Normalized defining polynomial
\( x^{14} + 2x^{12} + 28x^{10} + 80x^{8} + 272x^{6} + 544x^{4} + 704x^{2} - 128 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(24859454395438333952\) \(\medspace = 2^{21}\cdot 151^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(24.29\) | 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))
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Galois root discriminant: | $2^{3/2}151^{1/2}\approx 34.75629439396553$ | ||
Ramified primes: | \(2\), \(151\) | 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 \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{48}a^{8}-\frac{1}{3}$, $\frac{1}{48}a^{9}-\frac{1}{3}a$, $\frac{1}{96}a^{10}-\frac{1}{6}a^{2}$, $\frac{1}{96}a^{11}-\frac{1}{6}a^{3}$, $\frac{1}{2496}a^{12}-\frac{1}{208}a^{10}-\frac{1}{208}a^{8}+\frac{3}{52}a^{6}+\frac{2}{39}a^{4}-\frac{5}{13}$, $\frac{1}{2496}a^{13}-\frac{1}{208}a^{11}-\frac{1}{208}a^{9}+\frac{3}{52}a^{7}+\frac{2}{39}a^{5}-\frac{5}{13}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | 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: | $\frac{1}{416}a^{12}+\frac{1}{416}a^{10}+\frac{17}{312}a^{8}+\frac{5}{52}a^{6}+\frac{4}{13}a^{4}-\frac{25}{39}$, $\frac{1}{832}a^{12}+\frac{1}{156}a^{10}+\frac{5}{104}a^{8}+\frac{9}{52}a^{6}+\frac{17}{26}a^{4}+\frac{7}{6}a^{2}+\frac{11}{13}$, $\frac{1}{2496}a^{12}-\frac{1}{208}a^{10}+\frac{5}{312}a^{8}-\frac{7}{104}a^{6}+\frac{2}{39}a^{4}+\frac{11}{39}$, $\frac{11}{2496}a^{13}+\frac{1}{104}a^{11}+\frac{71}{624}a^{9}+\frac{5}{13}a^{7}+\frac{83}{78}a^{5}+\frac{5}{2}a^{3}+\frac{121}{39}a+1$, $\frac{1}{104}a^{13}+\frac{1}{832}a^{12}+\frac{25}{1248}a^{11}-\frac{5}{1248}a^{10}+\frac{27}{104}a^{9}+\frac{1}{156}a^{8}+\frac{79}{104}a^{7}-\frac{1}{13}a^{6}+\frac{129}{52}a^{5}-\frac{9}{26}a^{4}+\frac{13}{3}a^{3}-\frac{13}{6}a^{2}+\frac{62}{13}a-\frac{58}{39}$, $\frac{11}{1248}a^{13}+\frac{19}{2496}a^{12}+\frac{1}{52}a^{11}+\frac{29}{1248}a^{10}+\frac{155}{624}a^{9}+\frac{125}{624}a^{8}+\frac{10}{13}a^{7}+\frac{31}{52}a^{6}+\frac{371}{156}a^{5}+\frac{77}{39}a^{4}+\frac{11}{2}a^{3}+\frac{13}{6}a^{2}+\frac{346}{39}a+\frac{118}{39}$, $\frac{41}{1248}a^{13}+\frac{1}{64}a^{12}+\frac{31}{416}a^{11}+\frac{1}{32}a^{10}+\frac{191}{208}a^{9}+\frac{7}{16}a^{8}+\frac{297}{104}a^{7}+\frac{5}{4}a^{6}+\frac{359}{39}a^{5}+\frac{17}{4}a^{4}+\frac{39}{2}a^{3}+9a^{2}+\frac{331}{13}a+10$ | 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: | \( 21290.231629680748 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 21290.231629680748 \cdot 1}{2\cdot\sqrt{24859454395438333952}}\cr\approx \mathstrut & 0.525465002301094 \end{aligned}\]
Galois group
A solvable group of order 28 |
The 10 conjugacy class representatives for $D_{14}$ |
Character table for $D_{14}$ |
Intermediate fields
\(\Q(\sqrt{2}) \), 7.1.3442951.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 28 |
Degree 14 sibling: | deg 14 |
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 | ${\href{/padicField/3.2.0.1}{2} }^{7}$ | ${\href{/padicField/5.14.0.1}{14} }$ | ${\href{/padicField/7.2.0.1}{2} }^{6}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.14.0.1}{14} }$ | ${\href{/padicField/13.2.0.1}{2} }^{7}$ | ${\href{/padicField/17.7.0.1}{7} }^{2}$ | ${\href{/padicField/19.14.0.1}{14} }$ | ${\href{/padicField/23.2.0.1}{2} }^{6}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.14.0.1}{14} }$ | ${\href{/padicField/31.7.0.1}{7} }^{2}$ | ${\href{/padicField/37.14.0.1}{14} }$ | ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.14.0.1}{14} }$ | ${\href{/padicField/47.7.0.1}{7} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{7}$ | ${\href{/padicField/59.14.0.1}{14} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.14.21.34 | $x^{14} + 4 x^{13} + 14 x^{12} + 656 x^{11} + 1236 x^{10} - 43472 x^{9} - 244456 x^{8} + 434816 x^{7} + 8570160 x^{6} + 35893184 x^{5} + 77018272 x^{4} + 105671936 x^{3} + 121134528 x^{2} + 74194176 x - 52979584$ | $2$ | $7$ | $21$ | $C_{14}$ | $[3]^{7}$ |
\(151\) | $\Q_{151}$ | $x + 145$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{151}$ | $x + 145$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.151.2t1.a.a | $1$ | $ 151 $ | \(\Q(\sqrt{-151}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.1208.2t1.b.a | $1$ | $ 2^{3} \cdot 151 $ | \(\Q(\sqrt{-302}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 2.9664.14t3.a.a | $2$ | $ 2^{6} \cdot 151 $ | 14.2.24859454395438333952.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |
* | 2.151.7t2.a.b | $2$ | $ 151 $ | 7.1.3442951.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
* | 2.151.7t2.a.a | $2$ | $ 151 $ | 7.1.3442951.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
* | 2.151.7t2.a.c | $2$ | $ 151 $ | 7.1.3442951.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
* | 2.9664.14t3.a.c | $2$ | $ 2^{6} \cdot 151 $ | 14.2.24859454395438333952.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |
* | 2.9664.14t3.a.b | $2$ | $ 2^{6} \cdot 151 $ | 14.2.24859454395438333952.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |