Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^37 + 2*x - 1)
gp: K = bnfinit(y^37 + 2*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^37 + 2*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^37 + 2*x - 1)
\( x^{37} + 2x - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $37$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[1, 18]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(14621767283680774406142649554727584574959674913509640451717805466709\)
\(\medspace = 13\cdot 79\cdot 3203\cdot 230203\cdot 34944191\cdot 52002217\cdot 4124383231\cdot 25\!\cdots\!59\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(65.35\) | 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: | $13^{1/2}79^{1/2}3203^{1/2}230203^{1/2}34944191^{1/2}52002217^{1/2}4124383231^{1/2}2576354638936903184957678084959^{1/2}\approx 3.823841953282166e+33$ | ||
| Ramified primes: |
\(13\), \(79\), \(3203\), \(230203\), \(34944191\), \(52002217\), \(4124383231\), \(25763\!\cdots\!84959\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{14621\!\cdots\!66709}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $18$ | 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: |
$a^{36}+2$, $a^{24}-a^{12}+1$, $a^{8}-a^{4}+1$, $a^{36}+a^{3}-a^{2}+a+1$, $a^{36}+a^{5}-a^{4}+a^{3}-a^{2}+a+1$, $a^{36}+a^{35}-a^{30}+a^{25}-a^{20}+a^{15}-a^{10}+a^{5}+1$, $a^{36}+a^{35}+a^{33}-a^{32}+a^{31}-a^{26}+2a^{25}-2a^{24}+a^{23}-a^{22}-2a^{18}+a^{17}-2a^{16}+a^{15}-a^{14}-a^{13}-a^{10}+a^{9}-3a^{8}+a^{7}-2a^{5}+a^{4}-2a^{3}+a+1$, $a^{34}+a^{33}-a^{32}-3a^{31}-2a^{30}-a^{28}-2a^{27}-2a^{26}-a^{24}-4a^{23}-4a^{22}-2a^{21}-2a^{19}-3a^{18}-2a^{17}-a^{16}-3a^{15}-4a^{14}-3a^{13}-a^{11}-2a^{10}-2a^{9}-a^{8}-2a^{7}-3a^{6}-2a^{5}+2a^{4}-2a+2$, $2a^{36}+a^{35}+3a^{33}+2a^{32}+a^{31}-3a^{30}-4a^{29}-2a^{28}-a^{27}+a^{26}-2a^{25}-2a^{24}-a^{23}+3a^{22}+4a^{21}+3a^{20}-a^{19}-a^{18}-a^{17}+2a^{16}+a^{15}-3a^{14}-3a^{13}-4a^{12}+3a^{11}+2a^{10}+3a^{9}-a^{8}-a^{7}+2a^{6}+4a^{5}+3a^{4}-2a^{3}-5a^{2}-4a+4$, $2a^{36}+2a^{35}-a^{34}-4a^{33}-4a^{32}-2a^{31}+3a^{30}+5a^{29}+2a^{28}+a^{27}-a^{26}-5a^{25}-4a^{24}-a^{23}+2a^{22}+7a^{21}+4a^{20}-2a^{19}-a^{18}-3a^{17}-6a^{16}-a^{15}+4a^{14}+6a^{13}+4a^{12}-2a^{11}-3a^{10}-3a^{9}-6a^{8}+6a^{6}+4a^{5}+5a^{4}+a^{3}-8a^{2}-5a+3$, $3a^{36}+2a^{35}-a^{34}-2a^{33}+3a^{31}+3a^{30}-3a^{28}-4a^{27}-3a^{26}-2a^{25}-a^{24}+a^{22}-2a^{20}-2a^{19}+a^{18}+4a^{17}+3a^{16}-a^{15}-3a^{14}+4a^{12}+6a^{11}+3a^{10}-2a^{8}-2a^{7}-2a^{6}-2a^{5}+a^{3}-4a+1$, $3a^{36}+2a^{35}-7a^{34}+7a^{33}-4a^{32}-3a^{31}+7a^{30}-8a^{29}+4a^{28}+2a^{27}-7a^{26}+9a^{25}-5a^{24}-a^{23}+9a^{22}-10a^{21}+7a^{20}+2a^{19}-8a^{18}+12a^{17}-8a^{16}-a^{15}+10a^{14}-15a^{13}+9a^{12}+a^{11}-12a^{10}+15a^{9}-11a^{8}-a^{7}+13a^{6}-19a^{5}+12a^{4}+2a^{3}-15a^{2}+20a-6$, $3a^{36}+2a^{35}+2a^{34}+a^{33}-a^{32}-3a^{30}+a^{29}-3a^{28}+3a^{27}-a^{26}+4a^{25}+a^{24}+3a^{23}-a^{21}-a^{20}-4a^{19}+2a^{18}-2a^{17}+5a^{16}+5a^{14}-2a^{13}+3a^{12}-3a^{11}-a^{10}-a^{9}-3a^{8}+2a^{7}-2a^{6}+6a^{5}-2a^{4}+7a^{3}-5a^{2}+2a$, $a^{36}-2a^{35}-a^{34}+2a^{33}-2a^{32}+2a^{31}-a^{29}+2a^{27}-2a^{26}+a^{25}-a^{24}-a^{23}+a^{22}+2a^{21}-3a^{20}+3a^{19}-2a^{18}+2a^{16}-2a^{15}-a^{14}+2a^{13}-3a^{12}+2a^{11}+a^{10}-2a^{9}+a^{8}+a^{7}-3a^{6}+a^{5}+2a^{4}-5a^{3}+4a^{2}-a-1$, $5a^{36}+2a^{35}-5a^{34}-2a^{33}+4a^{32}+3a^{31}-4a^{30}-4a^{29}+4a^{28}+4a^{27}-4a^{26}-5a^{25}+3a^{24}+6a^{23}-3a^{22}-6a^{21}+a^{20}+6a^{19}+a^{18}-7a^{17}-2a^{16}+7a^{15}+3a^{14}-6a^{13}-4a^{12}+7a^{11}+5a^{10}-6a^{9}-5a^{8}+5a^{7}+8a^{6}-5a^{5}-10a^{4}+4a^{3}+10a^{2}-3a-2$, $6a^{36}+3a^{35}-6a^{33}-5a^{32}-7a^{31}-2a^{30}+a^{29}+5a^{28}+8a^{27}+5a^{26}+5a^{25}-3a^{24}-3a^{23}-8a^{22}-6a^{21}-3a^{20}-a^{19}+6a^{18}+4a^{17}+6a^{16}+a^{15}-3a^{14}-5a^{13}-9a^{12}-4a^{11}-4a^{10}+6a^{9}+6a^{8}+12a^{7}+8a^{6}+4a^{5}+a^{4}-10a^{3}-6a^{2}-13a+8$, $6a^{36}-3a^{35}-6a^{34}+4a^{33}-5a^{31}+2a^{30}+7a^{29}-2a^{28}-11a^{27}+5a^{26}+10a^{25}-a^{24}-9a^{23}+2a^{22}+6a^{21}-6a^{20}-a^{19}+7a^{18}+a^{17}-13a^{16}-a^{15}+15a^{14}+a^{13}-12a^{12}-2a^{11}+11a^{10}-a^{9}-8a^{8}+8a^{7}+3a^{6}-11a^{5}-9a^{4}+17a^{3}+8a^{2}-17a+6$, $a^{36}+4a^{35}+5a^{34}+2a^{33}+a^{32}+2a^{31}+2a^{30}+a^{29}+a^{28}+2a^{27}+a^{26}-2a^{25}-2a^{24}+2a^{23}+2a^{22}-4a^{21}-4a^{20}+a^{19}-a^{18}-5a^{17}-2a^{16}+a^{15}-6a^{14}-8a^{13}+2a^{12}+a^{11}-9a^{10}-7a^{9}+a^{8}-2a^{7}-8a^{6}-2a^{5}+a^{4}-8a^{3}-7a^{2}+2a+3$
| 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: | \( 9551296549801247000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{18}\cdot 9551296549801247000 \cdot 1}{2\cdot\sqrt{14621767283680774406142649554727584574959674913509640451717805466709}}\cr\approx \mathstrut & 0.581835245875516 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^37 + 2*x - 1)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^37 + 2*x - 1, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^37 + 2*x - 1);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^37 + 2*x - 1);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A non-solvable group of order 13763753091226345046315979581580902400000000 |
| The 21637 conjugacy class representatives for $S_{37}$ are not computed |
| Character table for $S_{37}$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $36{,}\,{\href{/padicField/2.1.0.1}{1} }$ | $29{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | $19{,}\,17{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/7.13.0.1}{13} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $28{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | R | $27{,}\,{\href{/padicField/17.10.0.1}{10} }$ | $21{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $33{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.5.0.1}{5} }$ | $32{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | $36{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $27{,}\,{\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | $15{,}\,{\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(13\)
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.8.0.1 | $x^{8} + 8 x^{4} + 12 x^{3} + 2 x^{2} + 3 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 13.9.0.1 | $x^{9} + 12 x^{4} + 8 x^{3} + 12 x^{2} + 12 x + 11$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
| 13.16.0.1 | $x^{16} + 3 x^{8} + 12 x^{7} + 8 x^{6} + 2 x^{5} + 12 x^{4} + 9 x^{3} + 12 x^{2} + 6 x + 2$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | |
|
\(79\)
| $\Q_{79}$ | $x + 76$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 79.3.0.1 | $x^{3} + 9 x + 76$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 79.3.0.1 | $x^{3} + 9 x + 76$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| Deg $28$ | $1$ | $28$ | $0$ | $C_{28}$ | $[\ ]^{28}$ | ||
|
\(3203\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ | ||
|
\(230203\)
| $\Q_{230203}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
| Deg $21$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ | ||
|
\(34944191\)
| $\Q_{34944191}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{34944191}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{34944191}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $32$ | $1$ | $32$ | $0$ | 32T33 | $[\ ]^{32}$ | ||
|
\(52002217\)
| $\Q_{52002217}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $28$ | $1$ | $28$ | $0$ | $C_{28}$ | $[\ ]^{28}$ | ||
|
\(4124383231\)
| $\Q_{4124383231}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $34$ | $1$ | $34$ | $0$ | $C_{34}$ | $[\ ]^{34}$ | ||
|
\(257\!\cdots\!959\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $32$ | $1$ | $32$ | $0$ | 32T33 | $[\ ]^{32}$ |