Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 13*x^18 - 6*x^17 - 20*x^16 + 26*x^15 + 21*x^14 - 44*x^13 - 40*x^12 + 110*x^11 - 45*x^10 - 8*x^9 - 19*x^8 + 54*x^7 + 7*x^6 - 14*x^5 + 3*x^4 + 2*x^3 + 3*x^2 - 1)
gp: K = bnfinit(y^20 - 6*y^19 + 13*y^18 - 6*y^17 - 20*y^16 + 26*y^15 + 21*y^14 - 44*y^13 - 40*y^12 + 110*y^11 - 45*y^10 - 8*y^9 - 19*y^8 + 54*y^7 + 7*y^6 - 14*y^5 + 3*y^4 + 2*y^3 + 3*y^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 6*x^19 + 13*x^18 - 6*x^17 - 20*x^16 + 26*x^15 + 21*x^14 - 44*x^13 - 40*x^12 + 110*x^11 - 45*x^10 - 8*x^9 - 19*x^8 + 54*x^7 + 7*x^6 - 14*x^5 + 3*x^4 + 2*x^3 + 3*x^2 - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 6*x^19 + 13*x^18 - 6*x^17 - 20*x^16 + 26*x^15 + 21*x^14 - 44*x^13 - 40*x^12 + 110*x^11 - 45*x^10 - 8*x^9 - 19*x^8 + 54*x^7 + 7*x^6 - 14*x^5 + 3*x^4 + 2*x^3 + 3*x^2 - 1)
\( x^{20} - 6 x^{19} + 13 x^{18} - 6 x^{17} - 20 x^{16} + 26 x^{15} + 21 x^{14} - 44 x^{13} - 40 x^{12} + \cdots - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[2, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-275524109311254364946432\)
\(\medspace = -\,2^{20}\cdot 47\cdot 8647^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(14.86\) | 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: |
\(2\), \(47\), \(8647\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-47}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{43\!\cdots\!37}a^{19}+\frac{11\!\cdots\!62}{43\!\cdots\!37}a^{18}+\frac{12\!\cdots\!93}{43\!\cdots\!37}a^{17}+\frac{12\!\cdots\!87}{43\!\cdots\!37}a^{16}+\frac{72\!\cdots\!59}{43\!\cdots\!37}a^{15}-\frac{13\!\cdots\!34}{43\!\cdots\!37}a^{14}-\frac{22\!\cdots\!24}{43\!\cdots\!37}a^{13}+\frac{19\!\cdots\!09}{43\!\cdots\!37}a^{12}+\frac{12\!\cdots\!47}{43\!\cdots\!37}a^{11}+\frac{12\!\cdots\!87}{43\!\cdots\!37}a^{10}-\frac{61\!\cdots\!07}{43\!\cdots\!37}a^{9}-\frac{25\!\cdots\!16}{43\!\cdots\!37}a^{8}-\frac{15\!\cdots\!03}{43\!\cdots\!37}a^{7}+\frac{66\!\cdots\!32}{43\!\cdots\!37}a^{6}-\frac{28\!\cdots\!47}{43\!\cdots\!37}a^{5}+\frac{13\!\cdots\!59}{43\!\cdots\!37}a^{4}-\frac{20\!\cdots\!10}{43\!\cdots\!37}a^{3}-\frac{20\!\cdots\!39}{43\!\cdots\!37}a^{2}+\frac{14\!\cdots\!50}{43\!\cdots\!37}a+\frac{16\!\cdots\!35}{43\!\cdots\!37}$
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
Trivial group, which has order $1$
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: | $10$ | 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: |
$\frac{24663538138342}{229333372494307}a^{19}+\frac{99398789448816}{229333372494307}a^{18}-\frac{60595630704575}{229333372494307}a^{17}-\frac{304523159702316}{229333372494307}a^{16}+\frac{432653501681666}{229333372494307}a^{15}+\frac{402827738282152}{229333372494307}a^{14}-\frac{11\!\cdots\!97}{229333372494307}a^{13}-\frac{526290572973110}{229333372494307}a^{12}+\frac{22\!\cdots\!88}{229333372494307}a^{11}+\frac{398065570937063}{229333372494307}a^{10}-\frac{25\!\cdots\!99}{229333372494307}a^{9}-\frac{709764267526713}{229333372494307}a^{8}+\frac{13\!\cdots\!89}{229333372494307}a^{7}+\frac{181923887411678}{229333372494307}a^{6}-\frac{21\!\cdots\!19}{229333372494307}a^{5}-\frac{17\!\cdots\!25}{229333372494307}a^{4}+\frac{400546121103475}{229333372494307}a^{3}-\frac{25974947308982}{229333372494307}a^{2}-\frac{169391815449069}{229333372494307}a-\frac{174764010664492}{229333372494307}$, $\frac{48\!\cdots\!66}{43\!\cdots\!37}a^{19}+\frac{42\!\cdots\!87}{43\!\cdots\!37}a^{18}-\frac{14\!\cdots\!80}{43\!\cdots\!37}a^{17}+\frac{19\!\cdots\!24}{43\!\cdots\!37}a^{16}+\frac{18\!\cdots\!06}{43\!\cdots\!37}a^{15}-\frac{36\!\cdots\!46}{43\!\cdots\!37}a^{14}+\frac{19\!\cdots\!99}{43\!\cdots\!37}a^{13}+\frac{50\!\cdots\!25}{43\!\cdots\!37}a^{12}-\frac{30\!\cdots\!99}{43\!\cdots\!37}a^{11}-\frac{11\!\cdots\!03}{43\!\cdots\!37}a^{10}+\frac{15\!\cdots\!03}{43\!\cdots\!37}a^{9}-\frac{47\!\cdots\!54}{43\!\cdots\!37}a^{8}+\frac{17\!\cdots\!38}{43\!\cdots\!37}a^{7}-\frac{72\!\cdots\!12}{43\!\cdots\!37}a^{6}+\frac{78\!\cdots\!50}{43\!\cdots\!37}a^{5}+\frac{95\!\cdots\!78}{43\!\cdots\!37}a^{4}-\frac{49\!\cdots\!29}{43\!\cdots\!37}a^{3}-\frac{28\!\cdots\!51}{43\!\cdots\!37}a^{2}-\frac{59\!\cdots\!45}{43\!\cdots\!37}a+\frac{53\!\cdots\!97}{43\!\cdots\!37}$, $\frac{70\!\cdots\!47}{43\!\cdots\!37}a^{19}+\frac{36\!\cdots\!10}{43\!\cdots\!37}a^{18}-\frac{62\!\cdots\!77}{43\!\cdots\!37}a^{17}-\frac{10\!\cdots\!60}{43\!\cdots\!37}a^{16}+\frac{13\!\cdots\!91}{43\!\cdots\!37}a^{15}-\frac{44\!\cdots\!87}{43\!\cdots\!37}a^{14}-\frac{25\!\cdots\!49}{43\!\cdots\!37}a^{13}+\frac{13\!\cdots\!64}{43\!\cdots\!37}a^{12}+\frac{50\!\cdots\!26}{43\!\cdots\!37}a^{11}-\frac{47\!\cdots\!83}{43\!\cdots\!37}a^{10}-\frac{21\!\cdots\!69}{43\!\cdots\!37}a^{9}+\frac{50\!\cdots\!78}{43\!\cdots\!37}a^{8}+\frac{43\!\cdots\!71}{43\!\cdots\!37}a^{7}-\frac{45\!\cdots\!85}{43\!\cdots\!37}a^{6}-\frac{24\!\cdots\!49}{43\!\cdots\!37}a^{5}-\frac{15\!\cdots\!65}{43\!\cdots\!37}a^{4}+\frac{11\!\cdots\!60}{43\!\cdots\!37}a^{3}-\frac{70\!\cdots\!06}{43\!\cdots\!37}a^{2}-\frac{56\!\cdots\!17}{43\!\cdots\!37}a-\frac{15\!\cdots\!22}{43\!\cdots\!37}$, $\frac{17\!\cdots\!25}{43\!\cdots\!37}a^{19}+\frac{10\!\cdots\!18}{43\!\cdots\!37}a^{18}-\frac{24\!\cdots\!93}{43\!\cdots\!37}a^{17}+\frac{15\!\cdots\!96}{43\!\cdots\!37}a^{16}+\frac{28\!\cdots\!55}{43\!\cdots\!37}a^{15}-\frac{45\!\cdots\!86}{43\!\cdots\!37}a^{14}-\frac{24\!\cdots\!55}{43\!\cdots\!37}a^{13}+\frac{66\!