Normalized defining polynomial
\( x^{30} - 6 x^{28} + 68 x^{26} - 536 x^{24} + 1904 x^{22} - 4192 x^{20} + 2240 x^{18} + 23936 x^{16} + \cdots - 32768 \)
Invariants
Degree: | $30$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 14]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(347418617918033867346576267434667527455579320614912\) \(\medspace = 2^{45}\cdot 439^{14}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(48.38\) | 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: | $2^{3/2}439^{1/2}\approx 59.262129560116215$ | ||
Ramified primes: | \(2\), \(439\) | 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 \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{48}a^{8}-\frac{1}{3}$, $\frac{1}{48}a^{9}-\frac{1}{3}a$, $\frac{1}{96}a^{10}-\frac{1}{6}a^{2}$, $\frac{1}{96}a^{11}-\frac{1}{6}a^{3}$, $\frac{1}{192}a^{12}-\frac{1}{12}a^{4}$, $\frac{1}{192}a^{13}-\frac{1}{12}a^{5}$, $\frac{1}{1152}a^{14}-\frac{1}{576}a^{12}+\frac{1}{288}a^{10}-\frac{1}{144}a^{8}-\frac{1}{72}a^{6}+\frac{1}{36}a^{4}-\frac{1}{18}a^{2}+\frac{1}{9}$, $\frac{1}{1152}a^{15}-\frac{1}{576}a^{13}+\frac{1}{288}a^{11}-\frac{1}{144}a^{9}-\frac{1}{72}a^{7}+\frac{1}{36}a^{5}-\frac{1}{18}a^{3}+\frac{1}{9}a$, $\frac{1}{2304}a^{16}+\frac{1}{144}a^{8}-\frac{2}{9}$, $\frac{1}{2304}a^{17}+\frac{1}{144}a^{9}-\frac{2}{9}a$, $\frac{1}{4608}a^{18}+\frac{1}{288}a^{10}-\frac{1}{9}a^{2}$, $\frac{1}{4608}a^{19}+\frac{1}{288}a^{11}-\frac{1}{9}a^{3}$, $\frac{1}{9216}a^{20}+\frac{1}{576}a^{12}-\frac{1}{18}a^{4}$, $\frac{1}{9216}a^{21}+\frac{1}{576}a^{13}-\frac{1}{18}a^{5}$, $\frac{1}{55296}a^{22}-\frac{1}{27648}a^{20}+\frac{1}{13824}a^{18}-\frac{1}{6912}a^{16}+\frac{1}{3456}a^{14}-\frac{1}{1728}a^{12}+\frac{1}{864}a^{10}-\frac{1}{432}a^{8}-\frac{1}{108}a^{6}+\frac{1}{54}a^{4}-\frac{1}{27}a^{2}+\frac{2}{27}$, $\frac{1}{55296}a^{23}-\frac{1}{27648}a^{21}+\frac{1}{13824}a^{19}-\frac{1}{6912}a^{17}+\frac{1}{3456}a^{15}-\frac{1}{1728}a^{13}+\frac{1}{864}a^{11}-\frac{1}{432}a^{9}-\frac{1}{108}a^{7}+\frac{1}{54}a^{5}-\frac{1}{27}a^{3}+\frac{2}{27}a$, $\frac{1}{331776}a^{24}+\frac{1}{165888}a^{22}+\frac{1}{41472}a^{20}+\frac{1}{41472}a^{18}+\frac{1}{10368}a^{16}-\frac{1}{5184}a^{14}-\frac{1}{1296}a^{12}-\frac{11}{2592}a^{10}+\frac{1}{648}a^{8}+\frac{1}{648}a^{6}+\frac{1}{162}a^{4}+\frac{5}{81}a^{2}-\frac{5}{81}$, $\frac{1}{331776}a^{25}+\frac{1}{165888}a^{23}+\frac{1}{41472}a^{21}+\frac{1}{41472}a^{19}+\frac{1}{10368}a^{17}-\frac{1}{5184}a^{15}-\frac{1}{1296}a^{13}-\frac{11}{2592}a^{11}+\frac{1}{648}a^{9}+\frac{1}{648}a^{7}+\frac{1}{162}a^{5}+\frac{5}{81}a^{3}-\frac{5}{81}a$, $\frac{1}{417374208}a^{26}-\frac{137}{208687104}a^{24}-\frac{65}{13042944}a^{22}+\frac{19}{1410048}a^{20}+\frac{329}{6521472}a^{18}-\frac{947}{13042944}a^{16}+\frac{941}{6521472}a^{14}-\frac{2683}{1630368}a^{12}-\frac{1849}{407592}a^{10}+\frac{197}{47952}a^{8}+\frac{719}{11988}a^{6}-\frac{9697}{101898}a^{4}+\frac{5024}{50949}a^{2}+\frac{5401}{16983}$, $\frac{1}{417374208}a^{27}-\frac{137}{208687104}a^{25}-\frac{65}{13042944}a^{23}+\frac{19}{1410048}a^{21}+\frac{329}{6521472}a^{19}-\frac{947}{13042944}a^{17}+\frac{941}{6521472}a^{15}-\frac{2683}{1630368}a^{13}-\frac{1849}{407592}a^{11}+\frac{197}{47952}a^{9}+\frac{719}{11988}a^{7}-\frac{9697}{101898}a^{5}+\frac{5024}{50949}a^{3}+\frac{5401}{16983}a$, $\frac{1}{323047636992}a^{28}-\frac{113}{161523818496}a^{26}+\frac{1}{65981952}a^{24}+\frac{185729}{40380954624}a^{22}+\frac{63673}{5047619328}a^{20}+\frac{364007}{10095238656}a^{18}-\frac{263437}{2523809664}a^{16}-\frac{304201}{2523809664}a^{14}-\frac{14015}{78869052}a^{12}-\frac{795607}{630952416}a^{10}+\frac{18031}{9278712}a^{8}+\frac{2867587}{78869052}a^{6}-\frac{1612129}{13144842}a^{4}-\frac{4847656}{19717263}a^{2}+\frac{4609462}{19717263}$, $\frac{1}{323047636992}a^{29}-\frac{113}{161523818496}a^{27}+\frac{1}{65981952}a^{25}+\frac{185729}{40380954624}a^{23}+\frac{63673}{5047619328}a^{21}+\frac{364007}{10095238656}a^{19}-\frac{263437}{2523809664}a^{17}-\frac{304201}{2523809664}a^{15}-\frac{14015}{78869052}a^{13}-\frac{795607}{630952416}a^{11}+\frac{18031}{9278712}a^{9}+\frac{2867587}{78869052}a^{7}-\frac{1612129}{13144842}a^{5}-\frac{4847656}{19717263}a^{3}+\frac{4609462}{19717263}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
not computed
Unit group
Rank: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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: | not computed | 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: | not computed | 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 $
Galois group
A solvable group of order 60 |
The 18 conjugacy class representatives for $D_{30}$ |
Character table for $D_{30}$ |
Intermediate fields
\(\Q(\sqrt{2}) \), 3.1.439.1, 5.1.192721.1, 6.2.98673152.2, 10.2.1217048865701888.3, 15.1.3142328914862177479.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 30 sibling: | deg 30 |
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 | ${\href{/padicField/3.2.0.1}{2} }^{15}$ | ${\href{/padicField/5.10.0.1}{10} }^{3}$ | $15^{2}$ | $30$ | ${\href{/padicField/13.6.0.1}{6} }^{5}$ | ${\href{/padicField/17.2.0.1}{2} }^{14}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }^{3}$ | ${\href{/padicField/23.2.0.1}{2} }^{14}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | $30$ | ${\href{/padicField/31.2.0.1}{2} }^{14}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{15}$ | ${\href{/padicField/41.2.0.1}{2} }^{14}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{15}$ | ${\href{/padicField/47.2.0.1}{2} }^{14}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $30$ | ${\href{/padicField/59.2.0.1}{2} }^{15}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $30$ | $2$ | $15$ | $45$ | |||
\(439\) | $\Q_{439}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{439}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
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) | $1$ | $1$ |
1.439.2t1.a.a | $1$ | $ 439 $ | \(\Q(\sqrt{-439}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.3512.2t1.b.a | $1$ | $ 2^{3} \cdot 439 $ | \(\Q(\sqrt{-878}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 2.28096.6t3.a.a | $2$ | $ 2^{6} \cdot 439 $ | 6.2.98673152.2 | $D_{6}$ (as 6T3) | $1$ | $0$ |
* | 2.439.3t2.a.a | $2$ | $ 439 $ | 3.1.439.1 | $S_3$ (as 3T2) | $1$ | $0$ |
* | 2.439.5t2.a.a | $2$ | $ 439 $ | 5.1.192721.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
* | 2.439.5t2.a.b | $2$ | $ 439 $ | 5.1.192721.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
* | 2.28096.10t3.a.b | $2$ | $ 2^{6} \cdot 439 $ | 10.2.1217048865701888.3 | $D_{10}$ (as 10T3) | $1$ | $0$ |
* | 2.28096.10t3.a.a | $2$ | $ 2^{6} \cdot 439 $ | 10.2.1217048865701888.3 | $D_{10}$ (as 10T3) | $1$ | $0$ |
* | 2.439.15t2.a.c | $2$ | $ 439 $ | 15.1.3142328914862177479.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
* | 2.439.15t2.a.b | $2$ | $ 439 $ | 15.1.3142328914862177479.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
* | 2.439.15t2.a.a | $2$ | $ 439 $ | 15.1.3142328914862177479.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
* | 2.439.15t2.a.d | $2$ | $ 439 $ | 15.1.3142328914862177479.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
* | 2.28096.30t14.a.a | $2$ | $ 2^{6} \cdot 439 $ | 30.2.347418617918033867346576267434667527455579320614912.1 | $D_{30}$ (as 30T14) | $1$ | $0$ |
* | 2.28096.30t14.a.c | $2$ | $ 2^{6} \cdot 439 $ | 30.2.347418617918033867346576267434667527455579320614912.1 | $D_{30}$ (as 30T14) | $1$ | $0$ |
* | 2.28096.30t14.a.d | $2$ | $ 2^{6} \cdot 439 $ | 30.2.347418617918033867346576267434667527455579320614912.1 | $D_{30}$ (as 30T14) | $1$ | $0$ |
* | 2.28096.30t14.a.b | $2$ | $ 2^{6} \cdot 439 $ | 30.2.347418617918033867346576267434667527455579320614912.1 | $D_{30}$ (as 30T14) | $1$ | $0$ |