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Properties

Label 10.0.31512565339032283.3
Degree $10$
Signature $[0, 5]$
Discriminant $-3.151\times 10^{16}$
Root discriminant \(44.65\)
Ramified primes $11,43$
Class number $400$
Class group [2, 2, 10, 10]
Galois group $C_{10}$ (as 10T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 3*x^9 + 49*x^8 - 112*x^7 + 1110*x^6 - 1844*x^5 + 13964*x^4 - 15153*x^3 + 97561*x^2 - 51749*x + 305293)
 
gp: K = bnfinit(y^10 - 3*y^9 + 49*y^8 - 112*y^7 + 1110*y^6 - 1844*y^5 + 13964*y^4 - 15153*y^3 + 97561*y^2 - 51749*y + 305293, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 3*x^9 + 49*x^8 - 112*x^7 + 1110*x^6 - 1844*x^5 + 13964*x^4 - 15153*x^3 + 97561*x^2 - 51749*x + 305293);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - 3*x^9 + 49*x^8 - 112*x^7 + 1110*x^6 - 1844*x^5 + 13964*x^4 - 15153*x^3 + 97561*x^2 - 51749*x + 305293)
 

\( x^{10} - 3 x^{9} + 49 x^{8} - 112 x^{7} + 1110 x^{6} - 1844 x^{5} + 13964 x^{4} - 15153 x^{3} + \cdots + 305293 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $10$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 5]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-31512565339032283\) \(\medspace = -\,11^{8}\cdot 43^{5}\) 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:  \(44.65\)
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:  $11^{4/5}43^{1/2}\approx 44.65276699099921$
Ramified primes:   \(11\), \(43\) 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:  \(\Q(\sqrt{-43}) \)
$\card{ \Gal(K/\Q) }$:  $10$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(473=11\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{473}(1,·)$, $\chi_{473}(130,·)$, $\chi_{473}(388,·)$, $\chi_{473}(257,·)$, $\chi_{473}(42,·)$, $\chi_{473}(300,·)$, $\chi_{473}(386,·)$, $\chi_{473}(302,·)$, $\chi_{473}(214,·)$, $\chi_{473}(345,·)$$\rbrace$
This is a CM field.
Reflex fields:  \(\Q(\sqrt{-43}) \), 10.0.31512565339032283.3$^{15}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{55\!\cdots\!27}a^{9}+\frac{14\!\cdots\!47}{55\!\cdots\!27}a^{8}+\frac{74\!\cdots\!79}{55\!\cdots\!27}a^{7}+\frac{25\!\cdots\!77}{55\!\cdots\!27}a^{6}-\frac{24\!\cdots\!04}{55\!\cdots\!27}a^{5}+\frac{20\!\cdots\!88}{55\!\cdots\!27}a^{4}+\frac{23\!\cdots\!84}{55\!\cdots\!27}a^{3}-\frac{10\!\cdots\!48}{55\!\cdots\!27}a^{2}-\frac{17\!\cdots\!24}{55\!\cdots\!27}a-\frac{20\!\cdots\!39}{55\!\cdots\!27}$ 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

