Normalized defining polynomial
\( x^{15} + 29 x^{13} - 15 x^{12} + 365 x^{11} + 708 x^{10} + 1338 x^{9} + 15525 x^{8} - 1890 x^{7} + \cdots - 295769 \)
Invariants
Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[3, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(122463012708006247915836261129\) \(\medspace = 3^{5}\cdot 11^{12}\cdot 107^{7}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(86.94\) | 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: | $3^{1/2}11^{4/5}107^{1/2}\approx 122.00191969369786$ | ||
Ramified primes: | \(3\), \(11\), \(107\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{321}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{62}a^{13}+\frac{9}{62}a^{12}-\frac{10}{31}a^{11}+\frac{15}{31}a^{10}+\frac{11}{62}a^{9}+\frac{7}{62}a^{8}-\frac{29}{62}a^{7}+\frac{1}{62}a^{6}+\frac{29}{62}a^{5}+\frac{14}{31}a^{4}+\frac{3}{31}a^{3}+\frac{23}{62}a^{2}-\frac{7}{62}a+\frac{21}{62}$, $\frac{1}{23\!\cdots\!38}a^{14}-\frac{14\!\cdots\!81}{23\!\cdots\!38}a^{13}-\frac{19\!\cdots\!31}{11\!\cdots\!19}a^{12}+\frac{43\!\cdots\!01}{11\!\cdots\!19}a^{11}-\frac{15\!\cdots\!61}{23\!\cdots\!38}a^{10}-\frac{62\!\cdots\!05}{23\!\cdots\!38}a^{9}+\frac{65\!\cdots\!09}{23\!\cdots\!38}a^{8}+\frac{11\!\cdots\!99}{23\!\cdots\!38}a^{7}+\frac{10\!\cdots\!77}{23\!\cdots\!38}a^{6}-\frac{56\!\cdots\!14}{11\!\cdots\!19}a^{5}-\frac{56\!\cdots\!30}{11\!\cdots\!19}a^{4}+\frac{72\!\cdots\!85}{23\!\cdots\!38}a^{3}+\frac{95\!\cdots\!19}{23\!\cdots\!38}a^{2}+\frac{82\!\cdots\!07}{23\!\cdots\!38}a-\frac{58\!\cdots\!30}{11\!\cdots\!19}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{5}$, which has order $5$ (assuming GRH)
Unit group
Rank: | $8$ | 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: | $\frac{11\!\cdots\!87}{23\!\cdots\!31}a^{14}+\frac{13\!\cdots\!61}{23\!\cdots\!31}a^{13}+\frac{34\!\cdots\!21}{23\!\cdots\!31}a^{12}+\frac{22\!\cdots\!21}{23\!\cdots\!31}a^{11}+\frac{43\!\cdots\!41}{23\!\cdots\!31}a^{10}+\frac{12\!\cdots\!63}{23\!\cdots\!31}a^{9}+\frac{29\!\cdots\!18}{23\!\cdots\!31}a^{8}+\frac{20\!\cdots\!62}{23\!\cdots\!31}a^{7}+\frac{20\!\cdots\!23}{23\!\cdots\!31}a^{6}+\frac{86\!\cdots\!37}{23\!\cdots\!31}a^{5}-\frac{10\!\cdots\!28}{23\!\cdots\!31}a^{4}-\frac{14\!\cdots\!98}{23\!\cdots\!31}a^{3}-\frac{23\!\cdots\!79}{23\!\cdots\!31}a^{2}-\frac{50\!\cdots\!86}{23\!\cdots\!31}a+\frac{27\!\cdots\!10}{23\!\cdots\!31}$, $\frac{57\!\cdots\!74}{23\!\cdots\!31}a^{14}-\frac{70\!\cdots\!00}{23\!\cdots\!31}a^{13}+\frac{14\!\cdots\!55}{23\!\cdots\!31}a^{12}-\frac{30\!\cdots\!51}{23\!\cdots\!31}a^{11}+\frac{16\!\cdots\!43}{23\!\cdots\!31}a^{10}+\frac{13\!\cdots\!24}{23\!\cdots\!31}a^{9}-\frac{28\!\cdots\!57}{23\!\cdots\!31}a^{8}+\frac{59\!\cdots\!31}{23\!\cdots\!31}a^{7}-\frac{13\!\cdots\!03}{23\!\cdots\!31}a^{6}+\frac{29\!\cdots\!41}{23\!\cdots\!31}a^{5}-\frac{15\!\cdots\!68}{23\!\cdots\!31}a^{4}+\frac{12\!\cdots\!00}{23\!\cdots\!31}a^{3}+\frac{20\!\cdots\!46}{23\!\cdots\!31}a^{2}+\frac{29\!\cdots\!36}{23\!\cdots\!31}a-\frac{25\!\cdots\!01}{23\!\cdots\!31}$, $\frac{32\!\cdots\!17}{23\!