sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 12*x^23 - 40*x^22 + 401*x^21 - 930*x^20 + 247873*x^19 - 1200746*x^18 - 6559960*x^17 - 64580422*x^16 - 2923221605*x^15 - 12334220651*x^14 - 93433926183*x^13 - 127075389375*x^12 + 9477419064504*x^11 + 83412644565090*x^10 + 530594726847237*x^9 + 3434957990181416*x^8 + 5105977496091715*x^7 - 45579398797207878*x^6 - 385128811285350640*x^5 + 2216757763306056151*x^4 - 22438363614405112672*x^3 + 141472039110471732990*x^2 - 220230876669439407034*x + 777019087232248674575)
gp: K = bnfinit(y^24 - 12*y^23 - 40*y^22 + 401*y^21 - 930*y^20 + 247873*y^19 - 1200746*y^18 - 6559960*y^17 - 64580422*y^16 - 2923221605*y^15 - 12334220651*y^14 - 93433926183*y^13 - 127075389375*y^12 + 9477419064504*y^11 + 83412644565090*y^10 + 530594726847237*y^9 + 3434957990181416*y^8 + 5105977496091715*y^7 - 45579398797207878*y^6 - 385128811285350640*y^5 + 2216757763306056151*y^4 - 22438363614405112672*y^3 + 141472039110471732990*y^2 - 220230876669439407034*y + 777019087232248674575, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 12*x^23 - 40*x^22 + 401*x^21 - 930*x^20 + 247873*x^19 - 1200746*x^18 - 6559960*x^17 - 64580422*x^16 - 2923221605*x^15 - 12334220651*x^14 - 93433926183*x^13 - 127075389375*x^12 + 9477419064504*x^11 + 83412644565090*x^10 + 530594726847237*x^9 + 3434957990181416*x^8 + 5105977496091715*x^7 - 45579398797207878*x^6 - 385128811285350640*x^5 + 2216757763306056151*x^4 - 22438363614405112672*x^3 + 141472039110471732990*x^2 - 220230876669439407034*x + 777019087232248674575);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^24 - 12*x^23 - 40*x^22 + 401*x^21 - 930*x^20 + 247873*x^19 - 1200746*x^18 - 6559960*x^17 - 64580422*x^16 - 2923221605*x^15 - 12334220651*x^14 - 93433926183*x^13 - 127075389375*x^12 + 9477419064504*x^11 + 83412644565090*x^10 + 530594726847237*x^9 + 3434957990181416*x^8 + 5105977496091715*x^7 - 45579398797207878*x^6 - 385128811285350640*x^5 + 2216757763306056151*x^4 - 22438363614405112672*x^3 + 141472039110471732990*x^2 - 220230876669439407034*x + 777019087232248674575)
\( x^{24} - 12 x^{23} - 40 x^{22} + 401 x^{21} - 930 x^{20} + 247873 x^{19} - 1200746 x^{18} + \cdots + 77\!\cdots\!75 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Degree : $24$
sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
Signature : $(4, 10)$
sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
Discriminant :
\(1211192363043189952963494621037336955654250561565277166664600372314453125\)
\(\medspace = 5^{39}\cdot 109^{22}\)
sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant : \(1008.02\)
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: OK = ring_of_integers(K);
(1.0 * abs(discriminant(OK)))^(1/degree(K))
Galois root discriminant : $5^{39/20}109^{19/20}\approx 1988.5965494204843$
Ramified primes :
\(5\), \(109\)
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant(OK))
Discriminant root field : \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$ :
$C_4$
sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphism_group(K)
This field is not Galois over $\Q$. This is not a CM field . This field has no CM subfields.
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{109}a^{20}-\frac{10}{109}a^{19}-\frac{7}{109}a^{18}+\frac{21}{109}a^{17}-\frac{31}{109}a^{16}+\frac{6}{109}a^{15}+\frac{47}{109}a^{14}-\frac{47}{109}a^{13}+\frac{45}{109}a^{12}-\frac{30}{109}a^{11}-\frac{38}{109}a^{10}+\frac{47}{109}a^{9}-\frac{4}{109}a^{8}+\frac{18}{109}a^{7}+\frac{50}{109}a^{6}-\frac{10}{109}a^{5}+\frac{22}{109}a^{4}-\frac{9}{109}a^{3}+\frac{28}{109}a^{2}+\frac{10}{109}a+\frac{27}{109}$, $\frac{1}{109}a^{21}+\frac{2}{109}a^{19}-\frac{49}{109}a^{18}-\frac{39}{109}a^{17}+\frac{23}{109}a^{16}-\frac{2}{109}a^{15}-\frac{13}{109}a^{14}+\frac{11}{109}a^{13}-\frac{16}{109}a^{12}-\frac{11}{109}a^{11}-\frac{6}{109}a^{10}+\frac{30}{109}a^{9}-\frac{22}{109}a^{8}+\frac{12}{109}a^{7}+\frac{54}{109}a^{6}+\frac{31}{109}a^{5}-\frac{7}{109}a^{4}+\frac{47}{109}a^{3}-\frac{37}{109}a^{2}+\frac{18}{109}a+\frac{52}{109}$, $\frac{1}{545}a^{22}-\frac{1}{545}a^{21}-\frac{28}{109}a^{19}-\frac{17}{109}a^{18}+\frac{238}{545}a^{17}+\frac{37}{545}a^{16}+\frac{39}{109}a^{15}-\frac{14}{109}a^{14}-\frac{52}{109}a^{13}+\frac{24}{545}a^{12}-\frac{44}{545}a^{11}-\frac{43}{109}a^{10}-\frac{51}{109}a^{9}+\frac{52}{109}a^{8}-\frac{103}{545}a^{7}-\frac{232}{545}a^{6}+\frac{40}{109}a^{5}+\frac{2}{109}a^{4}-\frac{35}{109}a^{3}-\frac{219}{545}a^{2}+\frac{14}{545}a-\frac{43}{109}$, $\frac{1}{16\cdots 75}a^{23}-\frac{15\cdots 39}{16\cdots 75}a^{22}+\frac{27\cdots 63}{16\cdots 75}a^{21}-\frac{32\cdots 83}{67\cdots 55}a^{20}+\frac{11\cdots 94}{33\cdots 75}a^{19}+\frac{21\cdots 58}{16\cdots 75}a^{18}+\frac{78\cdots 13}{16\cdots 75}a^{17}-\frac{67\cdots 86}{16\cdots 75}a^{16}+\frac{39\cdots 48}{61\cdots 95}a^{15}+\frac{38\cdots 84}{33\cdots 75}a^{14}+\frac{72\cdots 59}{16\cdots 75}a^{13}+\frac{50\cdots 24}{16\cdots 75}a^{12}+\frac{45\cdots 27}{16\cdots 75}a^{11}-\frac{57\cdots 47}{67\cdots 55}a^{10}-\frac{39\cdots 37}{33\cdots 75}a^{9}-\frac{62\cdots 93}{16\cdots 75}a^{8}+\frac{57\cdots 29}{12\cdots 75}a^{7}+\frac{34\cdots 11}{16\cdots 75}a^{6}-\frac{56\cdots 89}{13\cdots 31}a^{5}-\frac{11\cdots 03}{33\cdots 75}a^{4}+\frac{65\cdots 81}{16\cdots 75}a^{3}-\frac{30\cdots 93}{12\cdots 75}a^{2}+\frac{40\cdots 33}{16\cdots 75}a-\frac{27\cdots 81}{67\cdots 55}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
Ideal class group : not computed
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Narrow class group : not computed
sage: K.narrow_class_group().invariants()
gp: bnfnarrow(K)
magma: NarrowClassGroup(K);
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
Rank : $13$
sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
Torsion generator :
\( -1 \)
(order $2$)
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 : not computed
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 : not computed
sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
Unit signature rank : not computed
\[
\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot R \cdot h}{2\cdot\sqrt{1211192363043189952963494621037336955654250561565277166664600372314453125}}\cr\mathstrut & \text{
some values not computed }
\end{aligned}\]
sage: # self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^24 - 12*x^23 - 40*x^22 + 401*x^21 - 930*x^20 + 247873*x^19 - 1200746*x^18 - 6559960*x^17 - 64580422*x^16 - 2923221605*x^15 - 12334220651*x^14 - 93433926183*x^13 - 127075389375*x^12 + 9477419064504*x^11 + 83412644565090*x^10 + 530594726847237*x^9 + 3434957990181416*x^8 + 5105977496091715*x^7 - 45579398797207878*x^6 - 385128811285350640*x^5 + 2216757763306056151*x^4 - 22438363614405112672*x^3 + 141472039110471732990*x^2 - 220230876669439407034*x + 777019087232248674575)
