Properties

Label 34.0.100...312.1
Degree $34$
Signature $[0, 17]$
Discriminant $-1.010\times 10^{62}$
Root discriminant \(66.63\)
Ramified primes $2,17$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2\times F_{17}$ (as 34T9)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^34 + 2)
 
gp: K = bnfinit(y^34 + 2, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^34 + 2);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^34 + 2)
 

\( x^{34} + 2 \) Copy content Toggle raw display

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

Invariants

Degree:  $34$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 17]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-100988759218878889187610619236040764506432004552857893115789312\) \(\medspace = -\,2^{67}\cdot 17^{34}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(66.63\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{67/34}17^{287/272}\approx 77.89520567226786$
Ramified primes:   \(2\), \(17\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-2}) \)
$\card{ \Aut(K/\Q) }$:  $2$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

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

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

Monogenic:  Yes
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);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $16$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $a^{2}+1$, $a^{18}-a^{2}+1$, $a^{20}-a^{18}-a^{4}+a^{2}-1$, $a^{30}+a^{26}+a^{22}+a^{18}+a^{14}+a^{10}+a^{6}+a^{2}-1$, $a^{33}-a^{32}+a^{30}-a^{29}+a^{27}-a^{26}+a^{24}-a^{23}+a^{21}-a^{20}+a^{18}-a^{17}+a^{15}-a^{14}+a^{12}-a^{11}+a^{9}-a^{8}+a^{6}-a^{5}+a^{3}-a^{2}+1$, $a^{32}+a^{24}-a^{20}-a^{18}+a^{14}+a^{6}+a^{4}-1$, $a^{32}-a^{28}-a^{26}-a^{24}+a^{22}+a^{20}+a^{18}-a^{14}-2a^{12}+a^{8}+2a^{6}+a^{4}-3$, $a^{26}+a^{20}+a^{18}+a^{14}+a^{12}+a^{10}+a^{6}+a^{2}-1$, $a^{32}-a^{30}+a^{28}+a^{24}+a^{22}-a^{20}+a^{18}-2a^{16}+a^{14}-2a^{12}+a^{10}+a^{4}-a^{2}+3$, $a^{33}+a^{30}-2a^{28}-a^{27}-3a^{26}-a^{25}-a^{24}-a^{23}-a^{22}-a^{21}-2a^{20}+a^{19}+a^{18}+a^{17}+3a^{16}+2a^{14}+2a^{13}+2a^{12}+3a^{11}+a^{10}-a^{9}+a^{8}-2a^{7}-a^{5}-2a^{4}-4a^{3}-2a^{2}-5a+1$, $a^{33}-4a^{31}+3a^{29}-2a^{28}+4a^{26}-3a^{25}-4a^{24}+3a^{23}-a^{22}-3a^{21}+5a^{20}+2a^{19}-5a^{18}+2a^{17}+a^{16}-6a^{15}+2a^{14}+5a^{13}-3a^{12}+4a^{10}-4a^{9}-4a^{8}+4a^{7}+a^{6}-2a^{5}+4a^{4}+2a^{3}-5a^{2}-a+3$, $2a^{32}+2a^{31}+2a^{30}-a^{27}-2a^{25}-a^{24}-2a^{23}-a^{21}+2a^{18}+2a^{17}+3a^{16}+a^{15}+a^{14}-2a^{13}-2a^{12}-3a^{11}-a^{10}-a^{9}-a^{7}+a^{6}+a^{5}+3a^{4}+2a^{3}+2a^{2}+2a+1$, $3a^{33}-3a^{32}+2a^{31}-a^{30}-2a^{29}-a^{28}+3a^{27}-5a^{26}+6a^{25}-a^{24}+a^{23}-2a^{22}+4a^{21}-9a^{20}+6a^{19}-6a^{18}+2a^{17}-a^{16}+7a^{15}-7a^{14}+9a^{13}-5a^{12}+a^{11}-2a^{10}+2a^{9}-8a^{8}+9a^{7}-3a^{6}+2a^{5}+3a^{4}+a^{3}-5a^{2}+4a-5$, $2a^{30}-2a^{29}+a^{28}+a^{27}-2a^{26}+4a^{25}-2a^{24}+4a^{22}-5a^{21}+3a^{20}+a^{19}-4a^{18}+5a^{17}-4a^{16}-a^{15}+4a^{14}-4a^{13}+a^{12}-a^{11}-2a^{10}+3a^{9}-a^{7}-a^{6}+a^{5}+a^{4}+a^{3}-2a+3$, $19a^{33}+a^{32}-18a^{31}+9a^{30}+17a^{29}-16a^{28}-11a^{27}+19a^{26}+2a^{25}-22a^{24}+4a^{23}+20a^{22}-11a^{21}-15a^{20}+19a^{19}+13a^{18}-21a^{17}-4a^{16}+26a^{15}-7a^{14}-29a^{13}+15a^{12}+17a^{11}-28a^{10}-10a^{9}+30a^{8}-27a^{6}+13a^{5}+27a^{4}-21a^{3}-21a^{2}+26a+9$, $4a^{33}+12a^{32}+22a^{31}+11a^{30}-2a^{29}-19a^{28}-24a^{27}-7a^{26}+7a^{25}+23a^{24}+19a^{23}+13a^{22}-19a^{21}-24a^{20}-21a^{19}-a^{18}+17a^{17}+28a^{16}+22a^{15}-6a^{14}-21a^{13}-34a^{12}-10a^{11}+3a^{10}+31a^{9}+31a^{8}+15a^{7}-20a^{6}-33a^{5}-26a^{4}-13a^{3}+27a^{2}+36a+33$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 585226461963806300 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{17}\cdot 585226461963806300 \cdot 1}{2\cdot\sqrt{100988759218878889187610619236040764506432004552857893115789312}}\cr\approx \mathstrut & 1.07948149905439 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^34 + 2)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^34 + 2, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^34 + 2);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^34 + 2);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_2\times F_{17}$ (as 34T9):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 544
The 34 conjugacy class representatives for $C_2\times F_{17}$
Character table for $C_2\times F_{17}$

Intermediate fields

\(\Q(\sqrt{-2}) \), 17.1.54214017802982966177103872.1

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Degree 34 sibling: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R $16^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ $16^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ $16^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ $16^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{8}{,}\,{\href{/padicField/13.2.0.1}{2} }$ R ${\href{/padicField/19.8.0.1}{8} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ $16^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ $16^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ $16^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ $16^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ $16^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{8}{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.8.0.1}{8} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.8.0.1}{8} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display Deg $34$$34$$1$$67$
\(17\) Copy content Toggle raw display 17.17.17.1$x^{17} + 17 x + 17$$17$$1$$17$$F_{17}$$[17/16]_{16}$
17.17.17.1$x^{17} + 17 x + 17$$17$$1$$17$$F_{17}$$[17/16]_{16}$