\cdots\!51}{43\!\cdots\!37}a^{12}+\frac{58\!\cdots\!70}{43\!\cdots\!37}a^{11}-\frac{17\!\cdots\!40}{43\!\cdots\!37}a^{10}+\frac{10\!\cdots\!78}{43\!\cdots\!37}a^{9}-\frac{46\!\cdots\!73}{43\!\cdots\!37}a^{8}+\frac{68\!\cdots\!64}{43\!\cdots\!37}a^{7}-\frac{96\!\cdots\!56}{43\!\cdots\!37}a^{6}-\frac{24\!\cdots\!50}{43\!\cdots\!37}a^{5}+\frac{78\!\cdots\!44}{43\!\cdots\!37}a^{4}-\frac{23\!\cdots\!26}{43\!\cdots\!37}a^{3}+\frac{23\!\cdots\!88}{43\!\cdots\!37}a^{2}-\frac{77\!\cdots\!15}{43\!\cdots\!37}a-\frac{25\!\cdots\!98}{43\!\cdots\!37}$, $\frac{494553192871001}{43\!\cdots\!37}a^{19}-\frac{900332180600071}{43\!\cdots\!37}a^{18}+\frac{12\!\cdots\!93}{43\!\cdots\!37}a^{17}-\frac{21\!\cdots\!91}{43\!\cdots\!37}a^{16}-\frac{16\!\cdots\!23}{43\!\cdots\!37}a^{15}+\frac{81\!\cdots\!53}{43\!\cdots\!37}a^{14}-\frac{52\!\cdots\!67}{43\!\cdots\!37}a^{13}-\frac{10\!\cdots\!00}{43\!\cdots\!37}a^{12}+\frac{10\!\cdots\!74}{43\!\cdots\!37}a^{11}+\frac{14\!\cdots\!58}{43\!\cdots\!37}a^{10}-\frac{18\!\cdots\!94}{43\!\cdots\!37}a^{9}-\frac{44\!\cdots\!03}{43\!\cdots\!37}a^{8}+\frac{11\!\cdots\!86}{43\!\cdots\!37}a^{7}-\frac{24\!\cdots\!26}{43\!\cdots\!37}a^{6}+\frac{63\!\cdots\!12}{43\!\cdots\!37}a^{5}-\frac{16\!\cdots\!12}{43\!\cdots\!37}a^{4}-\frac{67\!\cdots\!14}{43\!\cdots\!37}a^{3}-\frac{10\!\cdots\!12}{43\!\cdots\!37}a^{2}-\frac{15\!\cdots\!33}{43\!\cdots\!37}a+\frac{27\!\cdots\!40}{43\!\cdots\!37}$, $\frac{137633559194009}{229333372494307}a^{19}+\frac{827265162594358}{229333372494307}a^{18}-\frac{18\!\cdots\!90}{229333372494307}a^{17}+\frac{10\!\cdots\!73}{229333372494307}a^{16}+\frac{23\!\cdots\!82}{229333372494307}a^{15}-\frac{32\!\cdots\!22}{229333372494307}a^{14}-\frac{24\!\cdots\!00}{229333372494307}a^{13}+\frac{52\!\cdots\!58}{229333372494307}a^{12}+\frac{51\!\cdots\!99}{229333372494307}a^{11}-\frac{13\!\cdots\!71}{229333372494307}a^{10}+\frac{69\!\cdots\!62}{229333372494307}a^{9}-\frac{23\!\cdots\!45}{229333372494307}a^{8}+\frac{48\!\cdots\!37}{229333372494307}a^{7}-\frac{75\!\cdots\!41}{229333372494307}a^{6}+\frac{5364322023849}{229333372494307}a^{5}-\frac{218381963811263}{229333372494307}a^{4}+\frac{279160825138189}{229333372494307}a^{3}+\frac{48294631785437}{229333372494307}a^{2}-\frac{436049260046104}{229333372494307}a-\frac{49064818466245}{229333372494307}$, $\frac{48582439381236}{229333372494307}a^{19}+\frac{260030365093871}{229333372494307}a^{18}-\frac{452504388532368}{229333372494307}a^{17}-\frac{60617261085174}{229333372494307}a^{16}+\frac{10\!\cdots\!44}{229333372494307}a^{15}-\frac{603110747708415}{229333372494307}a^{14}-\frac{16\!\cdots\!58}{229333372494307}a^{13}+\frac{12\!\cdots\!08}{229333372494307}a^{12}+\frac{31\!\cdots\!83}{229333372494307}a^{11}-\frac{36\!\cdots\!89}{229333372494307}a^{10}-\frac{907072572633449}{229333372494307}a^{9}+\frac{861109352602291}{229333372494307}a^{8}+\frac{15\!\cdots\!46}{229333372494307}a^{7}-\frac{19\!\cdots\!25}{229333372494307}a^{6}-\frac{21\!\cdots\!13}{229333372494307}a^{5}+\frac{474536735518501}{229333372494307}a^{4}+\frac{23352128967702}{229333372494307}a^{3}-\frac{95401201034043}{229333372494307}a^{2}-\frac{174764010664492}{229333372494307}a-\frac{24663538138342}{229333372494307}$, $\frac{50\!\cdots\!70}{43\!\cdots\!37}a^{19}-\frac{32\!\cdots\!09}{43\!\cdots\!37}a^{18}+\frac{81\!\cdots\!14}{43\!\cdots\!37}a^{17}-\frac{77\!\cdots\!18}{43\!\cdots\!37}a^{16}-\frac{33\!\cdots\!50}{43\!\cdots\!37}a^{15}+\frac{10\!\cdots\!31}{43\!\cdots\!37}a^{14}+\frac{39\!\cdots\!45}{43\!\cdots\!37}a^{13}-\frac{16\!\cdots\!93}{43\!\cdots\!37}a^{12}-\frac{14\!\cdots\!87}{43\!\cdots\!37}a^{11}+\frac{50\!\cdots\!28}{43\!\cdots\!37}a^{10}-\frac{44\!\cdots\!53}{43\!\cdots\!37}a^{9}+\frac{40\!\cdots\!86}{43\!\cdots\!37}a^{8}-\frac{53\!\cdots\!24}{43\!\cdots\!37}a^{7}+\frac{61\!\cdots\!67}{43\!\cdots\!37}a^{6}-\frac{31\!\cdots\!41}{43\!\cdots\!37}a^{5}+\frac{24\!\cdots\!69}{43\!\cdots\!37}a^{4}-\frac{13\!\cdots\!22}{43\!\cdots\!37}a^{3}+\frac{81\!\cdots\!29}{43\!\cdots\!37}a^{2}+\frac{30\!\cdots\!36}{43\!\cdots\!37}a+\frac{17831063441412}{43\!\cdots\!37}$, $\frac{75\!\cdots\!46}{43\!\cdots\!37}a^{19}-\frac{33\!\cdots\!65}{43\!\cdots\!37}a^{18}+\frac{23\!\cdots\!99}{43\!\cdots\!37}a^{17}+\frac{13\!\cdots\!45}{43\!\cdots\!37}a^{16}-\frac{29\!\cdots\!12}{43\!\cdots\!37}a^{15}+\frac{24\!\cdots\!44}{43\!\cdots\!37}a^{14}+\frac{52\!\cdots\!76}{43\!\cdots\!37}a^{13}-\frac{25\!\cdots\!32}{43\!\cdots\!37}a^{12}-\frac{82\!\cdots\!33}{43\!\cdots\!37}a^{11}+\frac{60\!\cdots\!08}{43\!\cdots\!37}a^{10}+\frac{97\!\cdots\!48}{43\!\cdots\!37}a^{9}-\frac{11\!\cdots\!76}{43\!\cdots\!37}a^{8}+\frac{28\!\cdots\!61}{43\!\cdots\!37}a^{7}+\frac{10\!\cdots\!48}{43\!\cdots\!37}a^{6}+\frac{73\!\cdots\!93}{43\!\cdots\!37}a^{5}-\frac{34\!\cdots\!23}{43\!\cdots\!37}a^{4}+\frac{15\!\cdots\!86}{43\!\cdots\!37}a^{3}+\frac{13\!\cdots\!31}{43\!\cdots\!37}a^{2}+\frac{17\!\cdots\!31}{43\!\cdots\!37}a-\frac{69\!\cdots\!33}{43\!\cdots\!37}$, $\frac{15\!\cdots\!98}{43\!\cdots\!37}a^{19}+\frac{10\!\cdots\!62}{43\!\cdots\!37}a^{18}-\frac{19\!\cdots\!07}{43\!\cdots\!37}a^{17}-\frac{17\!\cdots\!27}{43\!\cdots\!37}a^{16}+\frac{11\!\cdots\!86}{43\!\cdots\!37}a^{15}-\frac{14\!\cdots\!77}{43\!\cdots\!37}a^{14}-\frac{60\!\cdots\!71}{43\!\cdots\!37}a^{13}+\frac{26\!\cdots\!12}{43\!\cdots\!37}a^{12}-\frac{45\!\cdots\!64}{43\!\cdots\!37}a^{11}-\frac{40\!\cdots\!05}{43\!\cdots\!37}a^{10}+\frac{13\!\cdots\!76}{43\!\cdots\!37}a^{9}+\frac{53\!\cdots\!81}{43\!\cdots\!37}a^{8}-\frac{66\!\cdots\!71}{43\!\cdots\!37}a^{7}+\frac{37\!\cdots\!38}{43\!\cdots\!37}a^{6}-\frac{25\!\cdots\!79}{43\!\cdots\!37}a^{5}+\frac{35\!\cdots\!37}{43\!\cdots\!37}a^{4}-\frac{16\!\cdots\!43}{43\!\cdots\!37}a^{3}-\frac{26\!\cdots\!85}{43\!\cdots\!37}a^{2}+\frac{22\!\cdots\!52}{43\!\cdots\!37}a-\frac{17\!\cdots\!18}{43\!\cdots\!37}$
| 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: | \( 3867.06506790925 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{9}\cdot 3867.06506790925 \cdot 1}{2\cdot\sqrt{275524109311254364946432}}\cr\approx \mathstrut & 0.224879821035016 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 13*x^18 - 6*x^17 - 20*x^16 + 26*x^15 + 21*x^14 - 44*x^13 - 40*x^12 + 110*x^11 - 45*x^10 - 8*x^9 - 19*x^8 + 54*x^7 + 7*x^6 - 14*x^5 + 3*x^4 + 2*x^3 + 3*x^2 - 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^20 - 6*x^19 + 13*x^18 - 6*x^17 - 20*x^16 + 26*x^15 + 21*x^14 - 44*x^13 - 40*x^12 + 110*x^11 - 45*x^10 - 8*x^9 - 19*x^8 + 54*x^7 + 7*x^6 - 14*x^5 + 3*x^4 + 2*x^3 + 3*x^2 - 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^20 - 6*x^19 + 13*x^18 - 6*x^17 - 20*x^16 + 26*x^15 + 21*x^14 - 44*x^13 - 40*x^12 + 110*x^11 - 45*x^10 - 8*x^9 - 19*x^8 + 54*x^7 + 7*x^6 - 14*x^5 + 3*x^4 + 2*x^3 + 3*x^2 - 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^20 - 6*x^19 + 13*x^18 - 6*x^17 - 20*x^16 + 26*x^15 + 21*x^14 - 44*x^13 - 40*x^12 + 110*x^11 - 45*x^10 - 8*x^9 - 19*x^8 + 54*x^7 + 7*x^6 - 14*x^5 + 3*x^4 + 2*x^3 + 3*x^2 - 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
$C_2^9.C_2^5.S_5$ (as 20T992):
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)
| A non-solvable group of order 1966080 |
| The 280 conjugacy class representatives for $C_2^9.C_2^5.S_5$ |
| Character table for $C_2^9.C_2^5.S_5$ |
Intermediate fields
| 5.3.8647.1, 10.2.76565103616.6 |
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]
Sibling fields
| Degree 20 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}{,}\,{\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.4.0.1}{4} }^{3}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.5.0.1}{5} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $20$ | $2$ | $10$ | $20$ | |||
|
\(47\)
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 47.2.1.1 | $x^{2} + 235$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.6.0.1 | $x^{6} + 2 x^{4} + 35 x^{3} + 9 x^{2} + 41 x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 47.6.0.1 | $x^{6} + 2 x^{4} + 35 x^{3} + 9 x^{2} + 41 x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(8647\)
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $8$ | $2$ | $4$ | $4$ |