# 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)

# Related objects

Show commands: SageMath / Pari/GP / Magma

## Normalizeddefining 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]);

$$x^{10} + 140x^{8} + 4540x^{6} - 24480x^{4} - 232920x^{2} + 56448$$

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: $$-272097792000000000000000000$$ -272097792000000000000000000 $$\medspace = -\,2^{27}\cdot 3^{12}\cdot 5^{18}$$ 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: $$2$$, $$3$$, $$5$$ 2, 3, 5 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 $$a$$)

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

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: $$-1$$ -1  (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $\frac{397}{766106}a^{8}+\frac{59821}{766106}a^{6}+\frac{1215318}{383053}a^{4}+\frac{7419935}{383053}a^{2}+\frac{8503981}{383053}$, $\frac{172042931}{4596636}a^{8}+\frac{23156899391}{4596636}a^{6}+\frac{54667950531}{383053}a^{4}-\frac{646218495661}{383053}a^{2}+\frac{13624168213}{34823}$, $\frac{7416395}{4596636}a^{8}+\frac{229982015}{2298318}a^{6}-\frac{687841481}{2298318}a^{4}-\frac{2120356746}{383053}a^{2}-\frac{1506297491}{383053}$, $\frac{2346988117}{2298318}a^{8}+\frac{658267833785}{4596636}a^{6}+\frac{3577746260405}{766106}a^{4}-\frac{9152248961784}{383053}a^{2}-\frac{93276602013937}{383053}$, $\frac{16\!\cdots\!49}{257411616}a^{9}-\frac{29\!\cdots\!15}{9193272}a^{8}+\frac{84\!\cdots\!19}{9193272}a^{7}-\frac{68\!\cdots\!53}{1532212}a^{6}+\frac{64\!\cdots\!13}{21450968}a^{5}-\frac{33\!\cdots\!79}{2298318}a^{4}-\frac{82\!\cdots\!71}{5362742}a^{3}+\frac{28\!\cdots\!50}{383053}a^{2}-\frac{16\!\cdots\!45}{10725484}a+\frac{26\!\cdots\!76}{34823}$, $\frac{34\!\cdots\!15}{128705808}a^{9}-\frac{35\!\cdots\!59}{4596636}a^{8}+\frac{30\!\cdots\!71}{766106}a^{7}-\frac{17\!\cdots\!35}{1532212}a^{6}+\frac{49\!\cdots\!01}{32176452}a^{5}-\frac{52\!\cdots\!59}{1149159}a^{4}+\frac{37\!\cdots\!31}{5362742}a^{3}-\frac{77\!\cdots\!14}{383053}a^{2}-\frac{92\!\cdots\!23}{5362742}a+\frac{19\!\cdots\!83}{383053}$ 397/766106*a^8 + 59821/766106*a^6 + 1215318/383053*a^4 + 7419935/383053*a^2 + 8503981/383053, 172042931/4596636*a^8 + 23156899391/4596636*a^6 + 54667950531/383053*a^4 - 646218495661/383053*a^2 + 13624168213/34823, 7416395/4596636*a^8 + 229982015/2298318*a^6 - 687841481/2298318*a^4 - 2120356746/383053*a^2 - 1506297491/383053, 2346988117/2298318*a^8 + 658267833785/4596636*a^6 + 3577746260405/766106*a^4 - 9152248961784/383053*a^2 - 93276602013937/383053, 1682917501708252779649/257411616*a^9 - 29243041302263956715/9193272*a^8 + 8428815391360238688919/9193272*a^7 - 683491367255219485053/1532212*a^6 + 641359421933758967654213/21450968*a^5 - 33433546289964220701979/2298318*a^4 - 820332228794052334624571/5362742*a^3 + 28508835666605237713950/383053*a^2 - 16721092920162163805734745/10725484*a + 26413848698174259868376/34823, 3404399783795406333204749468463495375227062322105316267217868203440986614260480076796989693264855379544053739514545794202552254160406044048215/128705808*a^9 - 358123636610636842996131767727138509810934959318950623329597509008609065681033493343066142632988488225415364405942952960101901676051054364559/4596636*a^8 + 3012805214947488536459600229544009904038129423011608724751177717569662760401777702578769191731528916094403421924012005765136862407204232405671/766106*a^7 - 17748085535160626831059892103738234626384674462926149385515140421350222850422086511743927757086827166168870946385216616050591872671119707977835/1532212*a^6 + 4961788811289907106160765622203040495322352613436496722647249485354595122800380719060378403714356192449680482545587090145444652227564707411062901/32176452*a^5 - 521952169557504162549211283033296158977840798923555659136406859388072300501412015041305017815709886658269475945600628866492858464274662018959159/1149159*a^4 + 3701951281758088250630128545801444037677351306467277248531398358379981953617533535563355124484326367906021666927253606830928006311058631068681931/5362742*a^3 - 778848748545384492959819520808694829230908505098410514884030685673868014783386900470473187133193378999408106524470467413445811612209786526334214/383053*a^2 - 922947389195846866560656121037694991383302653317249200063131415829137953125492506216265279775146016375416237098724006012983901212827213030976723/5362742*a + 194177709088336166278011442427466758809039389152541493456161618030595532552978856239712070050900919915950805562520033999316558269037527039322183/383053 (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

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