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Properties

Label 10.2.634...000.1
Degree $10$
Signature $[2, 4]$
Discriminant $6.346\times 10^{33}$
Root discriminant \(2400.22\)
Ramified primes $2,3,5,19$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $S_{10}$ (as 10T45)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 5700*x^7 + 10830000*x^4 - 6859000000*x - 47502004500)
 
gp: K = bnfinit(y^10 - 5700*y^7 + 10830000*y^4 - 6859000000*y - 47502004500, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 5700*x^7 + 10830000*x^4 - 6859000000*x - 47502004500);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^10 - 5700*x^7 + 10830000*x^4 - 6859000000*x - 47502004500)
 

\( x^{10} - 5700x^{7} + 10830000x^{4} - 6859000000x - 47502004500 \) 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:  $[2, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(6346238713644431953125000000000000\) \(\medspace = 2^{12}\cdot 3^{14}\cdot 5^{19}\cdot 19^{8}\) 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:  \(2400.22\)
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:  $2^{31/20}3^{235/108}5^{211/100}19^{8/9}\approx 13069.939519129885$
Ramified primes:   \(2\), \(3\), \(5\), \(19\) 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:  \(\Q(\sqrt{5}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{57}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{855}a^{4}-\frac{2}{9}a$, $\frac{1}{1710}a^{5}-\frac{1}{9}a^{2}$, $\frac{1}{97470}a^{6}+\frac{1}{5130}a^{5}-\frac{1}{2565}a^{4}-\frac{1}{513}a^{3}-\frac{1}{27}a^{2}+\frac{2}{27}a$, $\frac{1}{2924100}a^{7}-\frac{1}{7695}a^{4}-\frac{1}{2}a^{2}+\frac{1}{81}a$, $\frac{1}{55557900}a^{8}-\frac{2}{29241}a^{5}-\frac{1}{114}a^{3}+\frac{613}{1539}a^{2}+\frac{1}{3}a$, $\frac{1}{1166715900}a^{9}+\frac{1}{1166715900}a^{8}+\frac{1}{6140610}a^{7}-\frac{29}{6140610}a^{6}-\frac{173}{614061}a^{5}-\frac{101}{323190}a^{4}-\frac{25}{64638}a^{3}+\frac{11044}{32319}a^{2}-\frac{314}{1701}a-\frac{2}{7}$ 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

$C_{2}$, which has order $2$ (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:  $5$
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:   $\frac{70\!\cdots\!57}{194452650}a^{9}+\frac{21\!\cdots\!59}{388905300}a^{8}+\frac{16\!\cdots\!23}{20468700}a^{7}-\frac{17\!\cdots\!20}{204687}a^{6}-\frac{26\!\cdots\!33}{2046870}a^{5}-\frac{10\!\cdots\!91}{53865}a^{4}+\frac{23\!\cdots\!75}{21546}a^{3}+\frac{35\!\cdots\!23}{21546}a^{2}+\frac{13\!\cdots\!62}{567}a+\frac{80\!\cdots\!42}{7}$, $\frac{11\!\cdots\!97}{583357950}a^{9}-\frac{12\!\cdots\!53}{1166715900}a^{8}+\frac{36\!\cdots\!41}{61406100}a^{7}-\frac{36\!\cdots\!61}{3070305}a^{6}+\frac{19\!\cdots\!42}{3070305}a^{5}-\frac{55\!\cdots\!78}{161595}a^{4}+\frac{15\!\cdots\!27}{64638}a^{3}-\frac{82\!\cdots\!01}{64638}a^{2}+\frac{11\!\cdots\!00}{1701}a-\frac{12\!\cdots\!43}{7}$, $\frac{99\!\cdots\!17}{55557900}a^{9}-\frac{91\!\cdots\!23}{6173100}a^{8}-\frac{21\!\cdots\!57}{292410}a^{7}-\frac{17\!\cdots\!07}{29241}a^{6}+\frac{35\!\cdots\!73}{16245}a^{5}+\frac{23\!\cdots\!41}{3078}a^{4}+\frac{43\!\cdots\!21}{3078}a^{3}+\frac{35\!\cdots\!35}{171}a^{2}-\frac{53\!\cdots\!12}{81}a-43\!\cdots\!09$, $\frac{16\!\cdots\!17}{64817550}a^{9}+\frac{47\!\cdots\!97}{194452650}a^{8}+\frac{18\!\cdots\!67}{10234350}a^{7}-\frac{64\!\cdots\!95}{45486}a^{6}-\frac{31\!\cdots\!97}{2046870}a^{5}-\frac{78\!\cdots\!93}{53865}a^{4}+\frac{60\!\cdots\!19}{3591}a^{3}+\frac{23\!\cdots\!91}{10773}a^{2}+\frac{14\!\cdots\!76}{567}a+\frac{80\!\cdots\!16}{7}$, $\frac{41\!\cdots\!91}{194452650}a^{9}-\frac{22\!\cdots\!63}{388905300}a^{8}+\frac{10\!\cdots\!91}{1705725}a^{7}-\frac{12\!\cdots\!86}{1023435}a^{6}+\frac{43\!\cdots\!43}{2046870}a^{5}-\frac{86\!\cdots\!56}{5985}a^{4}+\frac{17\!\cdots\!91}{21546}a^{3}-\frac{76\!\cdots\!96}{10773}a^{2}-\frac{54\!\cdots\!93}{189}a+\frac{31\!\cdots\!88}{7}$ Copy content Toggle raw display (assuming GRH)
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:  \( 35499371082600 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 35499371082600 \cdot 2}{2\cdot\sqrt{6346238713644431953125000000000000}}\cr\approx \mathstrut & 2.77806192194579 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^10 - 5700*x^7 + 10830000*x^4 - 6859000000*x - 47502004500)
 
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 - 5700*x^7 + 10830000*x^4 - 6859000000*x - 47502004500, 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 - 5700*x^7 + 10830000*x^4 - 6859000000*x - 47502004500);
 
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 - 5700*x^7 + 10830000*x^4 - 6859000000*x - 47502004500);
 
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

$S_{10}$ (as 10T45):

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 non-solvable group of order 3628800
The 42 conjugacy class representatives for $S_{10}$
Character table for $S_{10}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
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
Degree 45 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 R R R ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ ${\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.10.0.1}{10} }$ R ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.4.0.1}{4} }$ ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.10.0.1}{10} }$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.10.12.3$x^{10} + 2 x^{3} + 2$$10$$1$$12$$(C_2^4 : C_5):C_4$$[8/5, 8/5, 8/5, 8/5]_{5}^{4}$
\(3\) Copy content Toggle raw display $\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.3.4.1$x^{3} + 6 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
3.6.10.9$x^{6} + 3 x^{5} + 9 x^{2} + 9 x + 12$$6$$1$$10$$C_3^2:D_4$$[9/4, 9/4]_{4}^{2}$
\(5\) Copy content Toggle raw display 5.10.19.8$x^{10} + 15 x^{5} + 25 x^{3} + 25 x^{2} + 25 x + 105$$10$$1$$19$$C_5^2 : C_4$$[7/4, 9/4]_{4}$
\(19\) Copy content Toggle raw display $\Q_{19}$$x + 17$$1$$1$$0$Trivial$[\ ]$
19.9.8.2$x^{9} + 57$$9$$1$$8$$C_9$$[\ ]_{9}$