Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 2*x^18 + 3*x^17 - 2*x^16 - 9*x^15 + 18*x^14 - 13*x^13 - x^12 + 6*x^11 + 10*x^10 - 26*x^9 + 20*x^8 - 6*x^7 + x^6 + 5*x^5 - 10*x^4 + 6*x^3 - 2*x + 1)
gp: K = bnfinit(y^20 - 3*y^19 + 2*y^18 + 3*y^17 - 2*y^16 - 9*y^15 + 18*y^14 - 13*y^13 - y^12 + 6*y^11 + 10*y^10 - 26*y^9 + 20*y^8 - 6*y^7 + y^6 + 5*y^5 - 10*y^4 + 6*y^3 - 2*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 3*x^19 + 2*x^18 + 3*x^17 - 2*x^16 - 9*x^15 + 18*x^14 - 13*x^13 - x^12 + 6*x^11 + 10*x^10 - 26*x^9 + 20*x^8 - 6*x^7 + x^6 + 5*x^5 - 10*x^4 + 6*x^3 - 2*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 3*x^19 + 2*x^18 + 3*x^17 - 2*x^16 - 9*x^15 + 18*x^14 - 13*x^13 - x^12 + 6*x^11 + 10*x^10 - 26*x^9 + 20*x^8 - 6*x^7 + x^6 + 5*x^5 - 10*x^4 + 6*x^3 - 2*x + 1)
\( x^{20} - 3 x^{19} + 2 x^{18} + 3 x^{17} - 2 x^{16} - 9 x^{15} + 18 x^{14} - 13 x^{13} - x^{12} + \cdots + 1 \)
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: |
\(3575480621700351753081\)
\(\medspace = 3^{10}\cdot 7^{10}\cdot 11^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.96\) | 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: | $3^{1/2}7^{1/2}11^{4/5}\approx 31.204971875395575$ | ||
| Ramified primes: |
\(3\), \(7\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $10$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2663593}a^{19}-\frac{605577}{2663593}a^{18}+\frac{865553}{2663593}a^{17}-\frac{1243914}{2663593}a^{16}-\frac{768917}{2663593}a^{15}+\frac{133054}{2663593}a^{14}-\frac{354728}{2663593}a^{13}+\frac{605595}{2663593}a^{12}-\frac{1111512}{2663593}a^{11}-\frac{501171}{2663593}a^{10}+\frac{1013558}{2663593}a^{9}+\frac{680637}{2663593}a^{8}-\frac{1035426}{2663593}a^{7}+\frac{1290760}{2663593}a^{6}-\frac{685238}{2663593}a^{5}+\frac{1163147}{2663593}a^{4}-\frac{394096}{2663593}a^{3}-\frac{978097}{2663593}a^{2}-\frac{1053511}{2663593}a+\frac{402138}{2663593}$
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: |
\( -\frac{5069577}{2663593} a^{19} + \frac{15210989}{2663593} a^{18} - \frac{13772404}{2663593} a^{17} - \frac{5212947}{2663593} a^{16} + \frac{3481178}{2663593} a^{15} + \frac{38083464}{2663593} a^{14} - \frac{89138870}{2663593} a^{13} + \frac{95969693}{2663593} a^{12} - \frac{52519168}{2663593} a^{11} + \frac{16837722}{2663593} a^{10} - \frac{63630828}{2663593} a^{9} + \frac{132396579}{2663593} a^{8} - \frac{146687436}{2663593} a^{7} + \frac{116295184}{2663593} a^{6} - \frac{79084843}{2663593} a^{5} + \frac{20213325}{2663593} a^{4} + \frac{18815882}{2663593} a^{3} - \frac{27982575}{2663593} a^{2} + \frac{18220722}{2663593} a - \frac{3418100}{2663593} \)
(order $6$)
| 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: |
$\frac{2748078}{2663593}a^{19}-\frac{10532873}{2663593}a^{18}+\frac{13280948}{2663593}a^{17}-\frac{12475}{2663593}a^{16}-\frac{7573847}{2663593}a^{15}-\frac{20604014}{2663593}a^{14}+\frac{68143181}{2663593}a^{13}-\frac{83835745}{2663593}a^{12}+\frac{49488162}{2663593}a^{11}-\frac{11611979}{2663593}a^{10}+\frac{35886575}{2663593}a^{9}-\frac{101908866}{2663593}a^{8}+\frac{127608160}{2663593}a^{7}-\frac{96191761}{2663593}a^{6}+\frac{62124064}{2663593}a^{5}-\frac{28761777}{2663593}a^{4}-\frac{8278839}{2663593}a^{3}+\frac{27420117}{2663593}a^{2}-\frac{15234705}{2663593}a+\frac{3163808}{2663593}$, $\frac{3844207}{2663593}a^{19}-\frac{12359962}{2663593}a^{18}+\frac{12516650}{2663593}a^{17}+\frac{3035319}{2663593}a^{16}-\frac{4390336}{2663593}a^{15}-\frac{29281542}{2663593}a^{14}+\frac{74465002}{2663593}a^{13}-\frac{83587078}{2663593}a^{12}+\frac{48352430}{2663593}a^{11}-\frac{14931532}{2663593}a^{10}+\frac{49554036}{2663593}a^{9}-\frac{111017586}{2663593}a^{8}+\frac{129446692}{2663593}a^{7}-\frac{101389868}{2663593}a^{6}+\frac{70124804}{2663593}a^{5}-\frac{22365601}{2663593}a^{4}-\frac{14811262}{2663593}a^{3}+\frac{24424848}{2663593}a^{2}-\frac{14430032}{2663593}a+\frac{5609226}{2663593}$, $\frac{3033068}{2663593}a^{19}-\frac{11740854}{2663593}a^{18}+\frac{14546281}{2663593}a^{17}-\frac{143779}{2663593}a^{16}-\frac{7433567}{2663593}a^{15}-\frac{23118095}{2663593}a^{14}+\frac{73546369}{2663593}a^{13}-\frac{93479502}{2663593}a^{12}+\frac{58135179}{2663593}a^{11}-\frac{15815016}{2663593}a^{10}+\frac{39101703}{2663593}a^{9}-\frac{111321240}{2663593}a^{8}+\frac{141020890}{2663593}a^{7}-\frac{113992184}{2663593}a^{6}+\frac{73448797}{2663593}a^{5}-\frac{30310690}{2663593}a^{4}-\frac{11299034}{2663593}a^{3}+\frac{27227130}{2663593}a^{2}-\frac{20458821}{2663593}a+\frac{4720010}{2663593}$, $\frac{5409575}{2663593}a^{19}-\frac{17104528}{2663593}a^{18}+\frac{15652286}{2663593}a^{17}+\frac{7564501}{2663593}a^{16}-\frac{7397580}{2663593}a^{15}-\frac{43780270}{2663593}a^{14}+\frac{103137331}{2663593}a^{13}-\frac{103937748}{2663593}a^{12}+\frac{45909846}{2663593}a^{11}-\frac{7949612}{2663593}a^{10}+\frac{68979337}{2663593}a^{9}-\frac{154110030}{2663593}a^{8}+\frac{158995877}{2663593}a^{7}-\frac{108732198}{2663593}a^{6}+\frac{73360657}{2663593}a^{5}-\frac{21539515}{2663593}a^{4}-\frac{26612604}{2663593}a^{3}+\frac{34535598}{2663593}a^{2}-\frac{15194246}{2663593}a+\frac{1977948}{2663593}$, $\frac{8939390}{2663593}a^{19}-\frac{23873574}{2663593}a^{18}+\frac{12563854}{2663593}a^{17}+\frac{23813034}{2663593}a^{16}-\frac{4817167}{2663593}a^{15}-\frac{77774101}{2663593}a^{14}+\frac{133991311}{2663593}a^{13}-\frac{92231078}{2663593}a^{12}+\frac{1433590}{2663593}a^{11}+\frac{16381868}{2663593}a^{10}+\frac{109637227}{2663593}a^{9}-\frac{200294654}{2663593}a^{8}+\frac{143241044}{2663593}a^{7}-\frac{67185228}{2663593}a^{6}+\frac{44212383}{2663593}a^{5}+\frac{17928869}{2663593}a^{4}-\frac{56467780}{2663593}a^{3}+\frac{31711876}{2663593}a^{2}-\frac{2683993}{2663593}a-\frac{7259142}{2663593}$, $\frac{1535220}{2663593}a^{19}-\frac{6739185}{2663593}a^{18}+\frac{11655192}{2663593}a^{17}-\frac{5331765}{2663593}a^{16}-\frac{5611000}{2663593}a^{15}-\frac{6448883}{2663593}a^{14}+\frac{47331329}{2663593}a^{13}-\frac{74834168}{2663593}a^{12}+\frac{58020298}{2663593}a^{11}-\frac{18250198}{2663593}a^{10}+\frac{16754020}{2663593}a^{9}-\frac{69252178}{2663593}a^{8}+\frac{116160842}{2663593}a^{7}-\frac{99306835}{2663593}a^{6}+\frac{58899222}{2663593}a^{5}-\frac{31490941}{2663593}a^{4}+\frac{3098051}{2663593}a^{3}+\frac{22458968}{2663593}a^{2}-\frac{18842669}{2663593}a+\frac{5378413}{2663593}$, $\frac{3632131}{2663593}a^{19}-\frac{9148605}{2663593}a^{18}+\frac{5683031}{2663593}a^{17}+\frac{5090063}{2663593}a^{16}+\frac{1951489}{2663593}a^{15}-\frac{26073811}{2663593}a^{14}+\frac{49267301}{2663593}a^{13}-\frac{47998315}{2663593}a^{12}+\frac{24082098}{2663593}a^{11}-\frac{11278422}{2663593}a^{10}+\frac{44855542}{2663593}a^{9}-\frac{72404247}{2663593}a^{8}+\frac{73245102}{2663593}a^{7}-\frac{63201123}{2663593}a^{6}+\frac{43052475}{2663593}a^{5}-\frac{3935892}{2663593}a^{4}-\frac{9401934}{2663593}a^{3}+\frac{14701380}{2663593}a^{2}-\frac{12212036}{2663593}a+\frac{2047819}{2663593}$, $\frac{74723}{2663593}a^{19}-\frac{4075880}{2663593}a^{18}+\frac{10005965}{2663593}a^{17}-\frac{5571680}{2663593}a^{16}-\frac{7411167}{2663593}a^{15}-\frac{998627}{2663593}a^{14}+\frac{31036715}{2663593}a^{13}-\frac{55841745}{2663593}a^{12}+\frac{45926831}{2663593}a^{11}-\frac{14864611}{2663593}a^{10}+\frac{4818258}{2663593}a^{9}-\frac{50014458}{2663593}a^{8}+\frac{81820256}{2663593}a^{7}-\frac{73823654}{2663593}a^{6}+\frac{49817432}{2663593}a^{5}-\frac{33833018}{2663593}a^{4}-\frac{2014793}{2663593}a^{3}+\frac{13304161}{2663593}a^{2}-\frac{14992896}{2663593}a+\frac{6292327}{2663593}$, $\frac{461180}{2663593}a^{19}-\frac{4938403}{2663593}a^{18}+\frac{9685560}{2663593}a^{17}-\frac{2243331}{2663593}a^{16}-\frac{10333156}{2663593}a^{15}-\frac{4675407}{2663593}a^{14}+\frac{39049729}{2663593}a^{13}-\frac{53350182}{2663593}a^{12}+\frac{28004620}{2663593}a^{11}+\frac{3240795}{2663593}a^{10}+\frac{4070056}{2663593}a^{9}-\frac{58871657}{2663593}a^{8}+\frac{80443778}{2663593}a^{7}-\frac{48329479}{2663593}a^{6}+\frac{22575796}{2663593}a^{5}-\frac{19505961}{2663593}a^{4}-\frac{6915704}{2663593}a^{3}+\frac{22008834}{2663593}a^{2}-\frac{8185408}{2663593}a+\frac{13029}{2663593}$
| 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: | \( 635.552521857 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 635.552521857 \cdot 1}{6\cdot\sqrt{3575480621700351753081}}\cr\approx \mathstrut & 0.169875858770 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 2*x^18 + 3*x^17 - 2*x^16 - 9*x^15 + 18*x^14 - 13*x^13 - x^12 + 6*x^11 + 10*x^10 - 26*x^9 + 20*x^8 - 6*x^7 + x^6 + 5*x^5 - 10*x^4 + 6*x^3 - 2*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 - 3*x^19 + 2*x^18 + 3*x^17 - 2*x^16 - 9*x^15 + 18*x^14 - 13*x^13 - x^12 + 6*x^11 + 10*x^10 - 26*x^9 + 20*x^8 - 6*x^7 + x^6 + 5*x^5 - 10*x^4 + 6*x^3 - 2*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 - 3*x^19 + 2*x^18 + 3*x^17 - 2*x^16 - 9*x^15 + 18*x^14 - 13*x^13 - x^12 + 6*x^11 + 10*x^10 - 26*x^9 + 20*x^8 - 6*x^7 + x^6 + 5*x^5 - 10*x^4 + 6*x^3 - 2*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 - 3*x^19 + 2*x^18 + 3*x^17 - 2*x^16 - 9*x^15 + 18*x^14 - 13*x^13 - x^12 + 6*x^11 + 10*x^10 - 26*x^9 + 20*x^8 - 6*x^7 + x^6 + 5*x^5 - 10*x^4 + 6*x^3 - 2*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
$C_5\times D_{10}$ (as 20T24):
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 100 |
| The 40 conjugacy class representatives for $C_5\times D_{10}$ |
| Character table for $C_5\times D_{10}$ |
Intermediate fields
| \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-3}, \sqrt{-7})\), 10.0.246071287.1 |
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 sibling: | 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 | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | R | R | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | ${\href{/padicField/23.10.0.1}{10} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{10}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.5.0.1}{5} }^{4}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{5}$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.20.10.1 | $x^{20} + 30 x^{18} + 409 x^{16} + 4 x^{15} + 3244 x^{14} - 60 x^{13} + 16162 x^{12} - 1250 x^{11} + 53008 x^{10} - 7102 x^{9} + 121150 x^{8} - 12140 x^{7} + 219264 x^{6} + 5736 x^{5} + 257465 x^{4} + 35250 x^{3} + 250183 x^{2} + 51502 x + 77812$ | $2$ | $10$ | $10$ | 20T3 | $[\ ]_{2}^{10}$ |
|
\(7\)
| 7.10.5.2 | $x^{10} + 35 x^{8} + 492 x^{6} + 8 x^{5} + 3360 x^{4} - 560 x^{3} + 11516 x^{2} + 1968 x + 17516$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
| 7.10.5.2 | $x^{10} + 35 x^{8} + 492 x^{6} + 8 x^{5} + 3360 x^{4} - 560 x^{3} + 11516 x^{2} + 1968 x + 17516$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
|
\(11\)
| 11.10.0.1 | $x^{10} + 7 x^{5} + 8 x^{4} + 10 x^{3} + 6 x^{2} + 6 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |
| 11.10.8.1 | $x^{10} - 77 x^{5} + 242$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |