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Properties

Label 10.10.531...000.1
Degree $10$
Signature $[10, 0]$
Discriminant $5.314\times 10^{20}$
Root discriminant \(118.18\)
Ramified primes $2,5,7,97$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $(C_2^4 : C_5):C_4$ (as 10T24)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 125*x^8 - 130*x^7 + 5570*x^6 + 9684*x^5 - 104680*x^4 - 225290*x^3 + 723545*x^2 + 1601880*x - 672615)
 
gp: K = bnfinit(y^10 - 125*y^8 - 130*y^7 + 5570*y^6 + 9684*y^5 - 104680*y^4 - 225290*y^3 + 723545*y^2 + 1601880*y - 672615, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 125*x^8 - 130*x^7 + 5570*x^6 + 9684*x^5 - 104680*x^4 - 225290*x^3 + 723545*x^2 + 1601880*x - 672615);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - 125*x^8 - 130*x^7 + 5570*x^6 + 9684*x^5 - 104680*x^4 - 225290*x^3 + 723545*x^2 + 1601880*x - 672615)
 

\( x^{10} - 125 x^{8} - 130 x^{7} + 5570 x^{6} + 9684 x^{5} - 104680 x^{4} - 225290 x^{3} + 723545 x^{2} + \cdots - 672615 \) 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:  $[10, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(531397009202500000000\) \(\medspace = 2^{8}\cdot 5^{10}\cdot 7^{4}\cdot 97^{4}\) 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:  \(118.18\)
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:  $2^{4/5}5^{23/20}7^{1/2}97^{1/2}\approx 288.7848385446545$
Ramified primes:   \(2\), \(5\), \(7\), \(97\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{42157043867976}a^{9}-\frac{1458421184159}{14052347955992}a^{8}-\frac{1091574167941}{21078521933988}a^{7}-\frac{1060748380343}{5269630483497}a^{6}+\frac{3056270641645}{21078521933988}a^{5}-\frac{358075871091}{7026173977996}a^{4}-\frac{9949759669709}{21078521933988}a^{3}+\frac{1381837382719}{10539260966994}a^{2}+\frac{6697539177881}{42157043867976}a+\frac{3915443424961}{14052347955992}$ 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

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:  $9$
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:   $\frac{70503050305}{3513086988998}a^{9}-\frac{608354577043}{7026173977996}a^{8}-\frac{15001898371717}{7026173977996}a^{7}+\frac{46386140391461}{7026173977996}a^{6}+\frac{585389565096769}{7026173977996}a^{5}-\frac{11\!\cdots\!15}{7026173977996}a^{4}-\frac{97\!\cdots\!15}{7026173977996}a^{3}+\frac{10\!\cdots\!81}{7026173977996}a^{2}+\frac{57\!\cdots\!89}{7026173977996}a-\frac{54\!\cdots\!76}{1756543494499}$, $\frac{199738489985}{14052347955992}a^{9}-\frac{860131125783}{14052347955992}a^{8}-\frac{2658848625225}{1756543494499}a^{7}+\frac{32801869628633}{7026173977996}a^{6}+\frac{103858930286744}{1756543494499}a^{5}-\frac{205032183415669}{1756543494499}a^{4}-\frac{34\!\cdots\!29}{3513086988998}a^{3}+\frac{73\!\cdots\!11}{7026173977996}a^{2}+\frac{81\!\cdots\!91}{14052347955992}a-\frac{31\!\cdots\!93}{14052347955992}$, $\frac{569962316827}{7026173977996}a^{9}-\frac{2458299641711}{7026173977996}a^{8}-\frac{15161331721652}{1756543494499}a^{7}+\frac{46868927851883}{1756543494499}a^{6}+\frac{591600004954398}{1756543494499}a^{5}-\frac{23\!\cdots\!41}{3513086988998}a^{4}-\frac{98\!\cdots\!66}{1756543494499}a^{3}+\frac{10\!\cdots\!03}{1756543494499}a^{2}+\frac{23\!\cdots\!97}{7026173977996}a-\frac{88\!\cdots\!99}{7026173977996}$, $\frac{68724451367}{7026173977996}a^{9}-\frac{147316748889}{3513086988998}a^{8}-\frac{3666349620247}{3513086988998}a^{7}+\frac{11246432025159}{3513086988998}a^{6}+\frac{71732260130320}{1756543494499}a^{5}-\frac{281521322498071}{3513086988998}a^{4}-\frac{11\!\cdots\!39}{1756543494499}a^{3}+\frac{12\!\cdots\!39}{1756543494499}a^{2}+\frac{28\!\cdots\!39}{7026173977996}a-\frac{54\!\cdots\!93}{3513086988998}$, $\frac{72353788937}{42157043867976}a^{9}-\frac{93219899841}{14052347955992}a^{8}-\frac{3873484437905}{21078521933988}a^{7}+\frac{2627321648537}{5269630483497}a^{6}+\frac{150141043896467}{21078521933988}a^{5}-\frac{87273432349617}{7026173977996}a^{4}-\frac{24\!\cdots\!47}{21078521933988}a^{3}+\frac{594292252266382}{5269630483497}a^{2}+\frac{28\!\cdots\!21}{42157043867976}a-\frac{36\!\cdots\!93}{14052347955992}$, $\frac{843743769}{1756543494499}a^{9}-\frac{4195049023}{1756543494499}a^{8}-\frac{324285943591}{7026173977996}a^{7}+\frac{1194573454383}{7026173977996}a^{6}+\frac{11729745812733}{7026173977996}a^{5}-\frac{27984751087639}{7026173977996}a^{4}-\frac{185130888661435}{7026173977996}a^{3}+\frac{234711688235565}{7026173977996}a^{2}+\frac{10\!\cdots\!65}{7026173977996}a-\frac{465315392196409}{7026173977996}$, $\frac{7438163209849}{21078521933988}a^{9}-\frac{2686510403363}{1756543494499}a^{8}-\frac{789984094489991}{21078521933988}a^{7}+\frac{24\!\cdots\!09}{21078521933988}a^{6}+\frac{30\!\cdots\!55}{21078521933988}a^{5}-\frac{20\!\cdots\!35}{7026173977996}a^{4}-\frac{51\!\cdots\!45}{21078521933988}a^{3}+\frac{54\!\cdots\!69}{21078521933988}a^{2}+\frac{75\!\cdots\!50}{5269630483497}a-\frac{38\!\cdots\!51}{7026173977996}$, $\frac{35867670097775}{21078521933988}a^{9}-\frac{51293997068497}{7026173977996}a^{8}-\frac{19\!\cdots\!99}{10539260966994}a^{7}+\frac{29\!\cdots\!35}{5269630483497}a^{6}+\frac{74\!\cdots\!65}{10539260966994}a^{5}-\frac{48\!\cdots\!15}{3513086988998}a^{4}-\frac{12\!\cdots\!67}{10539260966994}a^{3}+\frac{65\!\cdots\!11}{5269630483497}a^{2}+\frac{14\!\cdots\!23}{21078521933988}a-\frac{18\!\cdots\!29}{7026173977996}$, $\frac{224300894275}{14052347955992}a^{9}-\frac{616919644867}{14052347955992}a^{8}-\frac{3205379800104}{1756543494499}a^{7}+\frac{19461137259295}{7026173977996}a^{6}+\frac{265939806001397}{3513086988998}a^{5}-\frac{144015182638209}{3513086988998}a^{4}-\frac{23\!\cdots\!11}{1756543494499}a^{3}-\frac{15\!\cdots\!89}{7026173977996}a^{2}+\frac{11\!\cdots\!53}{14052347955992}a+\frac{69\!\cdots\!07}{14052347955992}$ 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:  \( 77523265.19793366 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{0}\cdot 77523265.19793366 \cdot 1}{2\cdot\sqrt{531397009202500000000}}\cr\approx \mathstrut & 1.72183869900256 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^10 - 125*x^8 - 130*x^7 + 5570*x^6 + 9684*x^5 - 104680*x^4 - 225290*x^3 + 723545*x^2 + 1601880*x - 672615)
 
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 - 125*x^8 - 130*x^7 + 5570*x^6 + 9684*x^5 - 104680*x^4 - 225290*x^3 + 723545*x^2 + 1601880*x - 672615, 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 - 125*x^8 - 130*x^7 + 5570*x^6 + 9684*x^5 - 104680*x^4 - 225290*x^3 + 723545*x^2 + 1601880*x - 672615);
 
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 - 125*x^8 - 130*x^7 + 5570*x^6 + 9684*x^5 - 104680*x^4 - 225290*x^3 + 723545*x^2 + 1601880*x - 672615);
 
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

$C_2^4:F_5$ (as 10T24):

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 320
The 11 conjugacy class representatives for $(C_2^4 : C_5):C_4$
Character table for $(C_2^4 : C_5):C_4$

Intermediate fields

5.5.2450000.1

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 10 sibling: 10.10.2656985046012500000000.1
Degree 16 sibling: deg 16
Degree 20 siblings: deg 20, deg 20, deg 20, deg 20, deg 20, deg 20
Degree 32 sibling: deg 32
Degree 40 siblings: deg 40, deg 40, deg 40, deg 40, deg 40, deg 40, deg 40, deg 40, deg 40, deg 40
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 ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ R R ${\href{/padicField/11.5.0.1}{5} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.5.0.1}{5} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.5.0.1}{5} }^{2}$

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
\(2\) Copy content Toggle raw display 2.5.4.1$x^{5} + 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} + 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
\(5\) Copy content Toggle raw display 5.5.5.1$x^{5} + 20 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
5.5.5.1$x^{5} + 20 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
\(7\) Copy content Toggle raw display $\Q_{7}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 4$$1$$1$$0$Trivial$[\ ]$
7.8.4.1$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(97\) Copy content Toggle raw display $\Q_{97}$$x + 92$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 92$$1$$1$$0$Trivial$[\ ]$
97.8.4.1$x^{8} + 20564 x^{7} + 158579686 x^{6} + 543510242244 x^{5} + 698570486224711 x^{4} + 55994198721100 x^{3} + 4236267101096262 x^{2} + 56043996888369552 x + 4548521252040853$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
1.35.4t1.a.a$1$ $ 5 \cdot 7 $ 4.4.6125.1 $C_4$ (as 4T1) $0$ $1$
1.35.4t1.a.b$1$ $ 5 \cdot 7 $ 4.4.6125.1 $C_4$ (as 4T1) $0$ $1$
* 4.2450000.5t3.c.a$4$ $ 2^{4} \cdot 5^{5} \cdot 7^{2}$ 5.5.2450000.1 $F_5$ (as 5T3) $1$ $4$
* 5.216...000.10t24.b.a$5$ $ 2^{4} \cdot 5^{5} \cdot 7^{2} \cdot 97^{4}$ 10.10.531397009202500000000.1 $(C_2^4 : C_5):C_4$ (as 10T24) $1$ $5$
5.108...000.10t25.b.a$5$ $ 2^{4} \cdot 5^{6} \cdot 7^{2} \cdot 97^{4}$ 10.10.531397009202500000000.1 $(C_2^4 : C_5):C_4$ (as 10T24) $1$ $5$
5.759...000.20t88.b.a$5$ $ 2^{4} \cdot 5^{6} \cdot 7^{3} \cdot 97^{4}$ 10.10.531397009202500000000.1 $(C_2^4 : C_5):C_4$ (as 10T24) $0$ $5$
5.759...000.20t88.b.b$5$ $ 2^{4} \cdot 5^{6} \cdot 7^{3} \cdot 97^{4}$ 10.10.531397009202500000000.1 $(C_2^4 : C_5):C_4$ (as 10T24) $0$ $5$
10.265...000.16t711.b.a$10$ $ 2^{8} \cdot 5^{11} \cdot 7^{4} \cdot 97^{4}$ 10.10.531397009202500000000.1 $(C_2^4 : C_5):C_4$ (as 10T24) $1$ $10$
10.650...000.20t77.b.a$10$ $ 2^{8} \cdot 5^{12} \cdot 7^{6} \cdot 97^{4}$ 10.10.531397009202500000000.1 $(C_2^4 : C_5):C_4$ (as 10T24) $1$ $10$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.