Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^18 + 20*x^16 - 8*x^14 - 145*x^12 + 486*x^10 - 688*x^8 + 468*x^6 - 146*x^4 + 21*x^2 - 1)
gp: K = bnfinit(y^20 - 7*y^18 + 20*y^16 - 8*y^14 - 145*y^12 + 486*y^10 - 688*y^8 + 468*y^6 - 146*y^4 + 21*y^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 7*x^18 + 20*x^16 - 8*x^14 - 145*x^12 + 486*x^10 - 688*x^8 + 468*x^6 - 146*x^4 + 21*x^2 - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 7*x^18 + 20*x^16 - 8*x^14 - 145*x^12 + 486*x^10 - 688*x^8 + 468*x^6 - 146*x^4 + 21*x^2 - 1)
\( x^{20} - 7x^{18} + 20x^{16} - 8x^{14} - 145x^{12} + 486x^{10} - 688x^{8} + 468x^{6} - 146x^{4} + 21x^{2} - 1 \)
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: | $[10, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-1242593181789399040000000000\)
\(\medspace = -\,2^{20}\cdot 5^{10}\cdot 3319^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(22.63\) | 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: | $2^{3/2}5^{1/2}3319^{1/2}\approx 364.3624569024641$ | ||
| Ramified primes: |
\(2\), \(5\), \(3319\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{327973}a^{18}-\frac{56422}{327973}a^{16}+\frac{69185}{327973}a^{14}+\frac{134890}{327973}a^{12}+\frac{138024}{327973}a^{10}+\frac{111492}{327973}a^{8}+\frac{44326}{327973}a^{6}+\frac{143303}{327973}a^{4}+\frac{95559}{327973}a^{2}-\frac{68763}{327973}$, $\frac{1}{327973}a^{19}-\frac{56422}{327973}a^{17}+\frac{69185}{327973}a^{15}+\frac{134890}{327973}a^{13}+\frac{138024}{327973}a^{11}+\frac{111492}{327973}a^{9}+\frac{44326}{327973}a^{7}+\frac{143303}{327973}a^{5}+\frac{95559}{327973}a^{3}-\frac{68763}{327973}a$
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 $1$ (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: | $14$ | 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: |
$\frac{84419}{327973}a^{19}-\frac{592885}{327973}a^{17}+\frac{1625196}{327973}a^{15}-\frac{271623}{327973}a^{13}-\frac{13167635}{327973}a^{11}+\frac{40542646}{327973}a^{9}-\frac{48427367}{327973}a^{7}+\frac{17266448}{327973}a^{5}+\frac{5090908}{327973}a^{3}-\frac{1421462}{327973}a$, $\frac{77043}{327973}a^{18}-\frac{621950}{327973}a^{16}+\frac{1970597}{327973}a^{14}-\frac{1462073}{327973}a^{12}-\frac{12220548}{327973}a^{10}+\frac{47949344}{327973}a^{8}-\frac{73968756}{327973}a^{6}+\frac{50773745}{327973}a^{4}-\frac{13304787}{327973}a^{2}+\frac{696006}{327973}$, $a$, $\frac{78995}{327973}a^{18}-\frac{558766}{327973}a^{16}+\frac{1566868}{327973}a^{14}-\frac{535193}{327973}a^{12}-\frac{11735560}{327973}a^{10}+\frac{38296439}{327973}a^{8}-\frac{51727112}{327973}a^{6}+\frac{33357663}{327973}a^{4}-\frac{10438499}{327973}a^{2}+\frac{1267533}{327973}$, $\frac{275193}{327973}a^{19}-\frac{2009518}{327973}a^{17}+\frac{5970596}{327973}a^{15}-\frac{3136046}{327973}a^{13}-\frac{40967069}{327973}a^{11}+\frac{145235845}{327973}a^{9}-\frac{212949348}{327973}a^{7}+\frac{142849241}{327973}a^{5}-\frac{37096175}{327973}a^{3}+\frac{3241652}{327973}a$, $\frac{77043}{327973}a^{18}-\frac{621950}{327973}a^{16}+\frac{1970597}{327973}a^{14}-\frac{1462073}{327973}a^{12}-\frac{12220548}{327973}a^{10}+\frac{47949344}{327973}a^{8}-\frac{73968756}{327973}a^{6}+\frac{50773745}{327973}a^{4}-\frac{13304787}{327973}a^{2}+\frac{1023979}{327973}$, $\frac{67320}{327973}a^{19}-\frac{401700}{327973}a^{17}+\frac{973546}{327973}a^{15}+\frac{206349}{327973}a^{13}-\frac{8882654}{327973}a^{11}+\frac{23593391}{327973}a^{9}-\frac{27749739}{327973}a^{7}+\frac{19182572}{327973}a^{5}-\frac{9669732}{327973}a^{3}+\frac{1853600}{327973}a$, $\frac{84419}{327973}a^{19}+\frac{6769}{327973}a^{18}-\frac{592885}{327973}a^{17}+\frac{168027}{327973}a^{16}+\frac{1625196}{327973}a^{15}-\frac{1016098}{327973}a^{14}-\frac{271623}{327973}a^{13}+\frac{2289389}{327973}a^{12}-\frac{13167635}{327973}a^{11}+\frac{1529244}{327973}a^{10}+\frac{40542646}{327973}a^{9}-\frac{24574500}{327973}a^{8}-\frac{48427367}{327973}a^{7}+\frac{53406998}{327973}a^{6}+\frac{17266448}{327973}a^{5}-\frac{40466806}{327973}a^{4}+\frac{5090908}{327973}a^{3}+\frac{7947467}{327973}a^{2}-\frac{1421462}{327973}a-\frac{391033}{327973}$, $\frac{275193}{327973}a^{19}-\frac{2009518}{327973}a^{17}+\frac{5970596}{327973}a^{15}-\frac{3136046}{327973}a^{13}-\frac{40967069}{327973}a^{11}+\frac{145235845}{327973}a^{9}-\frac{212949348}{327973}a^{7}+\frac{142849241}{327973}a^{5}-\frac{37096175}{327973}a^{3}+\frac{3241652}{327973}a+1$, $\frac{546264}{327973}a^{19}+\frac{60551}{327973}a^{18}-\frac{3980409}{327973}a^{17}-\frac{569727}{327973}a^{16}+\frac{11769159}{327973}a^{15}+\frac{1989644}{327973}a^{14}-\frac{6046464}{327973}a^{13}-\frac{2083040}{327973}a^{12}-\frac{81179965}{327973}a^{11}-\frac{10411898}{327973}a^{10}+\frac{286128190}{327973}a^{9}+\frac{48167891}{327973}a^{8}-\frac{417730182}{327973}a^{7}-\frac{81156737}{327973}a^{6}+\frac{283387078}{327973}a^{5}+\frac{59649378}{327973}a^{4}-\frac{80750435}{327973}a^{3}-\frac{17617199}{327973}a^{2}+\frac{9179502}{327973}a+\frac{2244633}{327973}$, $\frac{161912}{327973}a^{19}+\frac{372}{2089}a^{18}-\frac{1022841}{327973}a^{17}-\frac{2890}{2089}a^{16}+\frac{2587689}{327973}a^{15}+\frac{8696}{2089}a^{14}+\frac{259637}{327973}a^{13}-\frac{4967}{2089}a^{12}-\frac{22968469}{327973}a^{11}-\frac{59095}{2089}a^{10}+\frac{63885546}{327973}a^{9}+\frac{213096}{2089}a^{8}-\frac{73259826}{327973}a^{7}-\frac{304199}{2089}a^{6}+\frac{33806670}{327973}a^{5}+\frac{189624}{2089}a^{4}-\frac{2601251}{327973}a^{3}-\frac{44434}{2089}a^{2}-\frac{511371}{327973}a+\frac{4147}{2089}$, $\frac{728645}{327973}a^{19}+\frac{644226}{327973}a^{18}-\frac{4456289}{327973}a^{17}-\frac{3863404}{327973}a^{16}+\frac{10709496}{327973}a^{15}+\frac{9084300}{327973}a^{14}+\frac{3255140}{327973}a^{13}+\frac{3526763}{327973}a^{12}-\frac{102126762}{327973}a^{11}-\frac{88959127}{327973}a^{10}+\frac{265162343}{327973}a^{9}+\frac{224619697}{327973}a^{8}-\frac{276688063}{327973}a^{7}-\frac{228260696}{327973}a^{6}+\frac{112745164}{327973}a^{5}+\frac{95478716}{327973}a^{4}-\frac{10903454}{327973}a^{3}-\frac{15994362}{327973}a^{2}-\frac{692817}{327973}a+\frac{728645}{327973}$, $\frac{696006}{327973}a^{19}-\frac{374852}{327973}a^{18}-\frac{4794999}{327973}a^{17}+\frac{2528477}{327973}a^{16}+\frac{13298170}{327973}a^{15}-\frac{6886051}{327973}a^{14}-\frac{3597451}{327973}a^{13}+\frac{1450995}{327973}a^{12}-\frac{102382943}{327973}a^{11}+\frac{54267766}{327973}a^{10}+\frac{326038368}{327973}a^{9}-\frac{168633862}{327973}a^{8}-\frac{430902784}{327973}a^{7}+\frac{219820284}{327973}a^{6}+\frac{251762052}{327973}a^{5}-\frac{130891605}{327973}a^{4}-\frac{50843131}{327973}a^{3}+\frac{30246362}{327973}a^{2}+\frac{1311339}{327973}a-\frac{1417832}{327973}$, $\frac{352236}{327973}a^{19}-\frac{77043}{327973}a^{18}-\frac{2631468}{327973}a^{17}+\frac{621950}{327973}a^{16}+\frac{7941193}{327973}a^{15}-\frac{1970597}{327973}a^{14}-\frac{4598119}{327973}a^{13}+\frac{1462073}{327973}a^{12}-\frac{53187617}{327973}a^{11}+\frac{12220548}{327973}a^{10}+\frac{193185189}{327973}a^{9}-\frac{47949344}{327973}a^{8}-\frac{286918104}{327973}a^{7}+\frac{73968756}{327973}a^{6}+\frac{193622986}{327973}a^{5}-\frac{50773745}{327973}a^{4}-\frac{50400962}{327973}a^{3}+\frac{13304787}{327973}a^{2}+\frac{3937658}{327973}a-\frac{696006}{327973}$
| 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: | \( 1180484.37704 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{5}\cdot 1180484.37704 \cdot 1}{2\cdot\sqrt{1242593181789399040000000000}}\cr\approx \mathstrut & 0.167905554561 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^18 + 20*x^16 - 8*x^14 - 145*x^12 + 486*x^10 - 688*x^8 + 468*x^6 - 146*x^4 + 21*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 - 7*x^18 + 20*x^16 - 8*x^14 - 145*x^12 + 486*x^10 - 688*x^8 + 468*x^6 - 146*x^4 + 21*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 - 7*x^18 + 20*x^16 - 8*x^14 - 145*x^12 + 486*x^10 - 688*x^8 + 468*x^6 - 146*x^4 + 21*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 - 7*x^18 + 20*x^16 - 8*x^14 - 145*x^12 + 486*x^10 - 688*x^8 + 468*x^6 - 146*x^4 + 21*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
$D_{10}\wr C_2$ (as 20T168):
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 800 |
| The 44 conjugacy class representatives for $D_{10}\wr C_2$ |
| Character table for $D_{10}\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.2.400.1, 10.6.34424253125.1 |
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: | 20.2.1242593181789399040000000000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{5}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{5}$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.5.0.1}{5} }^{2}$ | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{9}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | $20$ | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{5}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | $20$ | ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# 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 $20$ | $2$ | $10$ | $20$ | |||
|
\(5\)
| Deg $20$ | $2$ | $10$ | $10$ | |||
|
\(3319\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |