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Properties

Label 40.0.11302165783...0000.1
Degree $40$
Signature $[0, 20]$
Discriminant $2^{40}\cdot 3^{20}\cdot 5^{20}\cdot 11^{36}$
Root discriminant $67.04$
Ramified primes $2, 3, 5, 11$
Class number Not computed
Class group Not computed
Galois group $C_2^2\times C_{10}$ (as 40T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![95367431640625, 0, 19073486328125, 0, 0, 0, -762939453125, 0, -152587890625, 0, 0, 0, 6103515625, 0, 1220703125, 0, 0, 0, -48828125, 0, -9765625, 0, -1953125, 0, 0, 0, 78125, 0, 15625, 0, 0, 0, -625, 0, -125, 0, 0, 0, 5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 + 5*x^38 - 125*x^34 - 625*x^32 + 15625*x^28 + 78125*x^26 - 1953125*x^22 - 9765625*x^20 - 48828125*x^18 + 1220703125*x^14 + 6103515625*x^12 - 152587890625*x^8 - 762939453125*x^6 + 19073486328125*x^2 + 95367431640625)
 
gp: K = bnfinit(x^40 + 5*x^38 - 125*x^34 - 625*x^32 + 15625*x^28 + 78125*x^26 - 1953125*x^22 - 9765625*x^20 - 48828125*x^18 + 1220703125*x^14 + 6103515625*x^12 - 152587890625*x^8 - 762939453125*x^6 + 19073486328125*x^2 + 95367431640625, 1)
 

Normalized defining polynomial

\( x^{40} + 5 x^{38} - 125 x^{34} - 625 x^{32} + 15625 x^{28} + 78125 x^{26} - 1953125 x^{22} - 9765625 x^{20} - 48828125 x^{18} + 1220703125 x^{14} + 6103515625 x^{12} - 152587890625 x^{8} - 762939453125 x^{6} + 19073486328125 x^{2} + 95367431640625 \)

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

Invariants

Degree:  $40$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 20]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(11302165783522556415463223790320401501047994110286233600000000000000000000=2^{40}\cdot 3^{20}\cdot 5^{20}\cdot 11^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.04$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(660=2^{2}\cdot 3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{660}(1,·)$, $\chi_{660}(259,·)$, $\chi_{660}(641,·)$, $\chi_{660}(521,·)$, $\chi_{660}(139,·)$, $\chi_{660}(499,·)$, $\chi_{660}(401,·)$, $\chi_{660}(19,·)$, $\chi_{660}(281,·)$, $\chi_{660}(541,·)$, $\chi_{660}(161,·)$, $\chi_{660}(419,·)$, $\chi_{660}(421,·)$, $\chi_{660}(41,·)$, $\chi_{660}(299,·)$, $\chi_{660}(301,·)$, $\chi_{660}(559,·)$, $\chi_{660}(179,·)$, $\chi_{660}(181,·)$, $\chi_{660}(439,·)$, $\chi_{660}(59,·)$, $\chi_{660}(61,·)$, $\chi_{660}(581,·)$, $\chi_{660}(199,·)$, $\chi_{660}(461,·)$, $\chi_{660}(79,·)$, $\chi_{660}(599,·)$, $\chi_{660}(601,·)$, $\chi_{660}(221,·)$, $\chi_{660}(479,·)$, $\chi_{660}(481,·)$, $\chi_{660}(101,·)$, $\chi_{660}(359,·)$, $\chi_{660}(361,·)$, $\chi_{660}(619,·)$, $\chi_{660}(239,·)$, $\chi_{660}(241,·)$, $\chi_{660}(659,·)$, $\chi_{660}(119,·)$, $\chi_{660}(379,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{5} a^{2}$, $\frac{1}{5} a^{3}$, $\frac{1}{25} a^{4}$, $\frac{1}{25} a^{5}$, $\frac{1}{125} a^{6}$, $\frac{1}{125} a^{7}$, $\frac{1}{625} a^{8}$, $\frac{1}{625} a^{9}$, $\frac{1}{3125} a^{10}$, $\frac{1}{3125} a^{11}$, $\frac{1}{15625} a^{12}$, $\frac{1}{15625} a^{13}$, $\frac{1}{78125} a^{14}$, $\frac{1}{78125} a^{15}$, $\frac{1}{390625} a^{16}$, $\frac{1}{390625} a^{17}$, $\frac{1}{1953125} a^{18}$, $\frac{1}{1953125} a^{19}$, $\frac{1}{9765625} a^{20}$, $\frac{1}{9765625} a^{21}$, $\frac{1}{48828125} a^{22}$, $\frac{1}{48828125} a^{23}$, $\frac{1}{244140625} a^{24}$, $\frac{1}{244140625} a^{25}$, $\frac{1}{1220703125} a^{26}$, $\frac{1}{1220703125} a^{27}$, $\frac{1}{6103515625} a^{28}$, $\frac{1}{6103515625} a^{29}$, $\frac{1}{30517578125} a^{30}$, $\frac{1}{30517578125} a^{31}$, $\frac{1}{152587890625} a^{32}$, $\frac{1}{152587890625} a^{33}$, $\frac{1}{762939453125} a^{34}$, $\frac{1}{762939453125} a^{35}$, $\frac{1}{3814697265625} a^{36}$, $\frac{1}{3814697265625} a^{37}$, $\frac{1}{19073486328125} a^{38}$, $\frac{1}{19073486328125} a^{39}$

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:  $19$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{1}{152587890625} a^{32} \) (order $66$)
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

$C_2^2\times C_{10}$ (as 40T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 40
The 40 conjugacy class representatives for $C_2^2\times C_{10}$
Character table for $C_2^2\times C_{10}$ is not computed

Intermediate fields

\(\Q(\sqrt{-165}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{55}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-5}, \sqrt{33})\), \(\Q(\sqrt{-11}, \sqrt{15})\), \(\Q(\sqrt{-3}, \sqrt{55})\), \(\Q(\sqrt{-5}, \sqrt{-11})\), \(\Q(\sqrt{-3}, \sqrt{-5})\), \(\Q(\sqrt{-3}, \sqrt{-11})\), \(\Q(\sqrt{15}, \sqrt{33})\), \(\Q(\zeta_{11})^+\), 8.0.189747360000.2, 10.0.1833540124521600000.1, 10.0.685948419200000.1, \(\Q(\zeta_{33})^+\), \(\Q(\zeta_{11})\), 10.10.166685465865600000.1, 10.10.7545432611200000.1, 10.0.52089208083.1, 20.0.3361869388230684433628866560000000000.7, 20.0.3361869388230684433628866560000000000.10, 20.0.3361869388230684433628866560000000000.9, 20.0.56933553290160450365440000000000.3, 20.0.27784044530832102757263360000000000.2, \(\Q(\zeta_{33})\), 20.20.3361869388230684433628866560000000000.3

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 R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
2Data not computed
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5Data not computed
11Data not computed