$C_{2}\times C_{2}\times C_{10}\times C_{10}$, which has order $400$

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:  $4$
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:   $\frac{788662427068}{55\!\cdots\!27}a^{9}+\frac{22204163532429}{55\!\cdots\!27}a^{8}-\frac{23549978748598}{55\!\cdots\!27}a^{7}+\frac{771655181250898}{55\!\cdots\!27}a^{6}-\frac{856954183872504}{55\!\cdots\!27}a^{5}+\frac{13\!\cdots\!39}{55\!\cdots\!27}a^{4}-\frac{10\!\cdots\!12}{55\!\cdots\!27}a^{3}+\frac{11\!\cdots\!19}{55\!\cdots\!27}a^{2}-\frac{43\!\cdots\!91}{55\!\cdots\!27}a+\frac{40\!\cdots\!79}{55\!\cdots\!27}$, $\frac{375111016344}{55\!\cdots\!27}a^{9}+\frac{1647344743920}{55\!\cdots\!27}a^{8}+\frac{12436716161422}{55\!\cdots\!27}a^{7}+\frac{109061475618234}{55\!\cdots\!27}a^{6}+\frac{186928585781118}{55\!\cdots\!27}a^{5}+\frac{35\!\cdots\!58}{55\!\cdots\!27}a^{4}+\frac{13\!\cdots\!74}{55\!\cdots\!27}a^{3}+\frac{46\!\cdots\!81}{55\!\cdots\!27}a^{2}+\frac{40\!\cdots\!25}{55\!\cdots\!27}a+\frac{26\!\cdots\!92}{55\!\cdots\!27}$, $\frac{650996223662}{55\!\cdots\!27}a^{9}-\frac{1765433162814}{55\!\cdots\!27}a^{8}+\frac{32722487331398}{55\!\cdots\!27}a^{7}-\frac{66693218969433}{55\!\cdots\!27}a^{6}+\frac{777136546073937}{55\!\cdots\!27}a^{5}-\frac{11\!\cdots\!69}{55\!\cdots\!27}a^{4}+\frac{10\!\cdots\!47}{55\!\cdots\!27}a^{3}-\frac{91\!\cdots\!99}{55\!\cdots\!27}a^{2}+\frac{58\!\cdots\!09}{55\!\cdots\!27}a-\frac{59\!\cdots\!89}{55\!\cdots\!27}$, $\frac{650996223662}{55\!\cdots\!27}a^{9}-\frac{1765433162814}{55\!\cdots\!27}a^{8}+\frac{32722487331398}{55\!\cdots\!27}a^{7}-\frac{66693218969433}{55\!\cdots\!27}a^{6}+\frac{777136546073937}{55\!\cdots\!27}a^{5}-\frac{11\!\cdots\!69}{55\!\cdots\!27}a^{4}+\frac{10\!\cdots\!47}{55\!\cdots\!27}a^{3}-\frac{91\!\cdots\!99}{55\!\cdots\!27}a^{2}+\frac{58\!\cdots\!09}{55\!\cdots\!27}a-\frac{39\!\cdots\!62}{55\!\cdots\!27}$ Copy content Toggle raw display
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:  \( 26.1711060094 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 26.1711060094 \cdot 400}{2\cdot\sqrt{31512565339032283}}\cr\approx \mathstrut & 0.288741716879 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^10 - 3*x^9 + 49*x^8 - 112*x^7 + 1110*x^6 - 1844*x^5 + 13964*x^4 - 15153*x^3 + 97561*x^2 - 51749*x + 305293)
 
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^10 - 3*x^9 + 49*x^8 - 112*x^7 + 1110*x^6 - 1844*x^5 + 13964*x^4 - 15153*x^3 + 97561*x^2 - 51749*x + 305293, 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^10 - 3*x^9 + 49*x^8 - 112*x^7 + 1110*x^6 - 1844*x^5 + 13964*x^4 - 15153*x^3 + 97561*x^2 - 51749*x + 305293);
 
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^10 - 3*x^9 + 49*x^8 - 112*x^7 + 1110*x^6 - 1844*x^5 + 13964*x^4 - 15153*x^3 + 97561*x^2 - 51749*x + 305293);
 
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

$C_{10}$ (as 10T1):

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 10
The 10 conjugacy class representatives for $C_{10}$
Character table for $C_{10}$

Intermediate fields

\(\Q(\sqrt{-43}) \), \(\Q(\zeta_{11})^+\)

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 ${\href{/padicField/2.10.0.1}{10} }$ ${\href{/padicField/3.10.0.1}{10} }$ ${\href{/padicField/5.10.0.1}{10} }$ ${\href{/padicField/7.10.0.1}{10} }$ R ${\href{/padicField/13.5.0.1}{5} }^{2}$ ${\href{/padicField/17.5.0.1}{5} }^{2}$ ${\href{/padicField/19.10.0.1}{10} }$ ${\href{/padicField/23.1.0.1}{1} }^{10}$ ${\href{/padicField/29.10.0.1}{10} }$ ${\href{/padicField/31.5.0.1}{5} }^{2}$ ${\href{/padicField/37.10.0.1}{10} }$ ${\href{/padicField/41.5.0.1}{5} }^{2}$ R ${\href{/padicField/47.5.0.1}{5} }^{2}$ ${\href{/padicField/53.5.0.1}{5} }^{2}$ ${\href{/padicField/59.5.0.1}{5} }^{2}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(11\) Copy content Toggle raw display 11.5.4.4$x^{5} + 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} + 11$$5$$1$$4$$C_5$$[\ ]_{5}$
\(43\) Copy content Toggle raw display 43.2.1.2$x^{2} + 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 43$$2$$1$$1$$C_2$$[\ ]_{2}$