# Properties

 Label 17.3.215...383.1 Degree $17$ Signature $[3, 7]$ Discriminant $-2.151\times 10^{18}$ Root discriminant $11.98$ Ramified primes $37, 433, 5531, 12953$ Class number $1$ Class group trivial Galois group $S_{17}$ (as 17T10)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 2*x^15 - 4*x^14 + 4*x^13 + 7*x^12 - 3*x^11 - 3*x^10 - 3*x^9 + 5*x^8 - 5*x^7 - 4*x^6 + 14*x^5 - 2*x^4 - 7*x^3 + 2*x + 1)

gp: K = bnfinit(x^17 - 2*x^15 - 4*x^14 + 4*x^13 + 7*x^12 - 3*x^11 - 3*x^10 - 3*x^9 + 5*x^8 - 5*x^7 - 4*x^6 + 14*x^5 - 2*x^4 - 7*x^3 + 2*x + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, 0, -7, -2, 14, -4, -5, 5, -3, -3, -3, 7, 4, -4, -2, 0, 1]);

$$x^{17} - 2 x^{15} - 4 x^{14} + 4 x^{13} + 7 x^{12} - 3 x^{11} - 3 x^{10} - 3 x^{9} + 5 x^{8} - 5 x^{7} - 4 x^{6} + 14 x^{5} - 2 x^{4} - 7 x^{3} + 2 x + 1$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $17$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[3, 7]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$-2151149236330118383$$$$\medspace = -\,37^{5}\cdot 433\cdot 5531\cdot 12953$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $11.98$ 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: $37, 433, 5531, 12953$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $1$ This field is not Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{228469} a^{16} + \frac{13230}{228469} a^{15} + \frac{25644}{228469} a^{14} - \frac{6349}{228469} a^{13} + \frac{79326}{228469} a^{12} - \frac{103599}{228469} a^{11} - \frac{29242}{228469} a^{10} - \frac{73646}{228469} a^{9} + \frac{83702}{228469} a^{8} - \frac{11778}{228469} a^{7} - \frac{7087}{228469} a^{6} - \frac{88724}{228469} a^{5} + \frac{55216}{228469} a^{4} + \frac{92285}{228469} a^{3} - \frac{7793}{228469} a^{2} - \frac{61871}{228469} a + \frac{51099}{228469}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

## Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $9$ 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 sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$154.682742806$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{3}\cdot(2\pi)^{7}\cdot 154.682742806 \cdot 1}{2\sqrt{2151149236330118383}}\approx 0.163089373269$

## Galois group

$S_{17}$ (as 17T10):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A non-solvable group of order 355687428096000 The 297 conjugacy class representatives for $S_{17}$ are not computed Character table for $S_{17}$ is not computed

## Intermediate fields

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

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type $17$ ${\href{/padicField/3.13.0.1}{13} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ ${\href{/padicField/5.11.0.1}{11} }{,}\,{\href{/padicField/5.6.0.1}{6} }$ ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ $17$ ${\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ $17$ ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.7.0.1}{7} }$ $17$ ${\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ R ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ $17$ ${\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ ${\href{/padicField/59.11.0.1}{11} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$

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
37Data not computed
433Data not computed
5531Data not computed
12953Data not computed