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))))
gp: \\ self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^24 - 12*x^23 - 40*x^22 + 401*x^21 - 930*x^20 + 247873*x^19 - 1200746*x^18 - 6559960*x^17 - 64580422*x^16 - 2923221605*x^15 - 12334220651*x^14 - 93433926183*x^13 - 127075389375*x^12 + 9477419064504*x^11 + 83412644565090*x^10 + 530594726847237*x^9 + 3434957990181416*x^8 + 5105977496091715*x^7 - 45579398797207878*x^6 - 385128811285350640*x^5 + 2216757763306056151*x^4 - 22438363614405112672*x^3 + 141472039110471732990*x^2 - 220230876669439407034*x + 777019087232248674575, 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)))]
magma: /* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 12*x^23 - 40*x^22 + 401*x^21 - 930*x^20 + 247873*x^19 - 1200746*x^18 - 6559960*x^17 - 64580422*x^16 - 2923221605*x^15 - 12334220651*x^14 - 93433926183*x^13 - 127075389375*x^12 + 9477419064504*x^11 + 83412644565090*x^10 + 530594726847237*x^9 + 3434957990181416*x^8 + 5105977496091715*x^7 - 45579398797207878*x^6 - 385128811285350640*x^5 + 2216757763306056151*x^4 - 22438363614405112672*x^3 + 141472039110471732990*x^2 - 220230876669439407034*x + 777019087232248674575);
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)));
oscar: # self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = polynomial_ring(QQ); K, a = number_field(x^24 - 12*x^23 - 40*x^22 + 401*x^21 - 930*x^20 + 247873*x^19 - 1200746*x^18 - 6559960*x^17 - 64580422*x^16 - 2923221605*x^15 - 12334220651*x^14 - 93433926183*x^13 - 127075389375*x^12 + 9477419064504*x^11 + 83412644565090*x^10 + 530594726847237*x^9 + 3434957990181416*x^8 + 5105977496091715*x^7 - 45579398797207878*x^6 - 385128811285350640*x^5 + 2216757763306056151*x^4 - 22438363614405112672*x^3 + 141472039110471732990*x^2 - 220230876669439407034*x + 777019087232248674575);
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))))
$\GL(2,5)$ (as 24T1353 ):
sage: K.galois_group()
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K);
degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
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(L)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
$p$
$2$
$3$
$5$
$7$
$11$
$13$
$17$
$19$
$23$
$29$
$31$
$37$
$41$
$43$
$47$
$53$
$59$
Cycle type
$24$
${\href{/padicField/3.8.0.1}{8} }^{3}$
R
$24$
${\href{/padicField/11.12.0.1}{12} }^{2}$
${\href{/padicField/13.4.0.1}{4} }^{5}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$
${\href{/padicField/17.4.0.1}{4} }^{5}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$
${\href{/padicField/19.5.0.1}{5} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$
$24$
${\href{/padicField/29.12.0.1}{12} }^{2}$
${\href{/padicField/31.4.0.1}{4} }^{6}$
${\href{/padicField/37.4.0.1}{4} }^{5}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$
$20{,}\,{\href{/padicField/41.4.0.1}{4} }$
$24$
${\href{/padicField/47.8.0.1}{8} }^{3}$
${\href{/padicField/53.4.0.1}{4} }^{5}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$
${\href{/padicField/59.3.0.1}{3} }^{8}$
In the table, R denotes a ramified prime.
Cycle lengths which are repeated in a cycle type are indicated by
exponents.
sage: # to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
gp: \\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
magma: // to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
oscar: # to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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]
(0) (0) (2) (3) (5) (7) (11) (13) (17) (19) (23) (29) (31) (37) (41) (43) (47) (53) (59)
