Normalized defining polynomial
\( x^{25} + 3x - 2 \)
Invariants
Degree: | $25$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(1131549247851241351828920999584417333817376768\) \(\medspace = 2^{24}\cdot 7\cdot 1777\cdot 19854823\cdot 273087660366642710101853209\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(63.41\) | 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: | not computed | ||
Ramified primes: | \(2\), \(7\), \(1777\), \(19854823\), \(273087660366642710101853209\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{67445\!\cdots\!90073}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $12$ | 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: | $4a^{24}+3a^{23}+2a^{22}+a^{21}+a^{20}+a^{19}+a^{12}-a^{10}+a^{9}-a^{7}+a^{4}-a^{3}+2a+11$, $a^{23}+a^{22}-a^{20}-a^{19}-a^{18}+a^{17}+2a^{16}+3a^{15}-a^{13}-3a^{12}+2a^{9}-a^{8}-3a^{6}-a^{5}-a^{4}+4a^{3}+3a^{2}+2a-3$, $9a^{24}+6a^{23}+4a^{22}+3a^{21}+2a^{20}+a^{19}-a^{18}-2a^{17}-3a^{16}-3a^{15}-4a^{14}-5a^{13}-6a^{12}-6a^{11}-5a^{10}-5a^{9}-5a^{8}-6a^{7}-5a^{6}-4a^{5}-3a^{4}-3a^{3}-4a^{2}-3a+25$, $a^{24}-a^{23}+2a^{22}-a^{21}+a^{19}-2a^{18}+2a^{17}-a^{16}+a^{14}-3a^{13}+2a^{12}-2a^{11}+2a^{9}-4a^{8}+3a^{7}-3a^{6}-a^{5}+3a^{4}-5a^{3}+5a^{2}-3a+1$, $2a^{24}+a^{23}-a^{22}-a^{21}-a^{20}-a^{19}+a^{17}+2a^{16}+a^{15}+a^{14}-a^{13}-a^{12}-a^{11}-2a^{10}-a^{9}+a^{8}+3a^{7}+a^{6}+2a^{5}-2a^{3}-2a^{2}-3a+5$, $10a^{24}+8a^{23}+5a^{22}+3a^{21}+3a^{20}+3a^{18}-2a^{17}+4a^{16}-4a^{15}+5a^{14}-6a^{13}+6a^{12}-7a^{11}+8a^{10}-8a^{9}+9a^{8}-9a^{7}+9a^{6}-9a^{5}+9a^{4}-8a^{3}+8a^{2}-7a+35$, $a^{24}+a^{23}+a^{22}+a^{20}+a^{19}+a^{18}+a^{16}+a^{15}+a^{12}-a^{10}-a^{9}-a^{7}-2a^{6}+a^{5}-a^{2}+2a+3$, $3a^{24}+3a^{23}+2a^{22}-a^{21}-a^{20}-a^{17}+2a^{15}+2a^{12}+2a^{11}+a^{10}-a^{9}+a^{8}-4a^{6}-a^{5}+a^{4}-a^{3}-3a^{2}+13$, $a^{24}-a^{23}-2a^{22}-2a^{21}+3a^{19}+2a^{18}-2a^{16}-2a^{15}-2a^{14}+2a^{13}+3a^{12}+a^{11}-2a^{9}-3a^{8}-2a^{7}+4a^{6}+2a^{5}+3a^{4}-a^{3}-4a^{2}-6a+5$, $14a^{24}+9a^{23}+5a^{22}+4a^{21}+3a^{20}+a^{19}+a^{18}+2a^{17}+2a^{14}+2a^{11}+2a^{8}+a^{7}-a^{6}+2a^{5}+2a^{4}-2a^{3}+2a^{2}+3a+39$, $66a^{24}+43a^{23}+30a^{22}+21a^{21}+13a^{20}+9a^{19}+8a^{18}+6a^{17}+3a^{16}+3a^{15}+5a^{14}+3a^{13}+3a^{11}+5a^{10}+a^{9}+3a^{7}+3a^{6}+3a^{3}+a^{2}-3a+199$, $5a^{24}+a^{23}+2a^{21}+5a^{20}+7a^{19}+6a^{18}+2a^{17}-3a^{16}-6a^{15}-6a^{14}-3a^{13}+a^{12}+4a^{11}+5a^{10}+4a^{9}+2a^{8}-2a^{6}-4a^{5}-5a^{4}-4a^{3}+5a+23$ (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: | \( 22439086804455.46 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{12}\cdot 22439086804455.46 \cdot 1}{2\cdot\sqrt{1131549247851241351828920999584417333817376768}}\cr\approx \mathstrut & 2.52538076794292 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 15511210043330985984000000 |
The 1958 conjugacy class representatives for $S_{25}$ are not computed |
Character table for $S_{25}$ is not computed |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | $20{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.8.0.1}{8} }^{3}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | R | $17{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ | $15{,}\,{\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $25$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.8.8.13 | $x^{8} + 2 x + 2$ | $8$ | $1$ | $8$ | $C_2^3:(C_7: C_3)$ | $[8/7, 8/7, 8/7]_{7}^{3}$ | |
Deg $16$ | $8$ | $2$ | $16$ | ||||
\(7\) | 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.23.0.1 | $x^{23} + 4 x^{2} + 4 x + 4$ | $1$ | $23$ | $0$ | $C_{23}$ | $[\ ]^{23}$ | |
\(1777\) | $\Q_{1777}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{1777}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{1777}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
\(19854823\) | $\Q_{19854823}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{19854823}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
\(273\!\cdots\!209\) | $\Q_{27\!\cdots\!09}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ |