeta_{20})^+\), 4.0.98000.1, 6.0.2100875.1, 6.6.300125.1, \(\Q(\zeta_{7})\), 8.0.9604000000.2, 12.0.4413675765625.1, 12.12.46118408000000000.1, 12.0.2259801992000000000.1

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 ${\href{/padicField/3.12.0.1}{12} }^{2}$ R R ${\href{/padicField/11.6.0.1}{6} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{6}$ ${\href{/padicField/17.12.0.1}{12} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{4}$ ${\href{/padicField/23.12.0.1}{12} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{12}$ ${\href{/padicField/31.6.0.1}{6} }^{4}$ ${\href{/padicField/37.12.0.1}{12} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{12}$ ${\href{/padicField/43.4.0.1}{4} }^{6}$ ${\href{/padicField/47.12.0.1}{12} }^{2}$ ${\href{/padicField/53.12.0.1}{12} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }^{4}$

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
\(2\) Copy content Toggle raw display 2.12.12.25$x^{12} + 12 x^{11} + 60 x^{10} + 160 x^{9} + 308 x^{8} + 736 x^{7} + 2272 x^{6} + 5632 x^{5} + 10608 x^{4} + 15040 x^{3} + 12224 x^{2} + 3584 x + 704$$2$$6$$12$$C_{12}$$[2]^{6}$
2.12.12.25$x^{12} + 12 x^{11} + 60 x^{10} + 160 x^{9} + 308 x^{8} + 736 x^{7} + 2272 x^{6} + 5632 x^{5} + 10608 x^{4} + 15040 x^{3} + 12224 x^{2} + 3584 x + 704$$2$$6$$12$$C_{12}$$[2]^{6}$
\(5\) Copy content Toggle raw display Deg $24$$4$$6$$18$
\(7\) Copy content Toggle raw display Deg $24$$6$$4$$20$