Properties

Label 35.35.876...281.1
Degree $35$
Signature $[35, 0]$
Discriminant $8.766\times 10^{62}$
Root discriminant $62.86$
Ramified prime $71$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{35}$ (as 35T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^35 - x^34 - 34*x^33 + 33*x^32 + 528*x^31 - 496*x^30 - 4960*x^29 + 4495*x^28 + 31465*x^27 - 27405*x^26 - 142506*x^25 + 118755*x^24 + 475020*x^23 - 376740*x^22 - 1184040*x^21 + 888030*x^20 + 2220075*x^19 - 1562275*x^18 - 3124550*x^17 + 2042975*x^16 + 3268760*x^15 - 1961256*x^14 - 2496144*x^13 + 1352078*x^12 + 1352078*x^11 - 646646*x^10 - 497420*x^9 + 203490*x^8 + 116280*x^7 - 38760*x^6 - 15504*x^5 + 3876*x^4 + 969*x^3 - 153*x^2 - 18*x + 1)
 
gp: K = bnfinit(x^35 - x^34 - 34*x^33 + 33*x^32 + 528*x^31 - 496*x^30 - 4960*x^29 + 4495*x^28 + 31465*x^27 - 27405*x^26 - 142506*x^25 + 118755*x^24 + 475020*x^23 - 376740*x^22 - 1184040*x^21 + 888030*x^20 + 2220075*x^19 - 1562275*x^18 - 3124550*x^17 + 2042975*x^16 + 3268760*x^15 - 1961256*x^14 - 2496144*x^13 + 1352078*x^12 + 1352078*x^11 - 646646*x^10 - 497420*x^9 + 203490*x^8 + 116280*x^7 - 38760*x^6 - 15504*x^5 + 3876*x^4 + 969*x^3 - 153*x^2 - 18*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -18, -153, 969, 3876, -15504, -38760, 116280, 203490, -497420, -646646, 1352078, 1352078, -2496144, -1961256, 3268760, 2042975, -3124550, -1562275, 2220075, 888030, -1184040, -376740, 475020, 118755, -142506, -27405, 31465, 4495, -4960, -496, 528, 33, -34, -1, 1]);
 

\( x^{35} - x^{34} - 34 x^{33} + 33 x^{32} + 528 x^{31} - 496 x^{30} - 4960 x^{29} + 4495 x^{28} + 31465 x^{27} - 27405 x^{26} - 142506 x^{25} + 118755 x^{24} + 475020 x^{23} - 376740 x^{22} - 1184040 x^{21} + 888030 x^{20} + 2220075 x^{19} - 1562275 x^{18} - 3124550 x^{17} + 2042975 x^{16} + 3268760 x^{15} - 1961256 x^{14} - 2496144 x^{13} + 1352078 x^{12} + 1352078 x^{11} - 646646 x^{10} - 497420 x^{9} + 203490 x^{8} + 116280 x^{7} - 38760 x^{6} - 15504 x^{5} + 3876 x^{4} + 969 x^{3} - 153 x^{2} - 18 x + 1 \)

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

Invariants

Degree:  $35$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[35, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(876\!\cdots\!281\)\(\medspace = 71^{34}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $62.86$
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:  $71$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $35$
This field is Galois and abelian over $\Q$.
Conductor:  \(71\)
Dirichlet character group:    $\lbrace$$\chi_{71}(1,·)$, $\chi_{71}(2,·)$, $\chi_{71}(3,·)$, $\chi_{71}(4,·)$, $\chi_{71}(5,·)$, $\chi_{71}(6,·)$, $\chi_{71}(8,·)$, $\chi_{71}(9,·)$, $\chi_{71}(10,·)$, $\chi_{71}(12,·)$, $\chi_{71}(15,·)$, $\chi_{71}(16,·)$, $\chi_{71}(18,·)$, $\chi_{71}(19,·)$, $\chi_{71}(20,·)$, $\chi_{71}(24,·)$, $\chi_{71}(25,·)$, $\chi_{71}(27,·)$, $\chi_{71}(29,·)$, $\chi_{71}(30,·)$, $\chi_{71}(32,·)$, $\chi_{71}(36,·)$, $\chi_{71}(37,·)$, $\chi_{71}(38,·)$, $\chi_{71}(40,·)$, $\chi_{71}(43,·)$, $\chi_{71}(45,·)$, $\chi_{71}(48,·)$, $\chi_{71}(49,·)$, $\chi_{71}(50,·)$, $\chi_{71}(54,·)$, $\chi_{71}(57,·)$, $\chi_{71}(58,·)$, $\chi_{71}(60,·)$, $\chi_{71}(64,·)$$\rbrace$
This is not a CM field.

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

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

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

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:  $34$
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:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 190122458612916620000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{35}\cdot(2\pi)^{0}\cdot 190122458612916620000 \cdot 1}{2\sqrt{876564456148583685580741416193317498080031692578600918378223281}}\approx 0.110321801792764$ (assuming GRH)

Galois group

$C_{35}$ (as 35T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 35
The 35 conjugacy class representatives for $C_{35}$
Character table for $C_{35}$ is not computed

Intermediate fields

5.5.25411681.1, 7.7.128100283921.1

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 $35$ $35$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{7}$ $35$ $35$ $35$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{7}$ $35$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{5}$ $35$ $35$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{5}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{5}$ $35$ $35$ $35$ $35$

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
71Data not computed