Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - x^18 + 5*x^17 - 2*x^16 - 3*x^15 + 7*x^14 - 24*x^13 + 51*x^12 - 67*x^11 + 79*x^10 - 95*x^9 + 96*x^8 - 81*x^7 + 68*x^6 - 53*x^5 + 34*x^4 - 20*x^3 + 11*x^2 - 4*x + 1)
gp: K = bnfinit(y^20 - 2*y^19 - y^18 + 5*y^17 - 2*y^16 - 3*y^15 + 7*y^14 - 24*y^13 + 51*y^12 - 67*y^11 + 79*y^10 - 95*y^9 + 96*y^8 - 81*y^7 + 68*y^6 - 53*y^5 + 34*y^4 - 20*y^3 + 11*y^2 - 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 2*x^19 - x^18 + 5*x^17 - 2*x^16 - 3*x^15 + 7*x^14 - 24*x^13 + 51*x^12 - 67*x^11 + 79*x^10 - 95*x^9 + 96*x^8 - 81*x^7 + 68*x^6 - 53*x^5 + 34*x^4 - 20*x^3 + 11*x^2 - 4*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 2*x^19 - x^18 + 5*x^17 - 2*x^16 - 3*x^15 + 7*x^14 - 24*x^13 + 51*x^12 - 67*x^11 + 79*x^10 - 95*x^9 + 96*x^8 - 81*x^7 + 68*x^6 - 53*x^5 + 34*x^4 - 20*x^3 + 11*x^2 - 4*x + 1)
\( x^{20} - 2 x^{19} - x^{18} + 5 x^{17} - 2 x^{16} - 3 x^{15} + 7 x^{14} - 24 x^{13} + 51 x^{12} + \cdots + 1 \)
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: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(921959696367613292449\)
\(\medspace = 79\cdot 151\cdot 278005859^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.17\) | 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: | $79^{1/2}151^{1/2}278005859^{1/2}\approx 1821079.8697506378$ | ||
| Ramified primes: |
\(79\), \(151\), \(278005859\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{11929}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{377347}a^{19}-\frac{75240}{377347}a^{18}-\frac{52575}{377347}a^{17}-\frac{90746}{377347}a^{16}-\frac{169072}{377347}a^{15}-\frac{105584}{377347}a^{14}+\frac{19955}{377347}a^{13}+\frac{89399}{377347}a^{12}+\frac{8364}{377347}a^{11}+\frac{124097}{377347}a^{10}-\frac{113186}{377347}a^{9}-\frac{78923}{377347}a^{8}+\frac{76378}{377347}a^{7}+\frac{89418}{377347}a^{6}+\frac{88247}{377347}a^{5}-\frac{107374}{377347}a^{4}-\frac{16877}{377347}a^{3}+\frac{19051}{377347}a^{2}+\frac{182126}{377347}a+\frac{182966}{377347}$
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: | $9$ | 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: |
$\frac{594214}{377347}a^{19}-\frac{966147}{377347}a^{18}-\frac{997614}{377347}a^{17}+\frac{2606738}{377347}a^{16}-\frac{84128}{377347}a^{15}-\frac{1778756}{377347}a^{14}+\frac{3184365}{377347}a^{13}-\frac{13225725}{377347}a^{12}+\frac{25628155}{377347}a^{11}-\frac{29654354}{377347}a^{10}+\frac{34830012}{377347}a^{9}-\frac{42351879}{377347}a^{8}+\frac{38710555}{377347}a^{7}-\frac{29859337}{377347}a^{6}+\frac{25613946}{377347}a^{5}-\frac{18661238}{377347}a^{4}+\frac{10012563}{377347}a^{3}-\frac{5699291}{377347}a^{2}+\frac{2850181}{377347}a-\frac{636263}{377347}$, $\frac{40925}{377347}a^{19}+\frac{331867}{377347}a^{18}-\frac{376628}{377347}a^{17}-\frac{685570}{377347}a^{16}+\frac{895033}{377347}a^{15}+\frac{352644}{377347}a^{14}-\frac{297880}{377347}a^{13}+\frac{1029604}{377347}a^{12}-\frac{5617234}{377347}a^{11}+\frac{9012780}{377347}a^{10}-\frac{9636300}{377347}a^{9}+\frac{12618996}{377347}a^{8}-\frac{14134737}{377347}a^{7}+\frac{11240854}{377347}a^{6}-\frac{9135990}{377347}a^{5}+\frac{7471805}{377347}a^{4}-\frac{4297032}{377347}a^{3}+\frac{2327355}{377347}a^{2}-\frac{1360782}{377347}a+\frac{187029}{377347}$, $\frac{767269}{377347}a^{19}-\frac{888765}{377347}a^{18}-\frac{1528069}{377347}a^{17}+\frac{2607807}{377347}a^{16}+\frac{647292}{377347}a^{15}-\frac{2098789}{377347}a^{14}+\frac{3771840}{377347}a^{13}-\frac{14640821}{377347}a^{12}+\frac{26311777}{377347}a^{11}-\frac{29243136}{377347}a^{10}+\frac{35132205}{377347}a^{9}-\frac{42674326}{377347}a^{8}+\frac{38971976}{377347}a^{7}-\frac{30627817}{377347}a^{6}+\frac{26720135}{377347}a^{5}-\frac{20079875}{377347}a^{4}+\frac{11538496}{377347}a^{3}-\frac{6463919}{377347}a^{2}+\frac{3511630}{377347}a-\frac{1019250}{377347}$, $\frac{186896}{377347}a^{19}-\frac{219085}{377347}a^{18}-\frac{318667}{377347}a^{17}+\frac{551193}{377347}a^{16}+\frac{157268}{377347}a^{15}-\frac{243246}{377347}a^{14}+\frac{566626}{377347}a^{13}-\frac{3628432}{377347}a^{12}+\frac{7396463}{377347}a^{11}-\frac{8702077}{377347}a^{10}+\frac{11005227}{377347}a^{9}-\frac{13860617}{377347}a^{8}+\frac{12912823}{377347}a^{7}-\frac{9888430}{377347}a^{6}+\frac{8984964}{377347}a^{5}-\frac{6872543}{377347}a^{4}+\frac{3773251}{377347}a^{3}-\frac{1977331}{377347}a^{2}+\frac{1544149}{377347}a-\frac{326298}{377347}$, $a$, $\frac{358795}{377347}a^{19}-\frac{708767}{377347}a^{18}-\frac{447942}{377347}a^{17}+\frac{1684213}{377347}a^{16}-\frac{639214}{377347}a^{15}-\frac{768603}{377347}a^{14}+\frac{2613676}{377347}a^{13}-\frac{9146511}{377347}a^{12}+\frac{17655998}{377347}a^{11}-\frac{22694317}{377347}a^{10}+\frac{27059601}{377347}a^{9}-\frac{31246665}{377347}a^{8}+\frac{30161089}{377347}a^{7}-\frac{25724920}{377347}a^{6}+\frac{20904374}{377347}a^{5}-\frac{15483592}{377347}a^{4}+\frac{9715116}{377347}a^{3}-\frac{5520218}{377347}a^{2}+\frac{2604915}{377347}a-\frac{903661}{377347}$, $\frac{242145}{377347}a^{19}-\frac{299293}{377347}a^{18}-\frac{217636}{377347}a^{17}+\frac{735028}{377347}a^{16}-\frac{431369}{377347}a^{15}-\frac{246389}{377347}a^{14}+\frac{1584528}{377347}a^{13}-\frac{5027352}{377347}a^{12}+\frac{9890453}{377347}a^{11}-\frac{14144772}{377347}a^{10}+\frac{17401296}{377347}a^{9}-\frac{19693064}{377347}a^{8}+\frac{20019037}{377347}a^{7}-\frac{17407212}{377347}a^{6}+\frac{13748391}{377347}a^{5}-\frac{10302605}{377347}a^{4}+\frac{6779091}{377347}a^{3}-\frac{3736150}{377347}a^{2}+\frac{1865768}{377347}a-\frac{763894}{377347}$, $\frac{15851}{377347}a^{19}-\frac{212720}{377347}a^{18}-\frac{184149}{377347}a^{17}+\frac{786612}{377347}a^{16}+\frac{335469}{377347}a^{15}-\frac{1210080}{377347}a^{14}-\frac{287428}{377347}a^{13}-\frac{629130}{377347}a^{12}+\frac{2770396}{377347}a^{11}-\frac{48364}{377347}a^{10}-\frac{2467730}{377347}a^{9}-\frac{103168}{377347}a^{8}-\frac{1370886}{377347}a^{7}+\frac{4577550}{377347}a^{6}-\frac{2663561}{377347}a^{5}+\frac{1359084}{377347}a^{4}-\frac{2619733}{377347}a^{3}+\frac{1231842}{377347}a^{2}-\frac{580018}{377347}a+\frac{282371}{377347}$, $\frac{49747}{377347}a^{19}+\frac{317960}{377347}a^{18}-\frac{433815}{377347}a^{17}-\frac{893795}{377347}a^{16}+\frac{994540}{377347}a^{15}+\frac{937686}{377347}a^{14}-\frac{475919}{377347}a^{13}+\frac{297658}{377347}a^{12}-\frac{5412691}{377347}a^{11}+\frac{7980826}{377347}a^{10}-\frac{5929560}{377347}a^{9}+\frac{9169382}{377347}a^{8}-\frac{12005681}{377347}a^{7}+\frac{7657750}{377347}a^{6}-\frac{6446388}{377347}a^{5}+\frac{7359347}{377347}a^{4}-\frac{3756514}{377347}a^{3}+\frac{1721168}{377347}a^{2}-\frac{2143430}{377347}a+\frac{399962}{377347}$
| 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: | \( 104.951823132 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 104.951823132 \cdot 1}{2\cdot\sqrt{921959696367613292449}}\cr\approx \mathstrut & 0.165730601653 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - x^18 + 5*x^17 - 2*x^16 - 3*x^15 + 7*x^14 - 24*x^13 + 51*x^12 - 67*x^11 + 79*x^10 - 95*x^9 + 96*x^8 - 81*x^7 + 68*x^6 - 53*x^5 + 34*x^4 - 20*x^3 + 11*x^2 - 4*x + 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 - 2*x^19 - x^18 + 5*x^17 - 2*x^16 - 3*x^15 + 7*x^14 - 24*x^13 + 51*x^12 - 67*x^11 + 79*x^10 - 95*x^9 + 96*x^8 - 81*x^7 + 68*x^6 - 53*x^5 + 34*x^4 - 20*x^3 + 11*x^2 - 4*x + 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 - 2*x^19 - x^18 + 5*x^17 - 2*x^16 - 3*x^15 + 7*x^14 - 24*x^13 + 51*x^12 - 67*x^11 + 79*x^10 - 95*x^9 + 96*x^8 - 81*x^7 + 68*x^6 - 53*x^5 + 34*x^4 - 20*x^3 + 11*x^2 - 4*x + 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 - 2*x^19 - x^18 + 5*x^17 - 2*x^16 - 3*x^15 + 7*x^14 - 24*x^13 + 51*x^12 - 67*x^11 + 79*x^10 - 95*x^9 + 96*x^8 - 81*x^7 + 68*x^6 - 53*x^5 + 34*x^4 - 20*x^3 + 11*x^2 - 4*x + 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
$C_2^{10}.S_{10}$ (as 20T1110):
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 3715891200 |
| The 481 conjugacy class representatives for $C_2^{10}.S_{10}$ are not computed |
| Character table for $C_2^{10}.S_{10}$ is not computed |
Intermediate fields
| 10.0.278005859.1 |
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
| 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 | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.8.0.1}{8} }$ | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.7.0.1}{7} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | $18{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.7.0.1}{7} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ | $18{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(79\)
| $\Q_{79}$ | $x + 76$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{79}$ | $x + 76$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 79.2.0.1 | $x^{2} + 78 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 79.2.1.1 | $x^{2} + 237$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 79.2.0.1 | $x^{2} + 78 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 79.4.0.1 | $x^{4} + 2 x^{2} + 66 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 79.8.0.1 | $x^{8} + 60 x^{3} + 59 x^{2} + 48 x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
|
\(151\)
| 151.2.1.1 | $x^{2} + 453$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 151.18.0.1 | $x^{18} - x + 61$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | |
|
\(278005859\)
| $\Q_{278005859}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{278005859}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ |