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Properties

Label 42.0.865...744.1
Degree $42$
Signature $[0, 21]$
Discriminant $-8.652\times 10^{85}$
Root discriminant $111.20$
Ramified primes $2, 43$
Class number not computed
Class group not computed
Galois group $C_{42}$ (as 42T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^42 + 86*x^40 + 3440*x^38 + 84968*x^36 + 1450992*x^34 + 18175584*x^32 + 172913664*x^30 + 1276267520*x^28 + 7402351616*x^26 + 33963730944*x^24 + 123504476160*x^22 + 355075368960*x^20 + 801783091200*x^18 + 1406204190720*x^16 + 1884175073280*x^14 + 1884175073280*x^12 + 1360793108480*x^10 + 677317836800*x^8 + 216741707776*x^6 + 39926104064*x^4 + 3471835136*x^2 + 90177536)
 
gp: K = bnfinit(x^42 + 86*x^40 + 3440*x^38 + 84968*x^36 + 1450992*x^34 + 18175584*x^32 + 172913664*x^30 + 1276267520*x^28 + 7402351616*x^26 + 33963730944*x^24 + 123504476160*x^22 + 355075368960*x^20 + 801783091200*x^18 + 1406204190720*x^16 + 1884175073280*x^14 + 1884175073280*x^12 + 1360793108480*x^10 + 677317836800*x^8 + 216741707776*x^6 + 39926104064*x^4 + 3471835136*x^2 + 90177536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![90177536, 0, 3471835136, 0, 39926104064, 0, 216741707776, 0, 677317836800, 0, 1360793108480, 0, 1884175073280, 0, 1884175073280, 0, 1406204190720, 0, 801783091200, 0, 355075368960, 0, 123504476160, 0, 33963730944, 0, 7402351616, 0, 1276267520, 0, 172913664, 0, 18175584, 0, 1450992, 0, 84968, 0, 3440, 0, 86, 0, 1]);
 

\( x^{42} + 86 x^{40} + 3440 x^{38} + 84968 x^{36} + 1450992 x^{34} + 18175584 x^{32} + 172913664 x^{30} + 1276267520 x^{28} + 7402351616 x^{26} + 33963730944 x^{24} + 123504476160 x^{22} + 355075368960 x^{20} + 801783091200 x^{18} + 1406204190720 x^{16} + 1884175073280 x^{14} + 1884175073280 x^{12} + 1360793108480 x^{10} + 677317836800 x^{8} + 216741707776 x^{6} + 39926104064 x^{4} + 3471835136 x^{2} + 90177536 \)

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

Invariants

Degree:  $42$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 21]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-86\!\cdots\!744\)\(\medspace = -\,2^{63}\cdot 43^{41}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $111.20$
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:  $2, 43$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $42$
This field is Galois and abelian over $\Q$.
Conductor:  \(344=2^{3}\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{344}(1,·)$, $\chi_{344}(17,·)$, $\chi_{344}(5,·)$, $\chi_{344}(9,·)$, $\chi_{344}(141,·)$, $\chi_{344}(337,·)$, $\chi_{344}(145,·)$, $\chi_{344}(205,·)$, $\chi_{344}(277,·)$, $\chi_{344}(25,·)$, $\chi_{344}(281,·)$, $\chi_{344}(29,·)$, $\chi_{344}(261,·)$, $\chi_{344}(289,·)$, $\chi_{344}(37,·)$, $\chi_{344}(305,·)$, $\chi_{344}(41,·)$, $\chi_{344}(45,·)$, $\chi_{344}(157,·)$, $\chi_{344}(49,·)$, $\chi_{344}(309,·)$, $\chi_{344}(57,·)$, $\chi_{344}(61,·)$, $\chi_{344}(193,·)$, $\chi_{344}(69,·)$, $\chi_{344}(97,·)$, $\chi_{344}(77,·)$, $\chi_{344}(333,·)$, $\chi_{344}(81,·)$, $\chi_{344}(85,·)$, $\chi_{344}(185,·)$, $\chi_{344}(93,·)$, $\chi_{344}(225,·)$, $\chi_{344}(273,·)$, $\chi_{344}(237,·)$, $\chi_{344}(285,·)$, $\chi_{344}(245,·)$, $\chi_{344}(169,·)$, $\chi_{344}(153,·)$, $\chi_{344}(121,·)$, $\chi_{344}(125,·)$, $\chi_{344}(149,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{512} a^{18}$, $\frac{1}{512} a^{19}$, $\frac{1}{1024} a^{20}$, $\frac{1}{1024} a^{21}$, $\frac{1}{2048} a^{22}$, $\frac{1}{2048} a^{23}$, $\frac{1}{4096} a^{24}$, $\frac{1}{4096} a^{25}$, $\frac{1}{8192} a^{26}$, $\frac{1}{8192} a^{27}$, $\frac{1}{16384} a^{28}$, $\frac{1}{16384} a^{29}$, $\frac{1}{32768} a^{30}$, $\frac{1}{32768} a^{31}$, $\frac{1}{65536} a^{32}$, $\frac{1}{65536} a^{33}$, $\frac{1}{131072} a^{34}$, $\frac{1}{131072} a^{35}$, $\frac{1}{262144} a^{36}$, $\frac{1}{262144} a^{37}$, $\frac{1}{524288} a^{38}$, $\frac{1}{524288} a^{39}$, $\frac{1}{1048576} a^{40}$, $\frac{1}{1048576} a^{41}$

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

Class group and class number

not computed

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $20$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

Galois group

$C_{42}$ (as 42T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 42
The 42 conjugacy class representatives for $C_{42}$
Character table for $C_{42}$ is not computed

Intermediate fields

\(\Q(\sqrt{-86}) \), 3.3.1849.1, 6.0.75268322816.1, 7.7.6321363049.1, 14.0.3603461044766854684853927936.1, \(\Q(\zeta_{43})^+\)

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

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R $21^{2}$ $21^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{7}$ ${\href{/LocalNumberField/11.14.0.1}{14} }^{3}$ $42$ $21^{2}$ $21^{2}$ $21^{2}$ $21^{2}$ $21^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{14}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{6}$ R ${\href{/LocalNumberField/47.7.0.1}{7} }^{6}$ $42$ ${\href{/LocalNumberField/59.14.0.1}{14} }^{3}$

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$2.14.21.33$x^{14} + 4 x^{13} + 4 x^{12} + 4 x^{11} - 3 x^{10} + 4 x^{9} - 2 x^{7} - x^{6} - 2 x^{5} + 2 x^{4} - 2 x^{3} + 3 x^{2} + 2 x + 1$$2$$7$$21$$C_{14}$$[3]^{7}$
2.14.21.33$x^{14} + 4 x^{13} + 4 x^{12} + 4 x^{11} - 3 x^{10} + 4 x^{9} - 2 x^{7} - x^{6} - 2 x^{5} + 2 x^{4} - 2 x^{3} + 3 x^{2} + 2 x + 1$$2$$7$$21$$C_{14}$$[3]^{7}$
2.14.21.33$x^{14} + 4 x^{13} + 4 x^{12} + 4 x^{11} - 3 x^{10} + 4 x^{9} - 2 x^{7} - x^{6} - 2 x^{5} + 2 x^{4} - 2 x^{3} + 3 x^{2} + 2 x + 1$$2$$7$$21$$C_{14}$$[3]^{7}$
43Data not computed