\cdots\!38}a^{14}-\frac{28\!\cdots\!35}{23\!\cdots\!38}a^{13}+\frac{97\!\cdots\!75}{23\!\cdots\!38}a^{12}-\frac{66\!\cdots\!58}{11\!\cdots\!19}a^{11}+\frac{13\!\cdots\!25}{23\!\cdots\!38}a^{10}+\frac{11\!\cdots\!23}{23\!\cdots\!38}a^{9}+\frac{19\!\cdots\!82}{11\!\cdots\!19}a^{8}+\frac{50\!\cdots\!63}{23\!\cdots\!38}a^{7}-\frac{25\!\cdots\!17}{11\!\cdots\!19}a^{6}+\frac{13\!\cdots\!13}{11\!\cdots\!19}a^{5}-\frac{77\!\cdots\!83}{23\!\cdots\!38}a^{4}+\frac{54\!\cdots\!00}{11\!\cdots\!19}a^{3}-\frac{73\!\cdots\!80}{11\!\cdots\!19}a^{2}+\frac{15\!\cdots\!41}{23\!\cdots\!38}a-\frac{75\!\cdots\!67}{23\!\cdots\!38}$, $\frac{22\!\cdots\!27}{77\!\cdots\!98}a^{14}-\frac{16\!\cdots\!95}{11\!\cdots\!19}a^{13}+\frac{20\!\cdots\!99}{23\!\cdots\!38}a^{12}-\frac{10\!\cdots\!11}{11\!\cdots\!19}a^{11}+\frac{28\!\cdots\!91}{23\!\cdots\!38}a^{10}+\frac{17\!\cdots\!39}{11\!\cdots\!19}a^{9}+\frac{48\!\cdots\!22}{11\!\cdots\!19}a^{8}+\frac{56\!\cdots\!69}{11\!\cdots\!19}a^{7}-\frac{39\!\cdots\!06}{11\!\cdots\!19}a^{6}+\frac{59\!\cdots\!09}{23\!\cdots\!38}a^{5}-\frac{83\!\cdots\!65}{11\!\cdots\!19}a^{4}+\frac{20\!\cdots\!57}{23\!\cdots\!38}a^{3}-\frac{17\!\cdots\!77}{11\!\cdots\!19}a^{2}+\frac{13\!\cdots\!71}{11\!\cdots\!19}a-\frac{94\!\cdots\!71}{23\!\cdots\!38}$, $\frac{10\!\cdots\!86}{11\!\cdots\!19}a^{14}+\frac{24\!\cdots\!53}{23\!\cdots\!38}a^{13}+\frac{31\!\cdots\!52}{11\!\cdots\!19}a^{12}+\frac{22\!\cdots\!68}{11\!\cdots\!19}a^{11}+\frac{39\!\cdots\!57}{11\!\cdots\!19}a^{10}+\frac{23\!\cdots\!55}{23\!\cdots\!38}a^{9}+\frac{28\!\cdots\!13}{11\!\cdots\!19}a^{8}+\frac{38\!\cdots\!13}{23\!\cdots\!38}a^{7}+\frac{20\!\cdots\!95}{11\!\cdots\!19}a^{6}+\frac{15\!\cdots\!77}{23\!\cdots\!38}a^{5}-\frac{16\!\cdots\!03}{23\!\cdots\!38}a^{4}-\frac{13\!\cdots\!69}{23\!\cdots\!38}a^{3}-\frac{13\!\cdots\!83}{11\!\cdots\!19}a^{2}-\frac{66\!\cdots\!27}{23\!\cdots\!38}a+\frac{23\!\cdots\!80}{11\!\cdots\!19}$, $\frac{60\!\cdots\!95}{11\!\cdots\!19}a^{14}-\frac{35\!\cdots\!01}{11\!\cdots\!19}a^{13}+\frac{37\!\cdots\!09}{23\!\cdots\!38}a^{12}-\frac{35\!\cdots\!97}{38\!\cdots\!49}a^{11}+\frac{32\!\cdots\!17}{11\!\cdots\!19}a^{10}-\frac{78\!\cdots\!05}{11\!\cdots\!19}a^{9}-\frac{18\!\cdots\!89}{23\!\cdots\!38}a^{8}+\frac{79\!\cdots\!49}{11\!\cdots\!19}a^{7}-\frac{10\!\cdots\!05}{23\!\cdots\!38}a^{6}+\frac{83\!\cdots\!61}{11\!\cdots\!19}a^{5}-\frac{59\!\cdots\!39}{23\!\cdots\!38}a^{4}+\frac{13\!\cdots\!71}{23\!\cdots\!38}a^{3}+\frac{71\!\cdots\!17}{23\!\cdots\!38}a^{2}-\frac{15\!\cdots\!96}{11\!\cdots\!19}a+\frac{12\!\cdots\!63}{23\!\cdots\!38}$, $\frac{15\!\cdots\!99}{23\!\cdots\!38}a^{14}+\frac{26\!\cdots\!01}{11\!\cdots\!19}a^{13}+\frac{22\!\cdots\!48}{11\!\cdots\!19}a^{12}-\frac{62\!\cdots\!85}{11\!\cdots\!19}a^{11}+\frac{52\!\cdots\!07}{23\!\cdots\!38}a^{10}+\frac{59\!\cdots\!40}{11\!\cdots\!19}a^{9}+\frac{17\!\cdots\!09}{23\!\cdots\!38}a^{8}+\frac{11\!\cdots\!96}{11\!\cdots\!19}a^{7}-\frac{35\!\cdots\!29}{23\!\cdots\!38}a^{6}+\frac{44\!\cdots\!45}{23\!\cdots\!38}a^{5}-\frac{34\!\cdots\!35}{23\!\cdots\!38}a^{4}-\frac{15\!\cdots\!06}{11\!