Normalized defining polynomial
\( x^{15} - x^{14} + 4 x^{13} + 4 x^{11} - 2 x^{10} - 6 x^{9} - 12 x^{8} - 25 x^{7} + 19 x^{6} - 6 x^{5} + 28 x^{4} + 32 x^{3} - 36 x^{2} + 4 x + 4 \)
Invariants
Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-677952124826430464\) \(\medspace = -\,2^{10}\cdot 131^{7}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(15.44\) | 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: | $2^{2/3}131^{1/2}\approx 18.168635476349255$ | ||
Ramified primes: | \(2\), \(131\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-131}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{8}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{12}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a$, $\frac{1}{232}a^{13}-\frac{9}{232}a^{12}+\frac{19}{232}a^{11}+\frac{13}{232}a^{10}-\frac{23}{232}a^{9}-\frac{37}{232}a^{8}-\frac{23}{232}a^{7}+\frac{19}{232}a^{6}+\frac{45}{116}a^{5}+\frac{7}{116}a^{4}-\frac{7}{58}a^{3}-\frac{3}{29}a^{2}+\frac{10}{29}a-\frac{1}{58}$, $\frac{1}{90712}a^{14}+\frac{35}{22678}a^{13}+\frac{3225}{45356}a^{12}+\frac{4293}{45356}a^{11}+\frac{24}{391}a^{10}-\frac{21}{45356}a^{9}-\frac{8829}{45356}a^{8}+\frac{10215}{45356}a^{7}+\frac{11447}{90712}a^{6}-\frac{10949}{45356}a^{5}+\frac{3092}{11339}a^{4}+\frac{4507}{11339}a^{3}-\frac{7979}{22678}a^{2}-\frac{1048}{11339}a+\frac{5535}{22678}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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)
| |
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: | $\frac{16}{493}a^{14}-\frac{33}{1972}a^{13}+\frac{61}{493}a^{12}+\frac{13}{1972}a^{11}+\frac{169}{986}a^{10}-\frac{111}{493}a^{9}-\frac{683}{1972}a^{8}-\frac{2109}{1972}a^{7}-\frac{686}{493}a^{6}-\frac{723}{1972}a^{5}+\frac{387}{1972}a^{4}+\frac{2331}{986}a^{3}+\frac{2105}{986}a^{2}+\frac{1553}{986}a-\frac{108}{493}$, $\frac{111}{45356}a^{14}-\frac{1373}{90712}a^{13}+\frac{449}{90712}a^{12}-\frac{109}{3128}a^{11}-\frac{2201}{90712}a^{10}-\frac{11279}{90712}a^{9}+\frac{2045}{90712}a^{8}-\frac{2095}{90712}a^{7}+\frac{19675}{90712}a^{6}-\frac{893}{22678}a^{5}+\frac{1267}{45356}a^{4}-\frac{16843}{22678}a^{3}-\frac{499}{22678}a^{2}-\frac{1575}{11339}a+\frac{13933}{22678}$, $\frac{1161}{90712}a^{14}-\frac{210}{11339}a^{13}+\frac{2155}{22678}a^{12}-\frac{295}{45356}a^{11}+\frac{2596}{11339}a^{10}+\frac{157}{1564}a^{9}+\frac{21883}{45356}a^{8}+\frac{5277}{45356}a^{7}+\frac{10043}{90712}a^{6}-\frac{8981}{45356}a^{5}-\frac{45583}{45356}a^{4}-\frac{9510}{11339}a^{3}-\frac{23507}{22678}a^{2}+\frac{8339}{22678}a-\frac{8943}{22678}$, $\frac{5303}{45356}a^{14}-\frac{2075}{11339}a^{13}+\frac{6760}{11339}a^{12}-\frac{16423}{45356}a^{11}+\frac{9471}{11339}a^{10}-\frac{8175}{11339}a^{9}+\frac{4517}{45356}a^{8}-\frac{63287}{45356}a^{7}-\frac{72817}{45356}a^{6}+\frac{68987}{22678}a^{5}-\frac{129439}{45356}a^{4}+\frac{104253}{22678}a^{3}-\frac{4091}{11339}a^{2}-\frac{88237}{22678}a+\frac{20458}{11339}$, $\frac{8603}{90712}a^{14}-\frac{65}{1564}a^{13}+\frac{14955}{45356}a^{12}+\frac{5079}{22678}a^{11}+\frac{18815}{45356}a^{10}+\frac{4671}{45356}a^{9}-\frac{27597}{45356}a^{8}-\frac{25195}{22678}a^{7}-\frac{9205}{3128}a^{6}+\frac{17399}{22678}a^{5}-\frac{427}{11339}a^{4}+\frac{27576}{11339}a^{3}+\frac{48600}{11339}a^{2}-\frac{18643}{11339}a-\frac{22617}{22678}$, $\frac{3019}{45356}a^{14}-\frac{687}{11339}a^{13}+\frac{2729}{11339}a^{12}+\frac{2135}{45356}a^{11}+\frac{4261}{22678}a^{10}-\frac{1621}{22678}a^{9}-\frac{25977}{45356}a^{8}-\frac{1029}{1564}a^{7}-\frac{80197}{45356}a^{6}+\frac{29043}{22678}a^{5}-\frac{3407}{45356}a^{4}+\frac{18043}{11339}a^{3}+\frac{30625}{11339}a^{2}-\frac{43161}{22678}a+\frac{2344}{11339}$, $\frac{1346}{11339}a^{14}-\frac{3607}{45356}a^{13}+\frac{19685}{45356}a^{12}+\frac{4239}{22678}a^{11}+\frac{16955}{45356}a^{10}+\frac{3389}{45356}a^{9}-\frac{22971}{22678}a^{8}-\frac{17454}{11339}a^{7}-\frac{179023}{45356}a^{6}+\frac{39581}{22678}a^{5}-\frac{10443}{45356}a^{4}+\frac{40729}{11339}a^{3}+\frac{2408}{391}a^{2}-\frac{81363}{22678}a-\frac{12137}{11339}$ | 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: | \( 1343.20606196 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 1343.20606196 \cdot 1}{2\cdot\sqrt{677952124826430464}}\cr\approx \mathstrut & 0.630670051922 \end{aligned}\]
Galois group
A solvable group of order 30 |
The 9 conjugacy class representatives for $D_{15}$ |
Character table for $D_{15}$ |
Intermediate fields
3.1.524.1, 5.1.17161.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 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 | $15$ | ${\href{/padicField/5.5.0.1}{5} }^{3}$ | $15$ | ${\href{/padicField/11.5.0.1}{5} }^{3}$ | $15$ | ${\href{/padicField/17.2.0.1}{2} }^{7}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.2.0.1}{2} }^{7}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.2.0.1}{2} }^{7}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.2.0.1}{2} }^{7}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.2.0.1}{2} }^{7}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.2.0.1}{2} }^{7}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $15$ | $15$ | ${\href{/padicField/47.2.0.1}{2} }^{7}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.3.0.1}{3} }^{5}$ | $15$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(131\) | $\Q_{131}$ | $x + 129$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
131.2.1.2 | $x^{2} + 131$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
131.2.1.2 | $x^{2} + 131$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
131.2.1.2 | $x^{2} + 131$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
131.2.1.2 | $x^{2} + 131$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
131.2.1.2 | $x^{2} + 131$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
131.2.1.2 | $x^{2} + 131$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
131.2.1.2 | $x^{2} + 131$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.131.2t1.a.a | $1$ | $ 131 $ | \(\Q(\sqrt{-131}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 2.524.3t2.a.a | $2$ | $ 2^{2} \cdot 131 $ | 3.1.524.1 | $S_3$ (as 3T2) | $1$ | $0$ |
* | 2.131.5t2.a.a | $2$ | $ 131 $ | 5.1.17161.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
* | 2.131.5t2.a.b | $2$ | $ 131 $ | 5.1.17161.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
* | 2.524.15t2.a.c | $2$ | $ 2^{2} \cdot 131 $ | 15.1.677952124826430464.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
* | 2.524.15t2.a.a | $2$ | $ 2^{2} \cdot 131 $ | 15.1.677952124826430464.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
* | 2.524.15t2.a.b | $2$ | $ 2^{2} \cdot 131 $ | 15.1.677952124826430464.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
* | 2.524.15t2.a.d | $2$ | $ 2^{2} \cdot 131 $ | 15.1.677952124826430464.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |