Normalized defining polynomial
\( x^{14} + 29x^{12} + 290x^{10} + 1247x^{8} + 2262x^{6} + 1566x^{4} + 377x^{2} + 29 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-168110140833113738264576\) \(\medspace = -\,2^{14}\cdot 29^{13}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(45.60\) | 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: | $2\cdot 29^{13/14}\approx 45.60064585163551$ | ||
Ramified primes: | \(2\), \(29\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-29}) \) | ||
$\card{ \Gal(K/\Q) }$: | $14$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(116=2^{2}\cdot 29\) | ||
Dirichlet character group: | $\lbrace$$\chi_{116}(1,·)$, $\chi_{116}(81,·)$, $\chi_{116}(67,·)$, $\chi_{116}(51,·)$, $\chi_{116}(65,·)$, $\chi_{116}(71,·)$, $\chi_{116}(45,·)$, $\chi_{116}(49,·)$, $\chi_{116}(115,·)$, $\chi_{116}(53,·)$, $\chi_{116}(25,·)$, $\chi_{116}(91,·)$, $\chi_{116}(35,·)$, $\chi_{116}(63,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{64}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{17}a^{10}-\frac{5}{17}a^{8}+\frac{3}{17}a^{6}-\frac{4}{17}a^{4}+\frac{7}{17}a^{2}-\frac{6}{17}$, $\frac{1}{17}a^{11}-\frac{5}{17}a^{9}+\frac{3}{17}a^{7}-\frac{4}{17}a^{5}+\frac{7}{17}a^{3}-\frac{6}{17}a$, $\frac{1}{41123}a^{12}-\frac{75}{41123}a^{10}+\frac{15347}{41123}a^{8}+\frac{13794}{41123}a^{6}-\frac{12361}{41123}a^{4}-\frac{16731}{41123}a^{2}-\frac{18212}{41123}$, $\frac{1}{41123}a^{13}-\frac{75}{41123}a^{11}+\frac{15347}{41123}a^{9}+\frac{13794}{41123}a^{7}-\frac{12361}{41123}a^{5}-\frac{16731}{41123}a^{3}-\frac{18212}{41123}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{6}$, which has order $48$ (assuming GRH)
Unit group
Rank: | $6$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -1 \) (order $2$) | 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: | $\frac{3793}{41123}a^{12}+\frac{109822}{41123}a^{10}+\frac{1093743}{41123}a^{8}+\frac{4649409}{41123}a^{6}+\frac{477811}{2419}a^{4}+\frac{4849458}{41123}a^{2}+\frac{644721}{41123}$, $\frac{1627}{41123}a^{12}+\frac{47305}{41123}a^{10}+\frac{477194}{41123}a^{8}+\frac{2101467}{41123}a^{6}+\frac{4008502}{41123}a^{4}+\frac{169026}{2419}a^{2}+\frac{400961}{41123}$, $\frac{7923}{41123}a^{12}+\frac{228235}{41123}a^{10}+\frac{2255335}{41123}a^{8}+\frac{9484341}{41123}a^{6}+\frac{16344614}{41123}a^{4}+\frac{9766990}{41123}a^{2}+\frac{1363990}{41123}$, $\frac{4362}{41123}a^{12}+\frac{125203}{41123}a^{10}+\frac{1229060}{41123}a^{8}+\frac{5105731}{41123}a^{6}+\frac{8588278}{41123}a^{4}+\frac{4906340}{41123}a^{2}+\frac{625737}{41123}$, $\frac{1811}{41123}a^{12}+\frac{52857}{41123}a^{10}+\frac{531287}{41123}a^{8}+\frac{2271362}{41123}a^{6}+\frac{3959558}{41123}a^{4}+\frac{141046}{2419}a^{2}+\frac{347050}{41123}$, $\frac{6296}{41123}a^{12}+\frac{180930}{41123}a^{10}+\frac{1778141}{41123}a^{8}+\frac{7382874}{41123}a^{6}+\frac{12336112}{41123}a^{4}+\frac{6893548}{41123}a^{2}+\frac{921906}{41123}$ (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: | \( 6020.98510015 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 6020.98510015 \cdot 48}{2\cdot\sqrt{168110140833113738264576}}\cr\approx \mathstrut & 0.136251382214 \end{aligned}\] (assuming GRH)
Galois group
A cyclic group of order 14 |
The 14 conjugacy class representatives for $C_{14}$ |
Character table for $C_{14}$ |
Intermediate fields
\(\Q(\sqrt{-29}) \), 7.7.594823321.1 |
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 | R | ${\href{/padicField/3.7.0.1}{7} }^{2}$ | ${\href{/padicField/5.7.0.1}{7} }^{2}$ | ${\href{/padicField/7.14.0.1}{14} }$ | ${\href{/padicField/11.7.0.1}{7} }^{2}$ | ${\href{/padicField/13.7.0.1}{7} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{7}$ | ${\href{/padicField/19.7.0.1}{7} }^{2}$ | ${\href{/padicField/23.14.0.1}{14} }$ | R | ${\href{/padicField/31.7.0.1}{7} }^{2}$ | ${\href{/padicField/37.14.0.1}{14} }$ | ${\href{/padicField/41.2.0.1}{2} }^{7}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}$ | ${\href{/padicField/47.7.0.1}{7} }^{2}$ | ${\href{/padicField/53.7.0.1}{7} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{7}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(2\) | 2.14.14.15 | $x^{14} + 14 x^{13} + 126 x^{12} + 784 x^{11} + 4300 x^{10} + 19592 x^{9} + 80680 x^{8} + 276608 x^{7} + 822832 x^{6} + 1982880 x^{5} + 3998112 x^{4} + 6222080 x^{3} + 7679040 x^{2} + 6275456 x + 3453824$ | $2$ | $7$ | $14$ | $C_{14}$ | $[2]^{7}$ |
\(29\) | 29.14.13.1 | $x^{14} + 29$ | $14$ | $1$ | $13$ | $C_{14}$ | $[\ ]_{14}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.116.2t1.a.a | $1$ | $ 2^{2} \cdot 29 $ | \(\Q(\sqrt{-29}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
* | 1.29.7t1.a.a | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.116.14t1.a.a | $1$ | $ 2^{2} \cdot 29 $ | 14.0.168110140833113738264576.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ |
* | 1.29.7t1.a.b | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.116.14t1.a.b | $1$ | $ 2^{2} \cdot 29 $ | 14.0.168110140833113738264576.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ |
* | 1.29.7t1.a.c | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.116.14t1.a.c | $1$ | $ 2^{2} \cdot 29 $ | 14.0.168110140833113738264576.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ |
* | 1.29.7t1.a.d | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.116.14t1.a.d | $1$ | $ 2^{2} \cdot 29 $ | 14.0.168110140833113738264576.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ |
* | 1.29.7t1.a.e | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.116.14t1.a.e | $1$ | $ 2^{2} \cdot 29 $ | 14.0.168110140833113738264576.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ |
* | 1.29.7t1.a.f | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.116.14t1.a.f | $1$ | $ 2^{2} \cdot 29 $ | 14.0.168110140833113738264576.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ |