sage: x = polygen(QQ); K.<a> = NumberField(x^39 - 468*x^37 - 312*x^36 + 94302*x^35 + 118638*x^34 - 10878699*x^33 - 19307574*x^32 + 805821744*x^31 + 1800372886*x^30 - 40680500211*x^29 - 108237264603*x^28 + 1446005621320*x^27 + 4459027282749*x^26 - 36740305683120*x^25 - 130171431648258*x^24 + 667319579204967*x^23 + 2739095618686947*x^22 - 8508852916909748*x^21 - 41760228787431102*x^20 + 72335202029643423*x^19 + 459027912672556227*x^18 - 350556005056037274*x^17 - 3583579320625784766*x^16 + 223936776371258899*x^15 + 19350977144085682407*x^14 + 8850830449501775865*x^13 - 69386406134656900614*x^12 - 59318124902891828415*x^11 + 155603230401068027877*x^10 + 183666623582194428403*x^9 - 199867410901172997567*x^8 - 299712026978452810761*x^7 + 126091751205245716982*x^6 + 247052928867978365847*x^5 - 26149175806607388015*x^4 - 89715453239778791906*x^3 - 3866411569558451472*x^2 + 10712801222330860887*x + 1727305433480995147)
gp: K = bnfinit(y^39 - 468*y^37 - 312*y^36 + 94302*y^35 + 118638*y^34 - 10878699*y^33 - 19307574*y^32 + 805821744*y^31 + 1800372886*y^30 - 40680500211*y^29 - 108237264603*y^28 + 1446005621320*y^27 + 4459027282749*y^26 - 36740305683120*y^25 - 130171431648258*y^24 + 667319579204967*y^23 + 2739095618686947*y^22 - 8508852916909748*y^21 - 41760228787431102*y^20 + 72335202029643423*y^19 + 459027912672556227*y^18 - 350556005056037274*y^17 - 3583579320625784766*y^16 + 223936776371258899*y^15 + 19350977144085682407*y^14 + 8850830449501775865*y^13 - 69386406134656900614*y^12 - 59318124902891828415*y^11 + 155603230401068027877*y^10 + 183666623582194428403*y^9 - 199867410901172997567*y^8 - 299712026978452810761*y^7 + 126091751205245716982*y^6 + 247052928867978365847*y^5 - 26149175806607388015*y^4 - 89715453239778791906*y^3 - 3866411569558451472*y^2 + 10712801222330860887*y + 1727305433480995147, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^39 - 468*x^37 - 312*x^36 + 94302*x^35 + 118638*x^34 - 10878699*x^33 - 19307574*x^32 + 805821744*x^31 + 1800372886*x^30 - 40680500211*x^29 - 108237264603*x^28 + 1446005621320*x^27 + 4459027282749*x^26 - 36740305683120*x^25 - 130171431648258*x^24 + 667319579204967*x^23 + 2739095618686947*x^22 - 8508852916909748*x^21 - 41760228787431102*x^20 + 72335202029643423*x^19 + 459027912672556227*x^18 - 350556005056037274*x^17 - 3583579320625784766*x^16 + 223936776371258899*x^15 + 19350977144085682407*x^14 + 8850830449501775865*x^13 - 69386406134656900614*x^12 - 59318124902891828415*x^11 + 155603230401068027877*x^10 + 183666623582194428403*x^9 - 199867410901172997567*x^8 - 299712026978452810761*x^7 + 126091751205245716982*x^6 + 247052928867978365847*x^5 - 26149175806607388015*x^4 - 89715453239778791906*x^3 - 3866411569558451472*x^2 + 10712801222330860887*x + 1727305433480995147);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^39 - 468*x^37 - 312*x^36 + 94302*x^35 + 118638*x^34 - 10878699*x^33 - 19307574*x^32 + 805821744*x^31 + 1800372886*x^30 - 40680500211*x^29 - 108237264603*x^28 + 1446005621320*x^27 + 4459027282749*x^26 - 36740305683120*x^25 - 130171431648258*x^24 + 667319579204967*x^23 + 2739095618686947*x^22 - 8508852916909748*x^21 - 41760228787431102*x^20 + 72335202029643423*x^19 + 459027912672556227*x^18 - 350556005056037274*x^17 - 3583579320625784766*x^16 + 223936776371258899*x^15 + 19350977144085682407*x^14 + 8850830449501775865*x^13 - 69386406134656900614*x^12 - 59318124902891828415*x^11 + 155603230401068027877*x^10 + 183666623582194428403*x^9 - 199867410901172997567*x^8 - 299712026978452810761*x^7 + 126091751205245716982*x^6 + 247052928867978365847*x^5 - 26149175806607388015*x^4 - 89715453239778791906*x^3 - 3866411569558451472*x^2 + 10712801222330860887*x + 1727305433480995147)
\( x^{39} - 468 x^{37} - 312 x^{36} + 94302 x^{35} + 118638 x^{34} - 10878699 x^{33} + \cdots + 17\!\cdots\!47 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Degree : $39$
sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
Signature : $[39, 0]$
sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
Discriminant :
\(174\!\cdots\!249\)
\(\medspace = 3^{52}\cdot 13^{74}\)
sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant : \(562.08\)
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 : $3^{4/3}13^{74/39}\approx 562.0795404019609$
Ramified primes :
\(3\), \(13\)
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant(OK))
Discriminant root field : \(\Q\)
$\Aut(K/\Q)$
$=$
$\Gal(K/\Q)$ :
$C_{39}$
sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
This field is Galois and abelian over $\Q$. Conductor : \(1521=3^{2}\cdot 13^{2}\) Dirichlet character group :
$\lbrace$$\chi_{1521}(256,·)$ , $\chi_{1521}(1,·)$ , $\chi_{1521}(133,·)$ , $\chi_{1521}(1288,·)$ , $\chi_{1521}(139,·)$ , $\chi_{1521}(1420,·)$ , $\chi_{1521}(16,·)$ , $\chi_{1521}(1426,·)$ , $\chi_{1521}(1171,·)$ , $\chi_{1521}(22,·)$ , $\chi_{1521}(1303,·)$ , $\chi_{1521}(1309,·)$ , $\chi_{1521}(1054,·)$ , $\chi_{1521}(1186,·)$ , $\chi_{1521}(1192,·)$ , $\chi_{1521}(937,·)$ , $\chi_{1521}(1069,·)$ , $\chi_{1521}(1075,·)$ , $\chi_{1521}(820,·)$ , $\chi_{1521}(952,·)$ , $\chi_{1521}(958,·)$ , $\chi_{1521}(703,·)$ , $\chi_{1521}(835,·)$ , $\chi_{1521}(841,·)$ , $\chi_{1521}(586,·)$ , $\chi_{1521}(718,·)$ , $\chi_{1521}(724,·)$ , $\chi_{1521}(469,·)$ , $\chi_{1521}(601,·)$ , $\chi_{1521}(607,·)$ , $\chi_{1521}(352,·)$ , $\chi_{1521}(484,·)$ , $\chi_{1521}(490,·)$ , $\chi_{1521}(235,·)$ , $\chi_{1521}(367,·)$ , $\chi_{1521}(373,·)$ , $\chi_{1521}(118,·)$ , $\chi_{1521}(250,·)$ , $\chi_{1521}(1405,·)$ $\rbrace$ This is not a CM field . This field has no CM subfields.
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $\frac{1}{19}a^{36}-\frac{1}{19}a^{34}+\frac{3}{19}a^{33}-\frac{6}{19}a^{32}+\frac{5}{19}a^{31}+\frac{3}{19}a^{30}+\frac{6}{19}a^{29}+\frac{7}{19}a^{28}-\frac{2}{19}a^{27}+\frac{1}{19}a^{26}-\frac{2}{19}a^{25}+\frac{5}{19}a^{24}-\frac{2}{19}a^{23}+\frac{3}{19}a^{22}+\frac{9}{19}a^{21}+\frac{9}{19}a^{20}-\frac{1}{19}a^{19}-\frac{1}{19}a^{18}+\frac{3}{19}a^{17}+\frac{2}{19}a^{16}-\frac{9}{19}a^{15}-\frac{5}{19}a^{14}-\frac{9}{19}a^{13}-\frac{3}{19}a^{12}-\frac{8}{19}a^{10}+\frac{1}{19}a^{9}-\frac{9}{19}a^{8}+\frac{4}{19}a^{7}+\frac{3}{19}a^{6}-\frac{3}{19}a^{5}+\frac{2}{19}a^{4}-\frac{8}{19}a^{3}-\frac{2}{19}a^{2}+\frac{2}{19}a+\frac{9}{19}$, $\frac{1}{19}a^{37}-\frac{1}{19}a^{35}+\frac{3}{19}a^{34}-\frac{6}{19}a^{33}+\frac{5}{19}a^{32}+\frac{3}{19}a^{31}+\frac{6}{19}a^{30}+\frac{7}{19}a^{29}-\frac{2}{19}a^{28}+\frac{1}{19}a^{27}-\frac{2}{19}a^{26}+\frac{5}{19}a^{25}-\frac{2}{19}a^{24}+\frac{3}{19}a^{23}+\frac{9}{19}a^{22}+\frac{9}{19}a^{21}-\frac{1}{19}a^{20}-\frac{1}{19}a^{19}+\frac{3}{19}a^{18}+\frac{2}{19}a^{17}-\frac{9}{19}a^{16}-\frac{5}{19}a^{15}-\frac{9}{19}a^{14}-\frac{3}{19}a^{13}-\frac{8}{19}a^{11}+\frac{1}{19}a^{10}-\frac{9}{19}a^{9}+\frac{4}{19}a^{8}+\frac{3}{19}a^{7}-\frac{3}{19}a^{6}+\frac{2}{19}a^{5}-\frac{8}{19}a^{4}-\frac{2}{19}a^{3}+\frac{2}{19}a^{2}+\frac{9}{19}a$, $\frac{1}{16\cdots 23}a^{38}-\frac{66\cdots 72}{16\cdots 23}a^{37}-\frac{38\cdots 26}{16\cdots 23}a^{36}-\frac{19\cdots 88}{16\cdots 23}a^{35}-\frac{51\cdots 41}{16\cdots 23}a^{34}-\frac{52\cdots 09}{16\cdots 23}a^{33}+\frac{14\cdots 29}{85\cdots 17}a^{32}-\frac{76\cdots 84}{16\cdots 23}a^{31}-\frac{19\cdots 14}{16\cdots 23}a^{30}-\frac{80\cdots 56}{16\cdots 23}a^{29}-\frac{10\cdots 98}{16\cdots 23}a^{28}-\frac{28\cdots 29}{16\cdots 23}a^{27}-\frac{71\cdots 33}{16\cdots 23}a^{26}+\frac{47\cdots 41}{16\cdots 23}a^{25}+\frac{31\cdots 14}{85\cdots 17}a^{24}-\frac{81\cdots 01}{16\cdots 23}a^{23}+\frac{84\cdots 54}{16\cdots 23}a^{22}+\frac{45\cdots 72}{16\cdots 23}a^{21}-\frac{42\cdots 24}{16\cdots 23}a^{20}-\frac{46\cdots 69}{16\cdots 23}a^{19}+\frac{10\cdots 55}{16\cdots 23}a^{18}-\frac{78\cdots 12}{16\cdots 23}a^{17}+\frac{85\cdots 33}{85\cdots 17}a^{16}+\frac{31\cdots 69}{16\cdots 23}a^{15}-\frac{76\cdots 18}{16\cdots 23}a^{14}-\frac{21\cdots 34}{16\cdots 23}a^{13}+\frac{39\cdots 10}{16\cdots 23}a^{12}+\frac{38\cdots 69}{16\cdots 23}a^{11}+\frac{25\cdots 23}{16\cdots 23}a^{10}-\frac{19\cdots 04}{16\cdots 23}a^{9}-\frac{11\cdots 09}{16\cdots 23}a^{8}-\frac{27\cdots 50}{16\cdots 23}a^{7}-\frac{30\cdots 99}{16\cdots 23}a^{6}-\frac{21\cdots 90}{16\cdots 23}a^{5}+\frac{25\cdots 52}{16\cdots 23}a^{4}-\frac{23\cdots 24}{16\cdots 23}a^{3}-\frac{59\cdots 84}{16\cdots 23}a^{2}+\frac{62\cdots 53}{16\cdots 23}a+\frac{29\cdots 01}{16\cdots 23}$
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 : $38$
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)
\[
\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{39}\cdot(2\pi)^{0}\cdot R \cdot h}{2\cdot\sqrt{174627773231779347132152827349937303698541390024823069427632498388746376232649719308860385493378089106300249}}\cr\mathstrut & \text{
some values not computed }
\end{aligned}\]
sage: # self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^39 - 468*x^37 - 312*x^36 + 94302*x^35 + 118638*x^34 - 10878699*x^33 - 19307574*x^32 + 805821744*x^31 + 1800372886*x^30 - 40680500211*x^29 - 108237264603*x^28 + 1446005621320*x^27 + 4459027282749*x^26 - 36740305683120*x^25 - 130171431648258*x^24 + 667319579204967*x^23 + 2739095618686947*x^22 - 8508852916909748*x^21 - 41760228787431102*x^20 + 72335202029643423*x^19 + 459027912672556227*x^18 - 350556005056037274*x^17 - 3583579320625784766*x^16 + 223936776371258899*x^15 + 19350977144085682407*x^14 + 8850830449501775865*x^13 - 69386406134656900614*x^12 - 59318124902891828415*x^11 + 155603230401068027877*x^10 + 183666623582194428403*x^9 - 199867410901172997567*x^8 - 299712026978452810761*x^7 + 126091751205245716982*x^6 + 247052928867978365847*x^5 - 26149175806607388015*x^4 - 89715453239778791906*x^3 - 3866411569558451472*x^2 + 10712801222330860887*x + 1727305433480995147)
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^39 - 468*x^37 - 312*x^36 + 94302*x^35 + 118638*x^34 - 10878699*x^33 - 19307574*x^32 + 805821744*x^31 + 1800372886*x^30 - 40680500211*x^29 - 108237264603*x^28 + 1446005621320*x^27 + 4459027282749*x^26 - 36740305683120*x^25 - 130171431648258*x^24 + 667319579204967*x^23 + 2739095618686947*x^22 - 8508852916909748*x^21 - 41760228787431102*x^20 + 72335202029643423*x^19 + 459027912672556227*x^18 - 350556005056037274*x^17 - 3583579320625784766*x^16 + 223936776371258899*x^15 + 19350977144085682407*x^14 + 8850830449501775865*x^13 - 69386406134656900614*x^12 - 59318124902891828415*x^11 + 155603230401068027877*x^10 + 183666623582194428403*x^9 - 199867410901172997567*x^8 - 299712026978452810761*x^7 + 126091751205245716982*x^6 + 247052928867978365847*x^5 - 26149175806607388015*x^4 - 89715453239778791906*x^3 - 3866411569558451472*x^2 + 10712801222330860887*x + 1727305433480995147, 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^39 - 468*x^37 - 312*x^36 + 94302*x^35 + 118638*x^34 - 10878699*x^33 - 19307574*x^32 + 805821744*x^31 + 1800372886*x^30 - 40680500211*x^29 - 108237264603*x^28 + 1446005621320*x^27 + 4459027282749*x^26 - 36740305683120*x^25 - 130171431648258*x^24 + 667319579204967*x^23 + 2739095618686947*x^22 - 8508852916909748*x^21 - 41760228787431102*x^20 + 72335202029643423*x^19 + 459027912672556227*x^18 - 350556005056037274*x^17 - 3583579320625784766*x^16 + 223936776371258899*x^15 + 19350977144085682407*x^14 + 8850830449501775865*x^13 - 69386406134656900614*x^12 - 59318124902891828415*x^11 + 155603230401068027877*x^10 + 183666623582194428403*x^9 - 199867410901172997567*x^8 - 299712026978452810761*x^7 + 126091751205245716982*x^6 + 247052928867978365847*x^5 - 26149175806607388015*x^4 - 89715453239778791906*x^3 - 3866411569558451472*x^2 + 10712801222330860887*x + 1727305433480995147);
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^39 - 468*x^37 - 312*x^36 + 94302*x^35 + 118638*x^34 - 10878699*x^33 - 19307574*x^32 + 805821744*x^31 + 1800372886*x^30 - 40680500211*x^29 - 108237264603*x^28 + 1446005621320*x^27 + 4459027282749*x^26 - 36740305683120*x^25 - 130171431648258*x^24 + 667319579204967*x^23 + 2739095618686947*x^22 - 8508852916909748*x^21 - 41760228787431102*x^20 + 72335202029643423*x^19 + 459027912672556227*x^18 - 350556005056037274*x^17 - 3583579320625784766*x^16 + 223936776371258899*x^15 + 19350977144085682407*x^14 + 8850830449501775865*x^13 - 69386406134656900614*x^12 - 59318124902891828415*x^11 + 155603230401068027877*x^10 + 183666623582194428403*x^9 - 199867410901172997567*x^8 - 299712026978452810761*x^7 + 126091751205245716982*x^6 + 247052928867978365847*x^5 - 26149175806607388015*x^4 - 89715453239778791906*x^3 - 3866411569558451472*x^2 + 10712801222330860887*x + 1727305433480995147);
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))))
$C_{39}$ (as 39T1 ):
sage: K.galois_group(type='pari')
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
${\href{/padicField/2.13.0.1}{13} }^{3}$
R
$39$
$39$
${\href{/padicField/11.13.0.1}{13} }^{3}$
R
$39$
${\href{/padicField/19.3.0.1}{3} }^{13}$
${\href{/padicField/23.3.0.1}{3} }^{13}$
${\href{/padicField/29.13.0.1}{13} }^{3}$
$39$
$39$
$39$
$39$
$39$
${\href{/padicField/53.13.0.1}{13} }^{3}$
${\href{/padicField/59.13.0.1}{13} }^{3}$
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)
