sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 1256355480*x^20 + 346321257364221748*x^18 - 281369885833667927777292253*x^16 + 211852233873305124874717191486598021*x^14 - 51988958787642502292357344580089958853671609*x^12 + 16164751063634090233408096678662142988343936090500554*x^10 - 12944956884362741410715058367642423772239748398213938868430054*x^8 + 2876793194724455010428571062539352506917449465867424287833700972460630*x^6 - 30897186782318789777365615471712373496217555545254574233072653398519233478514*x^4 + 15685554292006069651346777755707519381113345792414561710283876785994521259120839545704*x^2 - 3765085538799951525236302125814989854461975604609828445625128934755165972532492439340389009701)
gp: K = bnfinit(y^22 - 1256355480*y^20 + 346321257364221748*y^18 - 281369885833667927777292253*y^16 + 211852233873305124874717191486598021*y^14 - 51988958787642502292357344580089958853671609*y^12 + 16164751063634090233408096678662142988343936090500554*y^10 - 12944956884362741410715058367642423772239748398213938868430054*y^8 + 2876793194724455010428571062539352506917449465867424287833700972460630*y^6 - 30897186782318789777365615471712373496217555545254574233072653398519233478514*y^4 + 15685554292006069651346777755707519381113345792414561710283876785994521259120839545704*y^2 - 3765085538799951525236302125814989854461975604609828445625128934755165972532492439340389009701, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 1256355480*x^20 + 346321257364221748*x^18 - 281369885833667927777292253*x^16 + 211852233873305124874717191486598021*x^14 - 51988958787642502292357344580089958853671609*x^12 + 16164751063634090233408096678662142988343936090500554*x^10 - 12944956884362741410715058367642423772239748398213938868430054*x^8 + 2876793194724455010428571062539352506917449465867424287833700972460630*x^6 - 30897186782318789777365615471712373496217555545254574233072653398519233478514*x^4 + 15685554292006069651346777755707519381113345792414561710283876785994521259120839545704*x^2 - 3765085538799951525236302125814989854461975604609828445625128934755165972532492439340389009701);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^22 - 1256355480*x^20 + 346321257364221748*x^18 - 281369885833667927777292253*x^16 + 211852233873305124874717191486598021*x^14 - 51988958787642502292357344580089958853671609*x^12 + 16164751063634090233408096678662142988343936090500554*x^10 - 12944956884362741410715058367642423772239748398213938868430054*x^8 + 2876793194724455010428571062539352506917449465867424287833700972460630*x^6 - 30897186782318789777365615471712373496217555545254574233072653398519233478514*x^4 + 15685554292006069651346777755707519381113345792414561710283876785994521259120839545704*x^2 - 3765085538799951525236302125814989854461975604609828445625128934755165972532492439340389009701)
\( x^{22} - 1256355480 x^{20} + \cdots - 37\!\cdots\!01 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Degree : $22$
sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
Signature : $[6, 8]$
sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
Discriminant :
\(984\!\cdots\!504\)
\(\medspace = 2^{22}\cdot 971^{11}\cdot 25709231^{11}\)
sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant : \($315\,997$.87\)
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 : not computed
Ramified primes :
\(2\), \(971\), \(25709231\)
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
Discriminant root field : $\Q(\sqrt{24963663301}$)
$\Aut(K/\Q)$ :
$C_2$
sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
This field is not Galois over $\Q$. This is not a CM field . This field has no CM subfields.
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{24963663301}a^{6}-\frac{1256355480}{24963663301}a^{4}+\frac{6898162534}{24963663301}a^{2}$, $\frac{1}{24963663301}a^{7}-\frac{1256355480}{24963663301}a^{5}+\frac{6898162534}{24963663301}a^{3}$, $\frac{1}{62\cdots 01}a^{8}-\frac{1256355480}{62\cdots 01}a^{6}+\frac{34\cdots 48}{62\cdots 01}a^{4}-\frac{8857655950}{24963663301}a^{2}$, $\frac{1}{62\cdots 01}a^{9}-\frac{1256355480}{62\cdots 01}a^{7}+\frac{34\cdots 48}{62\cdots 01}a^{5}-\frac{8857655950}{24963663301}a^{3}$, $\frac{1}{15\cdots 01}a^{10}-\frac{1256355480}{15\cdots 01}a^{8}+\frac{34\cdots 48}{15\cdots 01}a^{6}-\frac{11\cdots 53}{62\cdots 01}a^{4}-\frac{4094823597}{24963663301}a^{2}$, $\frac{1}{15\cdots 01}a^{11}-\frac{1256355480}{15\cdots 01}a^{9}+\frac{34\cdots 48}{15\cdots 01}a^{7}-\frac{11\cdots 53}{62\cdots 01}a^{5}-\frac{4094823597}{24963663301}a^{3}$, $\frac{1}{38\cdots 01}a^{12}-\frac{1256355480}{38\cdots 01}a^{10}+\frac{34\cdots 48}{38\cdots 01}a^{8}-\frac{11\cdots 53}{15\cdots 01}a^{6}+\frac{339951072009421}{62\cdots 01}a^{4}+\frac{3286899825}{24963663301}a^{2}$, $\frac{1}{38\cdots 01}a^{13}-\frac{1256355480}{38\cdots 01}a^{11}+\frac{34\cdots 48}{38\cdots 01}a^{9}-\frac{11\cdots 53}{15\cdots 01}a^{7}+\frac{339951072009421}{62\cdots 01}a^{5}+\frac{3286899825}{24963663301}a^{3}$, $\frac{1}{96\cdots 01}a^{14}-\frac{1256355480}{96\cdots 01}a^{12}+\frac{34\cdots 48}{96\cdots 01}a^{10}-\frac{11\cdots 53}{38\cdots 01}a^{8}+\frac{339951072009421}{15\cdots 01}a^{6}-\frac{3341843982509}{62\cdots 01}a^{4}-\frac{8304098848}{24963663301}a^{2}$, $\frac{1}{96\cdots 01}a^{15}-\frac{1256355480}{96\cdots 01}a^{13}+\frac{34\cdots 48}{96\cdots 01}a^{11}-\frac{11\cdots 53}{38\cdots 01}a^{9}+\frac{339951072009421}{15\cdots 01}a^{7}-\frac{3341843982509}{62\cdots 01}a^{5}-\frac{8304098848}{24963663301}a^{3}$, $\frac{1}{24\cdots 01}a^{16}-\frac{1256355480}{24\cdots 01}a^{14}+\frac{34\cdots 48}{24\cdots 01}a^{12}-\frac{11\cdots 53}{96\cdots 01}a^{10}+\frac{339951072009421}{38\cdots 01}a^{8}-\frac{3341843982509}{15\cdots 01}a^{6}+\frac{41623227754}{62\cdots 01}a^{4}-\frac{1335239054}{24963663301}a^{2}$, $\frac{1}{24\cdots 01}a^{17}-\frac{1256355480}{24\cdots 01}a^{15}+\frac{34\cdots 48}{24\cdots 01}a^{13}-\frac{11\cdots 53}{96\cdots 01}a^{11}+\frac{339951072009421}{38\cdots 01}a^{9}-\frac{3341843982509}{15\cdots 01}a^{7}+\frac{41623227754}{62\cdots 01}a^{5}-\frac{1335239054}{24963663301}a^{3}$, $\frac{1}{60\cdots 01}a^{18}-\frac{1256355480}{60\cdots 01}a^{16}+\frac{34\cdots 48}{60\cdots 01}a^{14}-\frac{11\cdots 53}{24\cdots 01}a^{12}+\frac{339951072009421}{96\cdots 01}a^{10}-\frac{3341843982509}{38\cdots 01}a^{8}+\frac{41623227754}{15\cdots 01}a^{6}-\frac{1335239054}{62\cdots 01}a^{4}+\frac{11886630}{24963663301}a^{2}$, $\frac{1}{60\cdots 01}a^{19}-\frac{1256355480}{60\cdots 01}a^{17}+\frac{34\cdots 48}{60\cdots 01}a^{15}-\frac{11\cdots 53}{24\cdots 01}a^{13}+\frac{339951072009421}{96\cdots 01}a^{11}-\frac{3341843982509}{38\cdots 01}a^{9}+\frac{41623227754}{15\cdots 01}a^{7}-\frac{1335239054}{62\cdots 01}a^{5}+\frac{11886630}{24963663301}a^{3}$, $\frac{1}{19\cdots 99}a^{20}-\frac{71\cdots 47}{19\cdots 99}a^{18}-\frac{28\cdots 13}{19\cdots 99}a^{16}+\frac{39\cdots 67}{79\cdots 99}a^{14}-\frac{38\cdots 89}{31\cdots 99}a^{12}-\frac{37\cdots 00}{12\cdots 99}a^{10}-\frac{13\cdots 95}{51\cdots 99}a^{8}-\frac{49\cdots 04}{20\cdots 99}a^{6}-\frac{18\cdots 90}{82\cdots 99}a^{4}-\frac{27\cdots 79}{32\cdots 99}a^{2}+\frac{12\cdots 35}{13\cdots 99}$, $\frac{1}{19\cdots 99}a^{21}-\frac{71\cdots 47}{19\cdots 99}a^{19}-\frac{28\cdots 13}{19\cdots 99}a^{17}+\frac{39\cdots 67}{79\cdots 99}a^{15}-\frac{38\cdots 89}{31\cdots 99}a^{13}-\frac{37\cdots 00}{12\cdots 99}a^{11}-\frac{13\cdots 95}{51\cdots 99}a^{9}-\frac{49\cdots 04}{20\cdots 99}a^{7}-\frac{18\cdots 90}{82\cdots 99}a^{5}-\frac{27\cdots 79}{32\cdots 99}a^{3}+\frac{12\cdots 35}{13\cdots 99}a$
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)
\[
\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{6}\cdot(2\pi)^{8}\cdot R \cdot h}{2\cdot\sqrt{9841275385698713547061800803050974420965363933658706138702352603003020290397779358031956909097305281204793677693797269504}}\cr\mathstrut & \text{
some values not computed }
\end{aligned}\]
sage: # self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 1256355480*x^20 + 346321257364221748*x^18 - 281369885833667927777292253*x^16 + 211852233873305124874717191486598021*x^14 - 51988958787642502292357344580089958853671609*x^12 + 16164751063634090233408096678662142988343936090500554*x^10 - 12944956884362741410715058367642423772239748398213938868430054*x^8 + 2876793194724455010428571062539352506917449465867424287833700972460630*x^6 - 30897186782318789777365615471712373496217555545254574233072653398519233478514*x^4 + 15685554292006069651346777755707519381113345792414561710283876785994521259120839545704*x^2 - 3765085538799951525236302125814989854461975604609828445625128934755165972532492439340389009701)
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^22 - 1256355480*x^20 + 346321257364221748*x^18 - 281369885833667927777292253*x^16 + 211852233873305124874717191486598021*x^14 - 51988958787642502292357344580089958853671609*x^12 + 16164751063634090233408096678662142988343936090500554*x^10 - 12944956884362741410715058367642423772239748398213938868430054*x^8 + 2876793194724455010428571062539352506917449465867424287833700972460630*x^6 - 30897186782318789777365615471712373496217555545254574233072653398519233478514*x^4 + 15685554292006069651346777755707519381113345792414561710283876785994521259120839545704*x^2 - 3765085538799951525236302125814989854461975604609828445625128934755165972532492439340389009701, 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^22 - 1256355480*x^20 + 346321257364221748*x^18 - 281369885833667927777292253*x^16 + 211852233873305124874717191486598021*x^14 - 51988958787642502292357344580089958853671609*x^12 + 16164751063634090233408096678662142988343936090500554*x^10 - 12944956884362741410715058367642423772239748398213938868430054*x^8 + 2876793194724455010428571062539352506917449465867424287833700972460630*x^6 - 30897186782318789777365615471712373496217555545254574233072653398519233478514*x^4 + 15685554292006069651346777755707519381113345792414561710283876785994521259120839545704*x^2 - 3765085538799951525236302125814989854461975604609828445625128934755165972532492439340389009701);
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 = PolynomialRing(QQ); K, a = NumberField(x^22 - 1256355480*x^20 + 346321257364221748*x^18 - 281369885833667927777292253*x^16 + 211852233873305124874717191486598021*x^14 - 51988958787642502292357344580089958853671609*x^12 + 16164751063634090233408096678662142988343936090500554*x^10 - 12944956884362741410715058367642423772239748398213938868430054*x^8 + 2876793194724455010428571062539352506917449465867424287833700972460630*x^6 - 30897186782318789777365615471712373496217555545254574233072653398519233478514*x^4 + 15685554292006069651346777755707519381113345792414561710283876785994521259120839545704*x^2 - 3765085538799951525236302125814989854461975604609828445625128934755165972532492439340389009701);
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))))
$C_2^{10}.S_{11}$ (as 22T50 ):
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)
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]
$p$
$2$
$3$
$5$
$7$
$11$
$13$
$17$
$19$
$23$
$29$
$31$
$37$
$41$
$43$
$47$
$53$
$59$
Cycle type
R
${\href{/padicField/3.11.0.1}{11} }^{2}$
${\href{/padicField/5.11.0.1}{11} }^{2}$
$16{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$
${\href{/padicField/11.11.0.1}{11} }^{2}$
$16{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$
${\href{/padicField/17.9.0.1}{9} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }$
${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$
${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$
${\href{/padicField/29.14.0.1}{14} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}$
${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$
${\href{/padicField/37.9.0.1}{9} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$
$18{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$
${\href{/padicField/43.5.0.1}{5} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$
${\href{/padicField/47.5.0.1}{5} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$
${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$
${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$
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]
$p$ Label Polynomial
$e$ $f$ $c$ Galois group Slope content
\(2\)
2.5.2.10a6.2 $x^{10} + 2 x^{9} + 4 x^{7} + 2 x^{6} + 2 x^{5} + 5 x^{4} + 4 x^{2} + 7$ $2$ $5$ $10$ $C_2 \times (C_2^4 : C_5)$ $$[2, 2, 2, 2, 2]^{5}$$ 2.6.2.12a8.1 $x^{12} + 2 x^{11} + 2 x^{10} + 4 x^{9} + 5 x^{8} + 4 x^{7} + 7 x^{6} + 6 x^{5} + 4 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 3$ $2$ $6$ $12$ 12T134 $$[2, 2, 2, 2, 2, 2]^{6}$$
\(971\)
Deg $2$ $2$ $1$ $1$ $C_2$ $$[\ ]_{2}$$ Deg $2$ $2$ $1$ $1$ $C_2$ $$[\ ]_{2}$$ Deg $2$ $2$ $1$ $1$ $C_2$ $$[\ ]_{2}$$ Deg $2$ $2$ $1$ $1$ $C_2$ $$[\ ]_{2}$$ Deg $14$ $2$ $7$ $7$
\(25709231\)
Deg $4$ $2$ $2$ $2$ Deg $4$ $2$ $2$ $2$ Deg $6$ $2$ $3$ $3$ Deg $8$ $2$ $4$ $4$
(0) (0) (2) (3) (5) (7) (11) (13) (17) (19) (23) (29) (31) (37) (41) (43) (47) (53) (59)
