Properties

Label 10.10.233...000.1
Degree $10$
Signature $[10, 0]$
Discriminant $2.330\times 10^{25}$
Root discriminant \(344.14\)
Ramified primes see page
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $A_5 \wr C_2$ (as 10T40)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 2*x^9 - 345*x^8 + 720*x^7 + 33292*x^6 - 77348*x^5 - 795480*x^4 + 1563624*x^3 + 4991153*x^2 - 5404190*x - 10994849)
 
gp: K = bnfinit(x^10 - 2*x^9 - 345*x^8 + 720*x^7 + 33292*x^6 - 77348*x^5 - 795480*x^4 + 1563624*x^3 + 4991153*x^2 - 5404190*x - 10994849, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-10994849, -5404190, 4991153, 1563624, -795480, -77348, 33292, 720, -345, -2, 1]);
 

\( x^{10} - 2 x^{9} - 345 x^{8} + 720 x^{7} + 33292 x^{6} - 77348 x^{5} - 795480 x^{4} + 1563624 x^{3} + 4991153 x^{2} - 5404190 x - 10994849 \) Copy content Toggle raw display

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

Invariants

Degree:  $10$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[10, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:   \(23300317255158046392320000\) \(\medspace = 2^{21}\cdot 5^{4}\cdot 71^{4}\cdot 26449^{2}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  \(344.14\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:   \(2\), \(5\), \(71\), \(26449\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$\card{ \Aut(K/\Q) }$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{14}a^{8}+\frac{1}{7}a^{7}+\frac{1}{7}a^{6}+\frac{3}{7}a^{5}+\frac{1}{7}a^{4}+\frac{3}{7}a^{2}+\frac{1}{7}a-\frac{3}{14}$, $\frac{1}{24\!\cdots\!28}a^{9}+\frac{70\!\cdots\!33}{34\!\cdots\!04}a^{8}+\frac{20\!\cdots\!27}{12\!\cdots\!64}a^{7}-\frac{12\!\cdots\!65}{12\!\cdots\!64}a^{6}-\frac{11\!\cdots\!15}{12\!\cdots\!64}a^{5}-\frac{67\!\cdots\!41}{12\!\cdots\!64}a^{4}+\frac{12\!\cdots\!59}{12\!\cdots\!64}a^{3}-\frac{60\!\cdots\!21}{12\!\cdots\!64}a^{2}-\frac{67\!\cdots\!33}{49\!\cdots\!72}a+\frac{67\!\cdots\!29}{24\!\cdots\!28}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{3}$, which has order $3$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
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);
 
Fundamental units:   $\frac{529689496193}{43\!\cdots\!32}a^{9}-\frac{1119994967803}{61\!\cdots\!76}a^{8}-\frac{141387944528709}{21\!\cdots\!16}a^{7}+\frac{12\!\cdots\!55}{21\!\cdots\!16}a^{6}+\frac{25\!\cdots\!01}{21\!\cdots\!16}a^{5}-\frac{11\!\cdots\!41}{21\!\cdots\!16}a^{4}-\frac{15\!\cdots\!09}{21\!\cdots\!16}a^{3}+\frac{31\!\cdots\!07}{21\!\cdots\!16}a^{2}+\frac{57\!\cdots\!13}{61\!\cdots\!76}a+\frac{19\!\cdots\!85}{43\!\cdots\!32}$, $\frac{49\!\cdots\!79}{24\!\cdots\!28}a^{9}-\frac{26\!\cdots\!93}{34\!\cdots\!04}a^{8}-\frac{83\!\cdots\!83}{12\!\cdots\!64}a^{7}+\frac{31\!\cdots\!73}{12\!\cdots\!64}a^{6}+\frac{75\!\cdots\!47}{12\!\cdots\!64}a^{5}-\frac{30\!\cdots\!07}{12\!\cdots\!64}a^{4}-\frac{13\!\cdots\!19}{12\!\cdots\!64}a^{3}+\frac{58\!\cdots\!29}{12\!\cdots\!64}a^{2}+\frac{74\!\cdots\!01}{49\!\cdots\!72}a-\frac{29\!\cdots\!13}{24\!\cdots\!28}$, $\frac{37\!\cdots\!43}{30\!\cdots\!16}a^{9}+\frac{55\!\cdots\!47}{43\!\cdots\!88}a^{8}-\frac{64\!\cdots\!11}{15\!\cdots\!08}a^{7}-\frac{59\!\cdots\!31}{15\!\cdots\!08}a^{6}+\frac{61\!\cdots\!67}{15\!\cdots\!08}a^{5}+\frac{39\!\cdots\!25}{15\!\cdots\!08}a^{4}-\frac{13\!\cdots\!71}{15\!\cdots\!08}a^{3}-\frac{12\!\cdots\!55}{15\!\cdots\!08}a^{2}+\frac{15\!\cdots\!91}{43\!\cdots\!88}a+\frac{13\!\cdots\!67}{30\!\cdots\!16}$, $\frac{49\!\cdots\!09}{12\!\cdots\!64}a^{9}-\frac{35\!\cdots\!15}{17\!\cdots\!52}a^{8}-\frac{80\!\cdots\!53}{61\!\cdots\!32}a^{7}+\frac{41\!\cdots\!27}{61\!\cdots\!32}a^{6}+\frac{65\!\cdots\!49}{61\!\cdots\!32}a^{5}-\frac{36\!\cdots\!09}{61\!\cdots\!32}a^{4}-\frac{49\!\cdots\!89}{61\!\cdots\!32}a^{3}+\frac{38\!\cdots\!03}{61\!\cdots\!32}a^{2}+\frac{11\!\cdots\!25}{17\!\cdots\!52}a-\frac{19\!\cdots\!27}{12\!\cdots\!64}$, $\frac{16\!\cdots\!15}{30\!\cdots\!16}a^{9}-\frac{89\!\cdots\!85}{43\!\cdots\!88}a^{8}-\frac{28\!\cdots\!51}{15\!\cdots\!08}a^{7}+\frac{11\!\cdots\!41}{15\!\cdots\!08}a^{6}+\frac{26\!\cdots\!43}{15\!\cdots\!08}a^{5}-\frac{11\!\cdots\!19}{15\!\cdots\!08}a^{4}-\frac{48\!\cdots\!79}{15\!\cdots\!08}a^{3}+\frac{21\!\cdots\!77}{15\!\cdots\!08}a^{2}+\frac{11\!\cdots\!71}{43\!\cdots\!88}a-\frac{10\!\cdots\!17}{30\!\cdots\!16}$, $\frac{28\!\cdots\!05}{34\!\cdots\!04}a^{9}-\frac{11\!\cdots\!77}{34\!\cdots\!04}a^{8}-\frac{48\!\cdots\!49}{17\!\cdots\!52}a^{7}+\frac{28\!\cdots\!41}{24\!\cdots\!36}a^{6}+\frac{44\!\cdots\!77}{17\!\cdots\!52}a^{5}-\frac{19\!\cdots\!01}{17\!\cdots\!52}a^{4}-\frac{77\!\cdots\!45}{17\!\cdots\!52}a^{3}+\frac{37\!\cdots\!99}{17\!\cdots\!52}a^{2}+\frac{20\!\cdots\!15}{34\!\cdots\!04}a-\frac{16\!\cdots\!07}{34\!\cdots\!04}$, $\frac{22\!\cdots\!13}{12\!\cdots\!64}a^{9}+\frac{30\!\cdots\!65}{17\!\cdots\!52}a^{8}-\frac{28\!\cdots\!93}{61\!\cdots\!32}a^{7}-\frac{27\!\cdots\!69}{61\!\cdots\!32}a^{6}+\frac{12\!\cdots\!05}{61\!\cdots\!32}a^{5}+\frac{11\!\cdots\!83}{61\!\cdots\!32}a^{4}+\frac{26\!\cdots\!31}{61\!\cdots\!32}a^{3}-\frac{79\!\cdots\!69}{61\!\cdots\!32}a^{2}-\frac{33\!\cdots\!95}{17\!\cdots\!52}a-\frac{70\!\cdots\!91}{12\!\cdots\!64}$, $\frac{26\!\cdots\!77}{61\!\cdots\!32}a^{9}-\frac{51\!\cdots\!91}{87\!\cdots\!76}a^{8}-\frac{25\!\cdots\!53}{30\!\cdots\!16}a^{7}+\frac{38\!\cdots\!03}{30\!\cdots\!16}a^{6}+\frac{20\!\cdots\!33}{30\!\cdots\!16}a^{5}-\frac{10\!\cdots\!37}{30\!\cdots\!16}a^{4}+\frac{12\!\cdots\!43}{30\!\cdots\!16}a^{3}+\frac{63\!\cdots\!27}{30\!\cdots\!16}a^{2}-\frac{14\!\cdots\!63}{87\!\cdots\!76}a-\frac{26\!\cdots\!47}{61\!\cdots\!32}$, $\frac{19\!\cdots\!23}{61\!\cdots\!32}a^{9}+\frac{43\!\cdots\!07}{87\!\cdots\!76}a^{8}-\frac{33\!\cdots\!31}{30\!\cdots\!16}a^{7}-\frac{52\!\cdots\!31}{30\!\cdots\!16}a^{6}+\frac{31\!\cdots\!55}{30\!\cdots\!16}a^{5}+\frac{42\!\cdots\!37}{30\!\cdots\!16}a^{4}-\frac{69\!\cdots\!71}{30\!\cdots\!16}a^{3}-\frac{10\!\cdots\!75}{30\!\cdots\!16}a^{2}+\frac{64\!\cdots\!95}{87\!\cdots\!76}a+\frac{70\!\cdots\!59}{61\!\cdots\!32}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 4673862280.11 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{10}\cdot(2\pi)^{0}\cdot 4673862280.11 \cdot 3}{2\sqrt{23300317255158046392320000}}\approx 1.48725764438$ (assuming GRH)

Galois group

$A_5 \wr C_2$ (as 10T40):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 7200
The 20 conjugacy class representatives for $A_5 \wr C_2$
Character table for $A_5 \wr C_2$

Intermediate fields

\(\Q(\sqrt{2}) \)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 sibling: data not computed
Degree 20 sibling: data not computed
Degree 24 sibling: data not computed
Degree 25 sibling: data not computed
Degree 30 sibling: data not computed
Degree 36 sibling: data not computed
Degree 40 sibling: data not computed

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ R ${\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{3}$ ${\href{/padicField/11.10.0.1}{10} }$ ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.10.0.1}{10} }$ ${\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.5.0.1}{5} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.5.0.1}{5} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.5.0.1}{5} }^{2}$ ${\href{/padicField/53.10.0.1}{10} }$ ${\href{/padicField/59.10.0.1}{10} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.8.18.60$x^{8} + 4 x^{6} + 36$$8$$1$$18$$A_4\times C_2$$[2, 2, 3]^{3}$
\(5\) Copy content Toggle raw display 5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(71\) Copy content Toggle raw display $\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
71.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
71.5.4.4$x^{5} + 568$$5$$1$$4$$C_5$$[\ ]_{5}$
\(26449\) Copy content Toggle raw display $\Q_{26449}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{26449}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{26449}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.8.2t1.a.a$1$ $ 2^{3}$ \(\Q(\sqrt{2}) \) $C_2$ (as 2T1) $1$ $1$
6.722...000.24t9631.a.a$6$ $ 2^{15} \cdot 5^{4} \cdot 71^{2} \cdot 26449^{2}$ 10.10.23300317255158046392320000.1 $A_5 \wr C_2$ (as 10T40) $1$ $6$
6.722...000.24t9631.a.b$6$ $ 2^{15} \cdot 5^{4} \cdot 71^{2} \cdot 26449^{2}$ 10.10.23300317255158046392320000.1 $A_5 \wr C_2$ (as 10T40) $1$ $6$
* 8.291...000.10t40.a.a$8$ $ 2^{18} \cdot 5^{4} \cdot 71^{4} \cdot 26449^{2}$ 10.10.23300317255158046392320000.1 $A_5 \wr C_2$ (as 10T40) $1$ $8$
9.459...000.72.a.a$9$ $ 2^{24} \cdot 5^{4} \cdot 71^{6} \cdot 26449^{6}$ 10.10.23300317255158046392320000.1 $A_5 \wr C_2$ (as 10T40) $1$ $9$
9.459...000.72.a.b$9$ $ 2^{24} \cdot 5^{4} \cdot 71^{6} \cdot 26449^{6}$ 10.10.23300317255158046392320000.1 $A_5 \wr C_2$ (as 10T40) $1$ $9$
9.898...000.60.a.a$9$ $ 2^{15} \cdot 5^{4} \cdot 71^{6} \cdot 26449^{6}$ 10.10.23300317255158046392320000.1 $A_5 \wr C_2$ (as 10T40) $1$ $9$
9.898...000.60.a.b$9$ $ 2^{15} \cdot 5^{4} \cdot 71^{6} \cdot 26449^{6}$ 10.10.23300317255158046392320000.1 $A_5 \wr C_2$ (as 10T40) $1$ $9$
10.233...000.12t269.a.a$10$ $ 2^{21} \cdot 5^{4} \cdot 71^{4} \cdot 26449^{2}$ 10.10.23300317255158046392320000.1 $A_5 \wr C_2$ (as 10T40) $1$ $10$
16.418...000.25t88.a.a$16$ $ 2^{30} \cdot 5^{8} \cdot 71^{16} \cdot 26449^{8}$ 10.10.23300317255158046392320000.1 $A_5 \wr C_2$ (as 10T40) $1$ $16$
16.171...000.50.a.a$16$ $ 2^{42} \cdot 5^{8} \cdot 71^{16} \cdot 26449^{8}$ 10.10.23300317255158046392320000.1 $A_5 \wr C_2$ (as 10T40) $1$ $16$
18.412...000.60.a.a$18$ $ 2^{39} \cdot 5^{8} \cdot 71^{12} \cdot 26449^{12}$ 10.10.23300317255158046392320000.1 $A_5 \wr C_2$ (as 10T40) $1$ $18$
24.382...000.120.a.a$24$ $ 2^{54} \cdot 5^{12} \cdot 71^{20} \cdot 26449^{14}$ 10.10.23300317255158046392320000.1 $A_5 \wr C_2$ (as 10T40) $1$ $24$
24.382...000.120.a.b$24$ $ 2^{54} \cdot 5^{12} \cdot 71^{20} \cdot 26449^{14}$ 10.10.23300317255158046392320000.1 $A_5 \wr C_2$ (as 10T40) $1$ $24$
25.121...000.36t7075.a.a$25$ $ 2^{48} \cdot 5^{12} \cdot 71^{20} \cdot 26449^{10}$ 10.10.23300317255158046392320000.1 $A_5 \wr C_2$ (as 10T40) $1$ $25$
25.399...000.72.a.a$25$ $ 2^{63} \cdot 5^{12} \cdot 71^{20} \cdot 26449^{10}$ 10.10.23300317255158046392320000.1 $A_5 \wr C_2$ (as 10T40) $1$ $25$
30.275...000.144.a.a$30$ $ 2^{69} \cdot 5^{16} \cdot 71^{22} \cdot 26449^{16}$ 10.10.23300317255158046392320000.1 $A_5 \wr C_2$ (as 10T40) $1$ $30$
30.275...000.144.a.b$30$ $ 2^{69} \cdot 5^{16} \cdot 71^{22} \cdot 26449^{16}$ 10.10.23300317255158046392320000.1 $A_5 \wr C_2$ (as 10T40) $1$ $30$
40.209...000.60.a.a$40$ $ 2^{90} \cdot 5^{20} \cdot 71^{36} \cdot 26449^{18}$ 10.10.23300317255158046392320000.1 $A_5 \wr C_2$ (as 10T40) $1$ $40$

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 *.