Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^14 - 45*x^13 - 7*x^12 + 715*x^11 - 1559*x^10 + 1450*x^9 + 1680*x^8 - 4760*x^7 + 896*x^6 + 2230*x^5 - 1432*x^4 + 450*x^3 + 738*x^2 + 270*x + 81)
gp: K = bnfinit(y^16 - 7*y^14 - 45*y^13 - 7*y^12 + 715*y^11 - 1559*y^10 + 1450*y^9 + 1680*y^8 - 4760*y^7 + 896*y^6 + 2230*y^5 - 1432*y^4 + 450*y^3 + 738*y^2 + 270*y + 81, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 7*x^14 - 45*x^13 - 7*x^12 + 715*x^11 - 1559*x^10 + 1450*x^9 + 1680*x^8 - 4760*x^7 + 896*x^6 + 2230*x^5 - 1432*x^4 + 450*x^3 + 738*x^2 + 270*x + 81);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 7*x^14 - 45*x^13 - 7*x^12 + 715*x^11 - 1559*x^10 + 1450*x^9 + 1680*x^8 - 4760*x^7 + 896*x^6 + 2230*x^5 - 1432*x^4 + 450*x^3 + 738*x^2 + 270*x + 81)
\( x^{16} - 7 x^{14} - 45 x^{13} - 7 x^{12} + 715 x^{11} - 1559 x^{10} + 1450 x^{9} + 1680 x^{8} + \cdots + 81 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[8, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(81925410828566900634765625\)
\(\medspace = 5^{14}\cdot 41^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(41.65\) | 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: | $5^{7/8}41^{3/4}\approx 66.25011650349252$ | ||
| Ramified primes: |
\(5\), \(41\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{9}a^{12}+\frac{1}{9}a^{11}-\frac{1}{9}a^{10}+\frac{1}{9}a^{9}+\frac{1}{9}a^{8}+\frac{1}{9}a^{7}+\frac{1}{9}a^{6}+\frac{4}{9}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{2}{9}a^{2}-\frac{1}{3}a$, $\frac{1}{9}a^{13}+\frac{1}{9}a^{11}-\frac{1}{9}a^{10}+\frac{1}{3}a^{6}-\frac{1}{9}a^{5}-\frac{4}{9}a^{3}-\frac{2}{9}a^{2}+\frac{1}{3}a$, $\frac{1}{27}a^{14}-\frac{1}{27}a^{13}+\frac{1}{9}a^{11}-\frac{1}{27}a^{10}-\frac{4}{27}a^{9}-\frac{4}{27}a^{8}-\frac{10}{27}a^{7}+\frac{10}{27}a^{6}-\frac{2}{9}a^{5}+\frac{8}{27}a^{4}-\frac{10}{27}a^{3}+\frac{4}{9}a^{2}$, $\frac{1}{42\!\cdots\!23}a^{15}-\frac{14\!\cdots\!38}{14\!\cdots\!41}a^{14}-\frac{23\!\cdots\!71}{42\!\cdots\!23}a^{13}+\frac{21\!\cdots\!59}{14\!\cdots\!41}a^{12}+\frac{29\!\cdots\!48}{42\!\cdots\!23}a^{11}-\frac{41\!\cdots\!69}{42\!\cdots\!23}a^{10}-\frac{56\!\cdots\!74}{42\!\cdots\!23}a^{9}+\frac{25\!\cdots\!89}{42\!\cdots\!23}a^{8}+\frac{68\!\cdots\!01}{14\!\cdots\!41}a^{7}-\frac{63\!\cdots\!93}{42\!\cdots\!23}a^{6}+\frac{18\!\cdots\!22}{42\!\cdots\!23}a^{5}-\frac{13\!\cdots\!95}{42\!\cdots\!23}a^{4}+\frac{14\!\cdots\!52}{42\!\cdots\!23}a^{3}-\frac{13\!\cdots\!74}{14\!\cdots\!41}a^{2}-\frac{87\!\cdots\!90}{17\!\cdots\!61}a-\frac{34\!\cdots\!26}{15\!\cdots\!49}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
Trivial group, which has order $1$ (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: |
\( -1 \)
(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: |
$\frac{14\!\cdots\!16}{36\!\cdots\!17}a^{15}-\frac{22\!\cdots\!52}{12\!\cdots\!39}a^{14}-\frac{99\!\cdots\!94}{36\!\cdots\!17}a^{13}-\frac{19\!\cdots\!53}{12\!\cdots\!39}a^{12}+\frac{20\!\cdots\!36}{36\!\cdots\!17}a^{11}+\frac{10\!\cdots\!14}{36\!\cdots\!17}a^{10}-\frac{26\!\cdots\!60}{36\!\cdots\!17}a^{9}+\frac{31\!\cdots\!93}{36\!\cdots\!17}a^{8}+\frac{42\!\cdots\!50}{12\!\cdots\!39}a^{7}-\frac{77\!\cdots\!58}{36\!\cdots\!17}a^{6}+\frac{45\!\cdots\!11}{36\!\cdots\!17}a^{5}+\frac{20\!\cdots\!62}{36\!\cdots\!17}a^{4}-\frac{31\!\cdots\!59}{36\!\cdots\!17}a^{3}+\frac{74\!\cdots\!86}{12\!\cdots\!39}a^{2}+\frac{13\!\cdots\!37}{15\!\cdots\!19}a+\frac{35\!\cdots\!49}{13\!\cdots\!71}$, $\frac{10\!\cdots\!10}{42\!\cdots\!23}a^{15}-\frac{25\!\cdots\!36}{14\!\cdots\!41}a^{14}-\frac{87\!\cdots\!76}{42\!\cdots\!23}a^{13}-\frac{12\!\cdots\!05}{14\!\cdots\!41}a^{12}+\frac{34\!\cdots\!54}{42\!\cdots\!23}a^{11}+\frac{90\!\cdots\!89}{42\!\cdots\!23}a^{10}-\frac{67\!\cdots\!09}{42\!\cdots\!23}a^{9}+\frac{12\!\cdots\!87}{42\!\cdots\!23}a^{8}-\frac{36\!\cdots\!60}{14\!\cdots\!41}a^{7}-\frac{12\!\cdots\!39}{42\!\cdots\!23}a^{6}+\frac{26\!\cdots\!09}{42\!\cdots\!23}a^{5}-\frac{28\!\cdots\!21}{42\!\cdots\!23}a^{4}-\frac{85\!\cdots\!71}{42\!\cdots\!23}a^{3}+\frac{27\!\cdots\!33}{14\!\cdots\!41}a^{2}+\frac{23\!\cdots\!30}{52\!\cdots\!83}a-\frac{73\!\cdots\!80}{15\!\cdots\!49}$, $\frac{49\!\cdots\!96}{42\!\cdots\!23}a^{15}+\frac{43\!\cdots\!39}{14\!\cdots\!41}a^{14}+\frac{15\!\cdots\!71}{42\!\cdots\!23}a^{13}-\frac{30\!\cdots\!65}{14\!\cdots\!41}a^{12}-\frac{68\!\cdots\!33}{42\!\cdots\!23}a^{11}-\frac{68\!\cdots\!03}{42\!\cdots\!23}a^{10}+\frac{72\!\cdots\!78}{42\!\cdots\!23}a^{9}-\frac{81\!\cdots\!84}{42\!\cdots\!23}a^{8}+\frac{13\!\cdots\!16}{14\!\cdots\!41}a^{7}+\frac{32\!\cdots\!70}{42\!\cdots\!23}a^{6}-\frac{26\!\cdots\!66}{42\!\cdots\!23}a^{5}-\frac{15\!\cdots\!88}{42\!\cdots\!23}a^{4}+\frac{76\!\cdots\!58}{42\!\cdots\!23}a^{3}-\frac{60\!\cdots\!82}{14\!\cdots\!41}a^{2}-\frac{14\!\cdots\!81}{17\!\cdots\!61}a-\frac{11\!\cdots\!65}{15\!\cdots\!49}$, $\frac{73\!\cdots\!24}{42\!\cdots\!23}a^{15}-\frac{11\!\cdots\!08}{14\!\cdots\!41}a^{14}-\frac{49\!\cdots\!12}{42\!\cdots\!23}a^{13}-\frac{10\!\cdots\!37}{14\!\cdots\!41}a^{12}+\frac{92\!\cdots\!81}{42\!\cdots\!23}a^{11}+\frac{51\!\cdots\!49}{42\!\cdots\!23}a^{10}-\frac{14\!\cdots\!61}{42\!\cdots\!23}a^{9}+\frac{17\!\cdots\!01}{42\!\cdots\!23}a^{8}+\frac{24\!\cdots\!93}{14\!\cdots\!41}a^{7}-\frac{43\!\cdots\!02}{42\!\cdots\!23}a^{6}+\frac{33\!\cdots\!42}{42\!\cdots\!23}a^{5}+\frac{82\!\cdots\!92}{42\!\cdots\!23}a^{4}-\frac{28\!\cdots\!68}{42\!\cdots\!23}a^{3}+\frac{44\!\cdots\!62}{14\!\cdots\!41}a^{2}+\frac{72\!\cdots\!65}{17\!\cdots\!61}a+\frac{47\!\cdots\!75}{15\!\cdots\!49}$, $\frac{10\!\cdots\!57}{28\!\cdots\!73}a^{15}+\frac{29\!\cdots\!41}{93\!\cdots\!91}a^{14}-\frac{14\!\cdots\!92}{28\!\cdots\!73}a^{13}-\frac{22\!\cdots\!56}{93\!\cdots\!91}a^{12}-\frac{55\!\cdots\!80}{28\!\cdots\!73}a^{11}+\frac{11\!\cdots\!71}{28\!\cdots\!73}a^{10}-\frac{19\!\cdots\!63}{28\!\cdots\!73}a^{9}-\frac{46\!\cdots\!13}{28\!\cdots\!73}a^{8}+\frac{17\!\cdots\!43}{93\!\cdots\!91}a^{7}+\frac{12\!\cdots\!80}{28\!\cdots\!73}a^{6}-\frac{28\!\cdots\!58}{28\!\cdots\!73}a^{5}+\frac{12\!\cdots\!42}{28\!\cdots\!73}a^{4}+\frac{29\!\cdots\!19}{28\!\cdots\!73}a^{3}-\frac{11\!\cdots\!74}{93\!\cdots\!91}a^{2}-\frac{54\!\cdots\!34}{11\!\cdots\!11}a-\frac{32\!\cdots\!09}{10\!\cdots\!99}$, $\frac{28\!\cdots\!55}{14\!\cdots\!41}a^{15}-\frac{18\!\cdots\!95}{47\!\cdots\!47}a^{14}-\frac{23\!\cdots\!24}{14\!\cdots\!41}a^{13}-\frac{31\!\cdots\!57}{47\!\cdots\!47}a^{12}+\frac{24\!\cdots\!16}{14\!\cdots\!41}a^{11}+\frac{22\!\cdots\!73}{14\!\cdots\!41}a^{10}-\frac{79\!\cdots\!13}{14\!\cdots\!41}a^{9}+\frac{10\!\cdots\!83}{14\!\cdots\!41}a^{8}-\frac{23\!\cdots\!94}{47\!\cdots\!47}a^{7}-\frac{24\!\cdots\!35}{14\!\cdots\!41}a^{6}+\frac{19\!\cdots\!54}{14\!\cdots\!41}a^{5}+\frac{66\!\cdots\!17}{14\!\cdots\!41}a^{4}-\frac{94\!\cdots\!32}{14\!\cdots\!41}a^{3}+\frac{32\!\cdots\!50}{47\!\cdots\!47}a^{2}+\frac{65\!\cdots\!10}{52\!\cdots\!83}a+\frac{58\!\cdots\!76}{52\!\cdots\!83}$, $\frac{11\!\cdots\!30}{42\!\cdots\!23}a^{15}-\frac{19\!\cdots\!76}{14\!\cdots\!41}a^{14}-\frac{12\!\cdots\!12}{42\!\cdots\!23}a^{13}-\frac{63\!\cdots\!78}{14\!\cdots\!41}a^{12}+\frac{27\!\cdots\!14}{42\!\cdots\!23}a^{11}+\frac{11\!\cdots\!64}{42\!\cdots\!23}a^{10}-\frac{55\!\cdots\!00}{42\!\cdots\!23}a^{9}+\frac{80\!\cdots\!08}{42\!\cdots\!23}a^{8}-\frac{13\!\cdots\!72}{14\!\cdots\!41}a^{7}-\frac{16\!\cdots\!07}{42\!\cdots\!23}a^{6}+\frac{16\!\cdots\!70}{42\!\cdots\!23}a^{5}+\frac{32\!\cdots\!16}{42\!\cdots\!23}a^{4}-\frac{48\!\cdots\!90}{42\!\cdots\!23}a^{3}+\frac{23\!\cdots\!94}{14\!\cdots\!41}a^{2}+\frac{24\!\cdots\!91}{52\!\cdots\!83}a+\frac{37\!\cdots\!88}{15\!\cdots\!49}$, $\frac{51\!