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Properties

Label 20.0.645...125.1
Degree $20$
Signature $[0, 10]$
Discriminant $6.455\times 10^{33}$
Root discriminant \(49.03\)
Ramified primes $3,5,7,11$
Class number not computed
Class group not computed
Galois group $C_{10}\wr C_2$ (as 20T53)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 58*x^17 - 193*x^16 - 2188*x^15 - 465*x^14 - 7861*x^13 + 91673*x^12 - 34407*x^11 + 607603*x^10 + 3012761*x^9 + 12205932*x^8 - 10310764*x^7 + 30457244*x^6 - 323709400*x^5 + 3839375256*x^4 - 3757533269*x^3 + 15771143512*x^2 - 45579456448*x + 87786948061)
 
gp: K = bnfinit(y^20 - 2*y^19 + 58*y^17 - 193*y^16 - 2188*y^15 - 465*y^14 - 7861*y^13 + 91673*y^12 - 34407*y^11 + 607603*y^10 + 3012761*y^9 + 12205932*y^8 - 10310764*y^7 + 30457244*y^6 - 323709400*y^5 + 3839375256*y^4 - 3757533269*y^3 + 15771143512*y^2 - 45579456448*y + 87786948061, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 2*x^19 + 58*x^17 - 193*x^16 - 2188*x^15 - 465*x^14 - 7861*x^13 + 91673*x^12 - 34407*x^11 + 607603*x^10 + 3012761*x^9 + 12205932*x^8 - 10310764*x^7 + 30457244*x^6 - 323709400*x^5 + 3839375256*x^4 - 3757533269*x^3 + 15771143512*x^2 - 45579456448*x + 87786948061);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 2*x^19 + 58*x^17 - 193*x^16 - 2188*x^15 - 465*x^14 - 7861*x^13 + 91673*x^12 - 34407*x^11 + 607603*x^10 + 3012761*x^9 + 12205932*x^8 - 10310764*x^7 + 30457244*x^6 - 323709400*x^5 + 3839375256*x^4 - 3757533269*x^3 + 15771143512*x^2 - 45579456448*x + 87786948061)
 

\( x^{20} - 2 x^{19} + 58 x^{17} - 193 x^{16} - 2188 x^{15} - 465 x^{14} - 7861 x^{13} + 91673 x^{12} + \cdots + 87786948061 \) Copy content Toggle raw display

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:   \(6455313778301718076702695673828125\) \(\medspace = 3^{10}\cdot 5^{10}\cdot 7^{15}\cdot 11^{9}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(49.03\)
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:  $3^{1/2}5^{1/2}7^{3/4}11^{9/10}\approx 144.25224346166357$
Ramified primes:   \(3\), \(5\), \(7\), \(11\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{77}) \)
$\card{ \Aut(K/\Q) }$:  $10$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{4}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}+\frac{1}{4}a^{10}+\frac{1}{4}a^{9}+\frac{1}{4}a^{8}+\frac{1}{4}a^{7}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}$, $\frac{1}{16}a^{17}-\frac{1}{16}a^{16}+\frac{1}{4}a^{15}-\frac{1}{4}a^{14}-\frac{1}{4}a^{13}-\frac{1}{2}a^{12}+\frac{7}{16}a^{11}-\frac{1}{8}a^{10}+\frac{1}{8}a^{9}+\frac{3}{8}a^{8}-\frac{3}{16}a^{7}-\frac{3}{8}a^{6}-\frac{7}{16}a^{5}+\frac{3}{8}a^{4}-\frac{3}{16}a^{3}-\frac{1}{8}a^{2}-\frac{7}{16}a+\frac{1}{16}$, $\frac{1}{64}a^{18}+\frac{3}{64}a^{16}+\frac{1}{8}a^{14}+\frac{1}{16}a^{13}+\frac{15}{64}a^{12}-\frac{11}{64}a^{11}-\frac{1}{4}a^{10}+\frac{3}{8}a^{9}-\frac{13}{64}a^{8}+\frac{7}{64}a^{7}+\frac{19}{64}a^{6}+\frac{15}{64}a^{5}-\frac{13}{64}a^{4}-\frac{5}{64}a^{3}-\frac{25}{64}a^{2}-\frac{11}{32}a-\frac{31}{64}$, $\frac{1}{30\!\cdots\!88}a^{19}+\frac{82\!\cdots\!49}{30\!\cdots\!88}a^{18}+\frac{75\!\cdots\!75}{30\!\cdots\!88}a^{17}-\frac{29\!\cdots\!61}{30\!\cdots\!88}a^{16}+\frac{10\!\cdots\!09}{38\!\cdots\!36}a^{15}+\frac{26\!\cdots\!15}{76\!\cdots\!72}a^{14}+\frac{64\!\cdots\!27}{30\!\cdots\!88}a^{13}-\frac{77\!\cdots\!43}{76\!\cdots\!72}a^{12}+\frac{86\!\cdots\!17}{30\!\cdots\!88}a^{11}+\frac{79\!\cdots\!41}{38\!\cdots\!36}a^{10}+\frac{89\!\cdots\!07}{30\!\cdots\!88}a^{9}-\frac{48\!\cdots\!19}{15\!\cdots\!44}a^{8}+\frac{14\!\cdots\!33}{15\!\cdots\!44}a^{7}+\frac{31\!\cdots\!97}{15\!\cdots\!44}a^{6}-\frac{47\!\cdots\!55}{15\!\cdots\!44}a^{5}+\frac{23\!\cdots\!39}{15\!\cdots\!44}a^{4}+\frac{17\!\cdots\!17}{15\!\cdots\!44}a^{3}-\frac{76\!\cdots\!91}{30\!\cdots\!88}a^{2}+\frac{42\!\cdots\!41}{43\!\cdots\!28}a-\frac{14\!\cdots\!03}{30\!\cdots\!88}$ Copy content Toggle raw display

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

not computed

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 $2$) Copy content Toggle raw display
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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 58*x^17 - 193*x^16 - 2188*x^15 - 465*x^14 - 7861*x^13 + 91673*x^12 - 34407*x^11 + 607603*x^10 + 3012761*x^9 + 12205932*x^8 - 10310764*x^7 + 30457244*x^6 - 323709400*x^5 + 3839375256*x^4 - 3757533269*x^3 + 15771143512*x^2 - 45579456448*x + 87786948061)
 
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 + 58*x^17 - 193*x^16 - 2188*x^15 - 465*x^14 - 7861*x^13 + 91673*x^12 - 34407*x^11 + 607603*x^10 + 3012761*x^9 + 12205932*x^8 - 10310764*x^7 + 30457244*x^6 - 323709400*x^5 + 3839375256*x^4 - 3757533269*x^3 + 15771143512*x^2 - 45579456448*x + 87786948061, 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 + 58*x^17 - 193*x^16 - 2188*x^15 - 465*x^14 - 7861*x^13 + 91673*x^12 - 34407*x^11 + 607603*x^10 + 3012761*x^9 + 12205932*x^8 - 10310764*x^7 + 30457244*x^6 - 323709400*x^5 + 3839375256*x^4 - 3757533269*x^3 + 15771143512*x^2 - 45579456448*x + 87786948061);
 
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 + 58*x^17 - 193*x^16 - 2188*x^15 - 465*x^14 - 7861*x^13 + 91673*x^12 - 34407*x^11 + 607603*x^10 + 3012761*x^9 + 12205932*x^8 - 10310764*x^7 + 30457244*x^6 - 323709400*x^5 + 3839375256*x^4 - 3757533269*x^3 + 15771143512*x^2 - 45579456448*x + 87786948061);
 
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

$C_{10}\wr C_2$ (as 20T53):

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 200
The 65 conjugacy class representatives for $C_{10}\wr C_2$
Character table for $C_{10}\wr C_2$

Intermediate fields

\(\Q(\sqrt{-7}) \), 4.0.848925.1, 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 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 ${\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.5.0.1}{5} }^{2}$ R R R R ${\href{/padicField/13.10.0.1}{10} }^{2}$ ${\href{/padicField/17.10.0.1}{10} }^{2}$ ${\href{/padicField/19.10.0.1}{10} }^{2}$ ${\href{/padicField/23.10.0.1}{10} }^{2}$ ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }^{2}$ $20$ ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{5}$ ${\href{/padicField/41.10.0.1}{10} }^{2}$ ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }^{2}$ $20$ ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{5}$ $20$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display 3.20.10.2$x^{20} + 162 x^{12} - 486 x^{10} + 1458 x^{8} - 19683 x^{2} + 118098$$2$$10$$10$20T1$[\ ]_{2}^{10}$
\(5\) Copy content Toggle raw display Deg $20$$2$$10$$10$
\(7\) Copy content Toggle raw display 7.20.15.1$x^{20} + 39 x^{16} + 16 x^{15} - 344 x^{12} - 7232 x^{11} + 96 x^{10} + 4344 x^{8} + 131648 x^{7} + 38272 x^{6} + 256 x^{5} + 59536 x^{4} - 212224 x^{3} + 84096 x^{2} - 8704 x + 9328$$4$$5$$15$20T12$[\ ]_{4}^{10}$
\(11\) Copy content Toggle raw display 11.10.9.5$x^{10} + 88$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.0.1$x^{10} + 7 x^{5} + 8 x^{4} + 10 x^{3} + 6 x^{2} + 6 x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$