# Properties

 Label 43.1.33547731764...9584.1 Degree $43$ Signature $[1, 21]$ Discriminant $-\,2^{42}\cdot 5736251\cdot 13297659175117020385300439521039825046372945434356686517822982914329253832871$ Root discriminant $166.54$ Ramified primes $2, 5736251, 13297659175117020385300439521039825046372945434356686517822982914329253832871$ Class number Not computed Class group Not computed Galois group $S_{43}$ (as 43T10)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^43 - 3*x - 4)

gp: K = bnfinit(x^43 - 3*x - 4, 1)

## Normalizeddefining polynomial

$$x^{43} - 3 x - 4$$

magma: DefiningPolynomial(K);

sage: K.defining_polynomial()

gp: K.pol

## Invariants

 Degree: $43$ magma: Degree(K);  sage: K.degree()  gp: poldegree(K.pol) Signature: $[1, 21]$ magma: Signature(K);  sage: K.signature()  gp: K.sign Discriminant: $$-335477317645632815204821422239306587903706814127360732280247849979340897599838242672095300419584=-\,2^{42}\cdot 5736251\cdot 13297659175117020385300439521039825046372945434356686517822982914329253832871$$ magma: Discriminant(K);  sage: K.disc()  gp: K.disc Root discriminant: $166.54$ magma: Abs(Discriminant(K))^(1/Degree(K));  sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $2, 5736251, 13297659175117020385300439521039825046372945434356686517822982914329253832871$ magma: PrimeDivisors(Discriminant(K));  sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~ This field is not Galois over $\Q$. This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$

magma: IntegralBasis(K);

sage: K.integral_basis()

gp: K.zk

## Class group and class number

Not computed

magma: ClassGroup(K);

sage: K.class_group().invariants()

gp: K.clgp

## Unit group

magma: UK, f := UnitGroup(K);

sage: UK = K.unit_group()

 Rank: $21$ magma: UnitRank(K);  sage: UK.rank()  gp: K.fu Torsion generator: $$-1$$ (order $2$) magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);  sage: UK.torsion_generator()  gp: K.tu[2] Fundamental units: Not computed magma: [K!f(g): g in Generators(UK)];  sage: UK.fundamental_units()  gp: K.fu Regulator: Not computed magma: Regulator(K);  sage: K.regulator()  gp: K.reg

## Galois group

$S_{43}$ (as 43T10):

magma: GaloisGroup(K);

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

 A non-solvable group of order 60415263063373835637355132068513997507264512000000000 The 63261 conjugacy class representatives for $S_{43}$ are not computed Character table for $S_{43}$ is not computed

## Intermediate fields

 The extension is primitive: there are no intermediate fields between this field and $\Q$.

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R $42{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ $43$ $31{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.14.0.1}{14} }{,}\,{\href{/LocalNumberField/11.13.0.1}{13} }{,}\,{\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $22{,}\,15{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }$ $20{,}\,{\href{/LocalNumberField/17.11.0.1}{11} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $22{,}\,{\href{/LocalNumberField/19.14.0.1}{14} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $20{,}\,19{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ $27{,}\,{\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ $43$ $42{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $21{,}\,{\href{/LocalNumberField/41.14.0.1}{14} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ $42{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $20{,}\,{\href{/LocalNumberField/47.7.0.1}{7} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ $35{,}\,{\href{/LocalNumberField/53.8.0.1}{8} }$ $42{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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

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];

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]])

## Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$$\Q_{2}$$x + 1$$1$$1$$0Trivial[\ ] 2.2.2.1x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2} 2.6.6.5x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3} 2.12.12.25x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
5736251Data not computed
13297659175117020385300439521039825046372945434356686517822982914329253832871Data not computed