\cdots\!50}{14\!\cdots\!41}a^{15}-\frac{18\!\cdots\!82}{17\!\cdots\!61}a^{14}-\frac{42\!\cdots\!41}{14\!\cdots\!41}a^{13}-\frac{53\!\cdots\!24}{47\!\cdots\!47}a^{12}+\frac{66\!\cdots\!89}{14\!\cdots\!41}a^{11}+\frac{42\!\cdots\!48}{14\!\cdots\!41}a^{10}-\frac{17\!\cdots\!05}{14\!\cdots\!41}a^{9}+\frac{27\!\cdots\!52}{14\!\cdots\!41}a^{8}-\frac{26\!\cdots\!79}{17\!\cdots\!61}a^{7}-\frac{22\!\cdots\!54}{14\!\cdots\!41}a^{6}+\frac{17\!\cdots\!03}{14\!\cdots\!41}a^{5}+\frac{68\!\cdots\!60}{14\!\cdots\!41}a^{4}+\frac{42\!\cdots\!65}{14\!\cdots\!41}a^{3}+\frac{38\!\cdots\!04}{47\!\cdots\!47}a^{2}+\frac{42\!\cdots\!38}{52\!\cdots\!83}a+\frac{28\!\cdots\!02}{52\!\cdots\!83}$, $\frac{40\!\cdots\!25}{42\!\cdots\!23}a^{15}+\frac{81\!\cdots\!35}{14\!\cdots\!41}a^{14}-\frac{27\!\cdots\!89}{42\!\cdots\!23}a^{13}-\frac{66\!\cdots\!65}{14\!\cdots\!41}a^{12}-\frac{13\!\cdots\!85}{42\!\cdots\!23}a^{11}+\frac{28\!\cdots\!50}{42\!\cdots\!23}a^{10}-\frac{45\!\cdots\!46}{42\!\cdots\!23}a^{9}+\frac{21\!\cdots\!99}{42\!\cdots\!23}a^{8}+\frac{30\!\cdots\!07}{14\!\cdots\!41}a^{7}-\frac{12\!\cdots\!43}{42\!\cdots\!23}a^{6}-\frac{87\!\cdots\!11}{42\!\cdots\!23}a^{5}+\frac{87\!\cdots\!67}{42\!\cdots\!23}a^{4}+\frac{51\!\cdots\!00}{42\!\cdots\!23}a^{3}-\frac{16\!\cdots\!32}{14\!\cdots\!41}a^{2}-\frac{23\!\cdots\!92}{52\!\cdots\!83}a-\frac{30\!\cdots\!63}{15\!\cdots\!49}$, $\frac{11\!\cdots\!95}{42\!\cdots\!23}a^{15}-\frac{36\!\cdots\!10}{14\!\cdots\!41}a^{14}-\frac{86\!\cdots\!24}{42\!\cdots\!23}a^{13}-\frac{15\!\cdots\!81}{14\!\cdots\!41}a^{12}+\frac{42\!\cdots\!95}{42\!\cdots\!23}a^{11}+\frac{87\!\cdots\!84}{42\!\cdots\!23}a^{10}-\frac{25\!\cdots\!05}{42\!\cdots\!23}a^{9}+\frac{32\!\cdots\!17}{42\!\cdots\!23}a^{8}+\frac{98\!\cdots\!21}{14\!\cdots\!41}a^{7}-\frac{71\!\cdots\!73}{42\!\cdots\!23}a^{6}+\frac{52\!\cdots\!46}{42\!\cdots\!23}a^{5}+\frac{12\!\cdots\!20}{42\!\cdots\!23}a^{4}-\frac{24\!\cdots\!04}{42\!\cdots\!23}a^{3}+\frac{72\!\cdots\!95}{14\!\cdots\!41}a^{2}+\frac{16\!\cdots\!22}{17\!\cdots\!61}a+\frac{11\!\cdots\!68}{15\!\cdots\!49}$, $\frac{46\!\cdots\!75}{14\!\cdots\!41}a^{15}+\frac{10\!\cdots\!85}{15\!\cdots\!49}a^{14}-\frac{20\!\cdots\!01}{14\!\cdots\!41}a^{13}-\frac{87\!\cdots\!79}{47\!\cdots\!47}a^{12}-\frac{52\!\cdots\!04}{14\!\cdots\!41}a^{11}+\frac{26\!\cdots\!02}{14\!\cdots\!41}a^{10}-\frac{15\!\cdots\!51}{14\!\cdots\!41}a^{9}+\frac{33\!\cdots\!84}{14\!\cdots\!41}a^{8}+\frac{11\!\cdots\!42}{15\!\cdots\!49}a^{7}+\frac{71\!\cdots\!68}{14\!\cdots\!41}a^{6}-\frac{12\!\cdots\!46}{14\!\cdots\!41}a^{5}-\frac{64\!\cdots\!76}{14\!\cdots\!41}a^{4}-\frac{17\!\cdots\!35}{14\!\cdots\!41}a^{3}-\frac{22\!\cdots\!11}{47\!\cdots\!47}a^{2}-\frac{17\!\cdots\!83}{52\!\cdots\!83}a-\frac{29\!\cdots\!71}{52\!\cdots\!83}$
| 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: | \( 23009235.0124 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 23009235.0124 \cdot 1}{2\cdot\sqrt{81925410828566900634765625}}\cr\approx \mathstrut & 0.507133510661 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^14 - 45*x^13 - 7*x^12 + 715*x^11 - 1559*x^10 + 1450*x^9 + 1680*x^8 - 4760*x^7 + 896*x^6 + 2230*x^5 - 1432*x^4 + 450*x^3 + 738*x^2 + 270*x + 81)
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^16 - 7*x^14 - 45*x^13 - 7*x^12 + 715*x^11 - 1559*x^10 + 1450*x^9 + 1680*x^8 - 4760*x^7 + 896*x^6 + 2230*x^5 - 1432*x^4 + 450*x^3 + 738*x^2 + 270*x + 81, 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^16 - 7*x^14 - 45*x^13 - 7*x^12 + 715*x^11 - 1559*x^10 + 1450*x^9 + 1680*x^8 - 4760*x^7 + 896*x^6 + 2230*x^5 - 1432*x^4 + 450*x^3 + 738*x^2 + 270*x + 81);
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^16 - 7*x^14 - 45*x^13 - 7*x^12 + 715*x^11 - 1559*x^10 + 1450*x^9 + 1680*x^8 - 4760*x^7 + 896*x^6 + 2230*x^5 - 1432*x^4 + 450*x^3 + 738*x^2 + 270*x + 81);
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
$C_4\wr C_2$ (as 16T42):
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 solvable group of order 32 |
| The 14 conjugacy class representatives for $C_4\wr C_2$ |
| Character table for $C_4\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{205}) \), \(\Q(\sqrt{41}) \), 4.4.5125.1 x2, 4.4.210125.1 x2, \(\Q(\sqrt{5}, \sqrt{41})\), 8.4.9051265703125.2, 8.4.5384453125.2, 8.8.44152515625.1 |
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]
Sibling fields
| Galois closure: | deg 32 |
| Degree 8 siblings: | 8.4.9051265703125.2, 8.4.5384453125.2 |
| Degree 16 sibling: | 16.0.137716615602820959967041015625.4 |
| Minimal sibling: | 8.4.5384453125.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.8.7.2 | $x^{8} + 5$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
| 5.8.7.2 | $x^{8} + 5$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ | |
|
\(41\)
| 41.8.4.1 | $x^{8} + 3772 x^{7} + 5335658 x^{6} + 3354711230 x^{5} + 791201413052 x^{4} + 137665893766 x^{3} + 86143602389 x^{2} + 18207949968812 x + 5534365087156$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 41.8.6.2 | $x^{8} - 1804 x^{4} - 4557191$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |