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Properties

Label 24.0.784...625.1
Degree $24$
Signature $[0, 12]$
Discriminant $7.841\times 10^{32}$
Root discriminant \(23.47\)
Ramified primes $5,13$
Class number $4$ (GRH)
Class group [2, 2] (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 - x^23 + 2*x^22 - 3*x^21 + 5*x^20 - 8*x^19 + 13*x^18 - 21*x^17 + 34*x^16 - 55*x^15 + 89*x^14 - 144*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 1)
 
gp: K = bnfinit(y^24 - y^23 + 2*y^22 - 3*y^21 + 5*y^20 - 8*y^19 + 13*y^18 - 21*y^17 + 34*y^16 - 55*y^15 + 89*y^14 - 144*y^13 + 233*y^12 + 144*y^11 + 89*y^10 + 55*y^9 + 34*y^8 + 21*y^7 + 13*y^6 + 8*y^5 + 5*y^4 + 3*y^3 + 2*y^2 + y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - x^23 + 2*x^22 - 3*x^21 + 5*x^20 - 8*x^19 + 13*x^18 - 21*x^17 + 34*x^16 - 55*x^15 + 89*x^14 - 144*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - x^23 + 2*x^22 - 3*x^21 + 5*x^20 - 8*x^19 + 13*x^18 - 21*x^17 + 34*x^16 - 55*x^15 + 89*x^14 - 144*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 1)
 

\( x^{24} - x^{23} + 2 x^{22} - 3 x^{21} + 5 x^{20} - 8 x^{19} + 13 x^{18} - 21 x^{17} + 34 x^{16} + \cdots + 1 \) 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:   \(784140351063197047157560791015625\) \(\medspace = 5^{12}\cdot 13^{22}\) 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:  \(23.47\)
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:  $5^{1/2}13^{11/12}\approx 23.474683296117743$
Ramified primes:   \(5\), \(13\) 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:  \(65=5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{65}(64,·)$, $\chi_{65}(1,·)$, $\chi_{65}(4,·)$, $\chi_{65}(6,·)$, $\chi_{65}(9,·)$, $\chi_{65}(11,·)$, $\chi_{65}(14,·)$, $\chi_{65}(16,·)$, $\chi_{65}(19,·)$, $\chi_{65}(21,·)$, $\chi_{65}(24,·)$, $\chi_{65}(29,·)$, $\chi_{65}(31,·)$, $\chi_{65}(34,·)$, $\chi_{65}(36,·)$, $\chi_{65}(41,·)$, $\chi_{65}(44,·)$, $\chi_{65}(46,·)$, $\chi_{65}(49,·)$, $\chi_{65}(51,·)$, $\chi_{65}(54,·)$, $\chi_{65}(56,·)$, $\chi_{65}(59,·)$, $\chi_{65}(61,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{2048}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{233}a^{13}-\frac{89}{233}$, $\frac{1}{233}a^{14}-\frac{89}{233}a$, $\frac{1}{233}a^{15}-\frac{89}{233}a^{2}$, $\frac{1}{233}a^{16}-\frac{89}{233}a^{3}$, $\frac{1}{233}a^{17}-\frac{89}{233}a^{4}$, $\frac{1}{233}a^{18}-\frac{89}{233}a^{5}$, $\frac{1}{233}a^{19}-\frac{89}{233}a^{6}$, $\frac{1}{233}a^{20}-\frac{89}{233}a^{7}$, $\frac{1}{233}a^{21}-\frac{89}{233}a^{8}$, $\frac{1}{233}a^{22}-\frac{89}{233}a^{9}$, $\frac{1}{233}a^{23}-\frac{89}{233}a^{10}$ 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:   \( -\frac{5}{233} a^{18} - \frac{2584}{233} a^{5} \)  (order $26$) 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:   $\frac{1}{233}a^{16}+\frac{610}{233}a^{3}$, $\frac{8}{233}a^{19}+\frac{4181}{233}a^{6}+1$, $\frac{8}{233}a^{19}+\frac{2}{233}a^{16}+\frac{4181}{233}a^{6}+\frac{987}{233}a^{3}$, $\frac{5}{233}a^{18}+\frac{2584}{233}a^{5}+1$, $\frac{21}{233}a^{21}+\frac{3}{233}a^{17}+\frac{10946}{233}a^{8}+\frac{1597}{233}a^{4}+1$, $\frac{89}{233}a^{23}-\frac{144}{233}a^{22}+\frac{267}{233}a^{21}-\frac{432}{233}a^{20}+\frac{712}{233}a^{19}-\frac{1152}{233}a^{18}+\frac{1869}{233}a^{17}-\frac{3024}{233}a^{16}+\frac{4895}{233}a^{15}-\frac{7920}{233}a^{14}+\frac{12816}{233}a^{13}-89a^{12}+144a^{11}-\frac{7921}{233}a^{10}+\frac{12816}{233}a^{9}-\frac{3026}{233}a^{8}+\frac{4896}{233}a^{7}-\frac{1157}{233}a^{6}+\frac{1872}{233}a^{5}-\frac{445}{233}a^{4}+\frac{720}{233}a^{3}-\frac{178}{233}a^{2}+\frac{288}{233}a+\frac{144}{233}$, $\frac{55}{233}a^{23}+\frac{8}{233}a^{18}+\frac{28657}{233}a^{10}+\frac{4181}{233}a^{5}+1$, $\frac{13}{233}a^{20}-\frac{2}{233}a^{17}+\frac{1}{233}a^{14}+\frac{6765}{233}a^{7}-\frac{987}{233}a^{4}+\frac{377}{233}a$, $\frac{34}{233}a^{23}+\frac{55}{233}a^{22}+\frac{34}{233}a^{21}+\frac{21}{233}a^{20}+\frac{13}{233}a^{19}+\frac{8}{233}a^{18}+\frac{5}{233}a^{17}+\frac{1}{233}a^{16}+\frac{1}{233}a^{13}+\frac{17711}{233}a^{10}+\frac{28657}{233}a^{9}+\frac{17711}{233}a^{8}+\frac{10946}{233}a^{7}+\frac{6765}{233}a^{6}+\frac{4181}{233}a^{5}+\frac{2584}{233}a^{4}+\frac{610}{233}a^{3}+\frac{144}{233}$, $\frac{89}{233}a^{23}-\frac{110}{233}a^{22}+\frac{144}{233}a^{21}-\frac{267}{233}a^{20}+\frac{440}{233}a^{19}-\frac{715}{233}a^{18}+\frac{1152}{233}a^{17}-\frac{1870}{233}a^{16}+\frac{3025}{233}a^{15}-\frac{4895}{233}a^{14}+\frac{7920}{233}a^{13}-55a^{12}+89a^{11}+\frac{12816}{233}a^{10}-\frac{3025}{233}a^{9}-\frac{12816}{233}a^{8}+\frac{3026}{233}a^{7}-\frac{715}{233}a^{6}-\frac{440}{233}a^{5}-\frac{1872}{233}a^{4}-\frac{165}{233}a^{3}-\frac{110}{233}a^{2}+\frac{178}{233}a-\frac{288}{233}$, $\frac{89}{233}a^{23}-\frac{178}{233}a^{22}+a^{21}-2a^{20}+\frac{712}{233}a^{19}-\frac{1157}{233}a^{18}+\frac{1872}{233}a^{17}-\frac{3026}{233}a^{16}+21a^{15}-34a^{14}+55a^{13}-89a^{12}+144a^{11}-\frac{7921}{233}a^{10}-\frac{4895}{233}a^{9}-89a^{8}-55a^{7}-\frac{1157}{233}a^{6}-\frac{712}{233}a^{5}+\frac{1152}{233}a^{4}-\frac{267}{233}a^{3}-5a^{2}-3a-2$ 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:  \( 7346081.887826216 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 7346081.887826216 \cdot 4}{26\cdot\sqrt{784140351063197047157560791015625}}\cr\approx \mathstrut & 0.152793165125543 \end{aligned}\] (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 + 5*x^20 - 8*x^19 + 13*x^18 - 21*x^17 + 34*x^16 - 55*x^15 + 89*x^14 - 144*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 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 - x^23 + 2*x^22 - 3*x^21 + 5*x^20 - 8*x^19 + 13*x^18 - 21*x^17 + 34*x^16 - 55*x^15 + 89*x^14 - 144*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 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 - x^23 + 2*x^22 - 3*x^21 + 5*x^20 - 8*x^19 + 13*x^18 - 21*x^17 + 34*x^16 - 55*x^15 + 89*x^14 - 144*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 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 - x^23 + 2*x^22 - 3*x^21 + 5*x^20 - 8*x^19 + 13*x^18 - 21*x^17 + 34*x^16 - 55*x^15 + 89*x^14 - 144*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 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

$C_2\times C_{12}$ (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{65}) \), 3.3.169.1, \(\Q(\sqrt{5}, \sqrt{13})\), 4.0.2197.1, 4.0.54925.1, 6.6.3570125.1, \(\Q(\zeta_{13})^+\), 6.6.46411625.1, 8.0.3016755625.1, 12.12.2154038935140625.1, \(\Q(\zeta_{13})\), 12.0.28002506156828125.1

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 ${\href{/padicField/2.12.0.1}{12} }^{2}$ ${\href{/padicField/3.6.0.1}{6} }^{4}$ R ${\href{/padicField/7.12.0.1}{12} }^{2}$ ${\href{/padicField/11.12.0.1}{12} }^{2}$ R ${\href{/padicField/17.6.0.1}{6} }^{4}$ ${\href{/padicField/19.12.0.1}{12} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{4}$ ${\href{/padicField/29.3.0.1}{3} }^{8}$ ${\href{/padicField/31.4.0.1}{4} }^{6}$ ${\href{/padicField/37.12.0.1}{12} }^{2}$ ${\href{/padicField/41.12.0.1}{12} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{6}$ ${\href{/padicField/53.2.0.1}{2} }^{12}$ ${\href{/padicField/59.12.0.1}{12} }^{2}$

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 5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(13\) Copy content Toggle raw display Deg $24$$12$$2$$22$