\cdots\!19}a^{3}-\frac{42\!\cdots\!09}{23\!\cdots\!38}a^{2}-\frac{34\!\cdots\!99}{11\!\cdots\!19}a+\frac{30\!\cdots\!36}{11\!\cdots\!19}$, $\frac{59\!\cdots\!05}{23\!\cdots\!38}a^{14}+\frac{63\!\cdots\!02}{11\!\cdots\!19}a^{13}+\frac{85\!\cdots\!51}{11\!\cdots\!19}a^{12}+\frac{12\!\cdots\!18}{11\!\cdots\!19}a^{11}+\frac{20\!\cdots\!19}{23\!\cdots\!38}a^{10}+\frac{42\!\cdots\!95}{11\!\cdots\!19}a^{9}+\frac{18\!\cdots\!85}{23\!\cdots\!38}a^{8}+\frac{50\!\cdots\!68}{11\!\cdots\!19}a^{7}+\frac{16\!\cdots\!69}{23\!\cdots\!38}a^{6}+\frac{35\!\cdots\!23}{23\!\cdots\!38}a^{5}-\frac{43\!\cdots\!75}{23\!\cdots\!38}a^{4}-\frac{98\!\cdots\!99}{11\!\cdots\!19}a^{3}-\frac{29\!\cdots\!43}{23\!\cdots\!38}a^{2}-\frac{24\!\cdots\!63}{11\!\cdots\!19}a-\frac{15\!\cdots\!35}{11\!\cdots\!19}$ (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: | \( 66275507.7691 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 66275507.7691 \cdot 5}{2\cdot\sqrt{122463012708006247915836261129}}\cr\approx \mathstrut & 0.233055737937 \end{aligned}\] (assuming GRH)
Galois group
$C_5^2:D_6$ (as 15T18):
A solvable group of order 300 |
The 14 conjugacy class representatives for $((C_5^2 : C_3):C_2):C_2$ |
Character table for $((C_5^2 : C_3):C_2):C_2$ |
Intermediate fields
3.3.321.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 15 sibling: | data not computed |
Degree 25 sibling: | data not computed |
Degree 30 siblings: | data not computed |
Minimal sibling: | 15.3.6575202805284367132329.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}{,}\,{\href{/padicField/2.3.0.1}{3} }$ | R | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | R | ${\href{/padicField/13.3.0.1}{3} }^{5}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.3.0.1}{3} }^{5}$ | ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.5.0.1}{5} }$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }$ | ${\href{/padicField/31.2.0.1}{2} }^{7}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.3.0.1}{3} }^{5}$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.5.0.1}{5} }$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }$ | ${\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.5.0.1 | $x^{5} + 2 x + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
3.10.5.2 | $x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
\(11\) | 11.5.4.5 | $x^{5} + 22$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
11.10.8.3 | $x^{10} - 110 x^{5} - 16819$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ | |
\(107\) | $\Q_{107}$ | $x + 105$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
107.2.1.2 | $x^{2} + 107$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
107.2.1.2 | $x^{2} + 107$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
107.2.1.1 | $x^{2} + 214$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
107.4.2.1 | $x^{4} + 206 x^{3} + 10827 x^{2} + 22454 x + 1146188$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
107.4.2.1 | $x^{4} + 206 x^{3} + 10827 x^{2} + 22454 x + 1146188$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |