Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 55*x^18 + 1265*x^16 + 15950*x^14 + 121550*x^12 + 581625*x^10 + 1754500*x^8 + 3251875*x^6 + 3478750*x^4 + 1890625*x^2 + 378125)
gp: K = bnfinit(y^20 + 55*y^18 + 1265*y^16 + 15950*y^14 + 121550*y^12 + 581625*y^10 + 1754500*y^8 + 3251875*y^6 + 3478750*y^4 + 1890625*y^2 + 378125, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + 55*x^18 + 1265*x^16 + 15950*x^14 + 121550*x^12 + 581625*x^10 + 1754500*x^8 + 3251875*x^6 + 3478750*x^4 + 1890625*x^2 + 378125);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 + 55*x^18 + 1265*x^16 + 15950*x^14 + 121550*x^12 + 581625*x^10 + 1754500*x^8 + 3251875*x^6 + 3478750*x^4 + 1890625*x^2 + 378125)
\( x^{20} + 55 x^{18} + 1265 x^{16} + 15950 x^{14} + 121550 x^{12} + 581625 x^{10} + 1754500 x^{8} + \cdots + 378125 \)
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: |
\(177917354031751407392000000000000000\)
\(\medspace = 2^{20}\cdot 5^{15}\cdot 11^{18}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(57.88\) | 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: | $2\cdot 5^{3/4}11^{9/10}\approx 57.87765351369302$ | ||
| Ramified primes: |
\(2\), \(5\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $20$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(220=2^{2}\cdot 5\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{220}(1,·)$, $\chi_{220}(43,·)$, $\chi_{220}(69,·)$, $\chi_{220}(7,·)$, $\chi_{220}(9,·)$, $\chi_{220}(183,·)$, $\chi_{220}(141,·)$, $\chi_{220}(63,·)$, $\chi_{220}(81,·)$, $\chi_{220}(83,·)$, $\chi_{220}(87,·)$, $\chi_{220}(89,·)$, $\chi_{220}(167,·)$, $\chi_{220}(169,·)$, $\chi_{220}(107,·)$, $\chi_{220}(49,·)$, $\chi_{220}(181,·)$, $\chi_{220}(201,·)$, $\chi_{220}(123,·)$, $\chi_{220}(127,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{512}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5}a^{4}$, $\frac{1}{5}a^{5}$, $\frac{1}{5}a^{6}$, $\frac{1}{5}a^{7}$, $\frac{1}{25}a^{8}$, $\frac{1}{25}a^{9}$, $\frac{1}{275}a^{10}$, $\frac{1}{275}a^{11}$, $\frac{1}{1375}a^{12}$, $\frac{1}{1375}a^{13}$, $\frac{1}{1375}a^{14}$, $\frac{1}{1375}a^{15}$, $\frac{1}{6875}a^{16}$, $\frac{1}{6875}a^{17}$, $\frac{1}{75625}a^{18}-\frac{4}{275}a^{8}$, $\frac{1}{75625}a^{19}-\frac{4}{275}a^{9}$
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
$C_{2}\times C_{2}\times C_{2762}$, which has order $11048$ (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: | $9$ | 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: |
$\frac{7}{15125}a^{18}+\frac{32}{1375}a^{16}+\frac{653}{1375}a^{14}+\frac{1403}{275}a^{12}+\frac{1726}{55}a^{10}+\frac{6198}{55}a^{8}+\frac{1153}{5}a^{6}+251a^{4}+128a^{2}+22$, $\frac{3}{6875}a^{18}+\frac{152}{6875}a^{16}+\frac{57}{125}a^{14}+\frac{1368}{275}a^{12}+\frac{8607}{275}a^{10}+\frac{2907}{25}a^{8}+\frac{1254}{5}a^{6}+\frac{1482}{5}a^{4}+171a^{2}+37$, $\frac{1}{6875}a^{18}+\frac{52}{6875}a^{16}+\frac{222}{1375}a^{14}+\frac{507}{275}a^{12}+\frac{678}{55}a^{10}+\frac{1242}{25}a^{8}+\frac{597}{5}a^{6}+\frac{808}{5}a^{4}+106a^{2}+23$, $\frac{2}{6875}a^{18}+\frac{104}{6875}a^{16}+\frac{444}{1375}a^{14}+\frac{5067}{1375}a^{12}+\frac{6756}{275}a^{10}+\frac{2452}{25}a^{8}+\frac{1152}{5}a^{6}+\frac{1496}{5}a^{4}+188a^{2}+40$, $\frac{1}{6875}a^{18}+\frac{48}{6875}a^{16}+\frac{184}{1375}a^{14}+\frac{1816}{1375}a^{12}+\frac{1992}{275}a^{10}+\frac{556}{25}a^{8}+\frac{182}{5}a^{6}+28a^{4}+8a^{2}+1$, $\frac{3}{15125}a^{18}+\frac{64}{6875}a^{16}+\frac{236}{1375}a^{14}+\frac{2166}{1375}a^{12}+\frac{2038}{275}a^{10}+\frac{4142}{275}a^{8}-4a^{6}-\frac{314}{5}a^{4}-77a^{2}-23$, $\frac{4}{75625}a^{18}+\frac{17}{6875}a^{16}+\frac{63}{1375}a^{14}+\frac{599}{1375}a^{12}+\frac{646}{275}a^{10}+\frac{2173}{275}a^{8}+\frac{97}{5}a^{6}+\frac{184}{5}a^{4}+39a^{2}+11$, $\frac{2}{75625}a^{18}+\frac{8}{6875}a^{16}+\frac{26}{1375}a^{14}+\frac{7}{55}a^{12}+\frac{23}{275}a^{10}-\frac{987}{275}a^{8}-\frac{101}{5}a^{6}-\frac{227}{5}a^{4}-43a^{2}-15$, $\frac{42}{75625}a^{18}+\frac{196}{6875}a^{16}+\frac{824}{1375}a^{14}+\frac{9243}{1375}a^{12}+\frac{2423}{55}a^{10}+\frac{47737}{275}a^{8}+\frac{2032}{5}a^{6}+\frac{2671}{5}a^{4}+345a^{2}+78$
| 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: | \( 140644.599182 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 140644.599182 \cdot 11048}{2\cdot\sqrt{177917354031751407392000000000000000}}\cr\approx \mathstrut & 0.176630727468 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 + 55*x^18 + 1265*x^16 + 15950*x^14 + 121550*x^12 + 581625*x^10 + 1754500*x^8 + 3251875*x^6 + 3478750*x^4 + 1890625*x^2 + 378125)
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 + 55*x^18 + 1265*x^16 + 15950*x^14 + 121550*x^12 + 581625*x^10 + 1754500*x^8 + 3251875*x^6 + 3478750*x^4 + 1890625*x^2 + 378125, 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 + 55*x^18 + 1265*x^16 + 15950*x^14 + 121550*x^12 + 581625*x^10 + 1754500*x^8 + 3251875*x^6 + 3478750*x^4 + 1890625*x^2 + 378125);
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 + 55*x^18 + 1265*x^16 + 15950*x^14 + 121550*x^12 + 581625*x^10 + 1754500*x^8 + 3251875*x^6 + 3478750*x^4 + 1890625*x^2 + 378125);
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
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 cyclic group of order 20 |
| The 20 conjugacy class representatives for $C_{20}$ |
| Character table for $C_{20}$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.0.242000.2, \(\Q(\zeta_{11})^+\), 10.10.669871503125.1 |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $20$ | R | $20$ | R | $20$ | $20$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}$ | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | $20$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | $20$ | $20$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 $20$ | $2$ | $10$ | $20$ | |||
|
\(5\)
| Deg $20$ | $4$ | $5$ | $15$ | |||
|
\(11\)
| 11.10.9.1 | $x^{10} + 110$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |
| 11.10.9.1 | $x^{10} + 110$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |