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Properties

Label 10.4.272...000.48
Degree $10$
Signature $[4, 3]$
Discriminant $-2.721\times 10^{26}$
Root discriminant \(440.02\)
Ramified primes see page
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2 \wr S_5$ (as 10T39)

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Show commands: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 + 140*x^8 + 4540*x^6 - 24480*x^4 - 232920*x^2 + 56448)
 
gp: K = bnfinit(x^10 + 140*x^8 + 4540*x^6 - 24480*x^4 - 232920*x^2 + 56448, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![56448, 0, -232920, 0, -24480, 0, 4540, 0, 140, 0, 1]);
 

\( x^{10} + 140x^{8} + 4540x^{6} - 24480x^{4} - 232920x^{2} + 56448 \) 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:  $[4, 3]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:   \(-272097792000000000000000000\) \(\medspace = -\,2^{27}\cdot 3^{12}\cdot 5^{18}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  \(440.02\)
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:   \(2\), \(3\), \(5\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$\card{ \Aut(K/\Q) }$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{6}a^{5}+\frac{1}{6}a^{3}$, $\frac{1}{132}a^{6}-\frac{5}{33}a^{4}-\frac{3}{11}a^{2}+\frac{4}{11}$, $\frac{1}{132}a^{7}+\frac{1}{66}a^{5}-\frac{7}{66}a^{3}+\frac{4}{11}a$, $\frac{1}{9193272}a^{8}-\frac{2221}{2298318}a^{6}+\frac{194329}{2298318}a^{4}-\frac{106866}{383053}a^{2}+\frac{169855}{383053}$, $\frac{1}{257411616}a^{9}+\frac{29531}{9193272}a^{7}-\frac{13991}{5850264}a^{5}-\frac{828844}{8044113}a^{3}-\frac{4287489}{10725484}a$ 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

Trivial group, which has order $1$ (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:  $6$
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:   $\frac{397}{766106}a^{8}+\frac{59821}{766106}a^{6}+\frac{1215318}{383053}a^{4}+\frac{7419935}{383053}a^{2}+\frac{8503981}{383053}$, $\frac{172042931}{4596636}a^{8}+\frac{23156899391}{4596636}a^{6}+\frac{54667950531}{383053}a^{4}-\frac{646218495661}{383053}a^{2}+\frac{13624168213}{34823}$, $\frac{7416395}{4596636}a^{8}+\frac{229982015}{2298318}a^{6}-\frac{687841481}{2298318}a^{4}-\frac{2120356746}{383053}a^{2}-\frac{1506297491}{383053}$, $\frac{2346988117}{2298318}a^{8}+\frac{658267833785}{4596636}a^{6}+\frac{3577746260405}{766106}a^{4}-\frac{9152248961784}{383053}a^{2}-\frac{93276602013937}{383053}$, $\frac{16\!\cdots\!49}{257411616}a^{9}-\frac{29\!\cdots\!15}{9193272}a^{8}+\frac{84\!\cdots\!19}{9193272}a^{7}-\frac{68\!\cdots\!53}{1532212}a^{6}+\frac{64\!\cdots\!13}{21450968}a^{5}-\frac{33\!\cdots\!79}{2298318}a^{4}-\frac{82\!\cdots\!71}{5362742}a^{3}+\frac{28\!\cdots\!50}{383053}a^{2}-\frac{16\!\cdots\!45}{10725484}a+\frac{26\!\cdots\!76}{34823}$, $\frac{34\!\cdots\!15}{128705808}a^{9}-\frac{35\!\cdots\!59}{4596636}a^{8}+\frac{30\!\cdots\!71}{766106}a^{7}-\frac{17\!\cdots\!35}{1532212}a^{6}+\frac{49\!\cdots\!01}{32176452}a^{5}-\frac{52\!\cdots\!59}{1149159}a^{4}+\frac{37\!\cdots\!31}{5362742}a^{3}-\frac{77\!\cdots\!14}{383053}a^{2}-\frac{92\!\cdots\!23}{5362742}a+\frac{19\!\cdots\!83}{383053}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 51233907436.9 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{4}\cdot(2\pi)^{3}\cdot 51233907436.9 \cdot 1}{2\sqrt{272097792000000000000000000}}\approx 6.16345972689$ (assuming GRH)

Galois group

$C_2 \wr S_5$ (as 10T39):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 3840
The 36 conjugacy class representatives for $C_2 \wr S_5$
Character table for $C_2 \wr S_5$ is not computed

Intermediate fields

5.5.364500000000.2

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

Sibling fields

Degree 10 sibling: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 32 siblings: data not computed
Degree 40 siblings: 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 R R ${\href{/padicField/7.2.0.1}{2} }^{3}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ ${\href{/padicField/29.10.0.1}{10} }$ ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }$ ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.5.0.1}{5} }^{2}$

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

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 2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.8.25.47$x^{8} + 6 x^{4} + 12 x^{2} + 10$$8$$1$$25$$C_2 \wr C_2\wr C_2$$[2, 2, 3, 7/2, 4, 17/4]^{2}$
\(3\) Copy content Toggle raw display 3.3.5.2$x^{3} + 21$$3$$1$$5$$S_3$$[5/2]_{2}$
3.3.5.2$x^{3} + 21$$3$$1$$5$$S_3$$[5/2]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(5\) Copy content Toggle raw display 5.10.18.5$x^{10} + 10 x^{8} + 40 x^{6} + 85 x^{5} + 80 x^{4} - 75 x^{3} + 80 x^{2} + 75 x + 57$$5$$2$$18$$F_{5}\times C_2$$[9/4]_{4}^{2}$