Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 18*x^12 - 13*x^11 + 46*x^10 + 211*x^9 + 206*x^8 - 328*x^7 - 386*x^6 - 752*x^5 + 299*x^4 + 213*x^3 - 38*x^2 - 7*x + 1)
gp: K = bnfinit(y^14 - 18*y^12 - 13*y^11 + 46*y^10 + 211*y^9 + 206*y^8 - 328*y^7 - 386*y^6 - 752*y^5 + 299*y^4 + 213*y^3 - 38*y^2 - 7*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 18*x^12 - 13*x^11 + 46*x^10 + 211*x^9 + 206*x^8 - 328*x^7 - 386*x^6 - 752*x^5 + 299*x^4 + 213*x^3 - 38*x^2 - 7*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 18*x^12 - 13*x^11 + 46*x^10 + 211*x^9 + 206*x^8 - 328*x^7 - 386*x^6 - 752*x^5 + 299*x^4 + 213*x^3 - 38*x^2 - 7*x + 1)
\( x^{14} - 18 x^{12} - 13 x^{11} + 46 x^{10} + 211 x^{9} + 206 x^{8} - 328 x^{7} - 386 x^{6} - 752 x^{5} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[10, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(3598465942658427991657\)
\(\medspace = 13^{2}\cdot 577^{7}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(34.65\) | 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: | $13^{1/2}577^{1/2}\approx 86.6083136886985$ | ||
| Ramified primes: |
\(13\), \(577\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{577}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7}a^{12}-\frac{2}{7}a^{11}-\frac{2}{7}a^{10}+\frac{2}{7}a^{9}-\frac{3}{7}a^{8}+\frac{3}{7}a^{7}+\frac{3}{7}a^{6}+\frac{3}{7}a^{5}+\frac{1}{7}a^{4}+\frac{3}{7}a^{3}-\frac{3}{7}a^{2}+\frac{3}{7}a-\frac{3}{7}$, $\frac{1}{70\!\cdots\!95}a^{13}-\frac{76602592032608}{70\!\cdots\!95}a^{12}-\frac{687798148899109}{70\!\cdots\!95}a^{11}-\frac{18\!\cdots\!46}{70\!\cdots\!95}a^{10}+\frac{31\!\cdots\!14}{70\!\cdots\!95}a^{9}-\frac{19\!\cdots\!46}{70\!\cdots\!95}a^{8}+\frac{942885274984584}{70\!\cdots\!95}a^{7}+\frac{609473038081645}{14\!\cdots\!39}a^{6}+\frac{624877430214904}{70\!\cdots\!95}a^{5}+\frac{97508508264593}{10\!\cdots\!85}a^{4}+\frac{13\!\cdots\!11}{70\!\cdots\!95}a^{3}-\frac{39099058427785}{14\!\cdots\!39}a^{2}+\frac{13\!\cdots\!17}{70\!\cdots\!95}a-\frac{879203960528808}{70\!\cdots\!95}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
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: | $11$ | 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: |
$\frac{24\!\cdots\!17}{70\!\cdots\!95}a^{13}-\frac{431086630745004}{70\!\cdots\!95}a^{12}+\frac{43\!\cdots\!78}{70\!\cdots\!95}a^{11}+\frac{39\!\cdots\!17}{70\!\cdots\!95}a^{10}-\frac{10\!\cdots\!93}{70\!\cdots\!95}a^{9}-\frac{53\!\cdots\!73}{70\!\cdots\!95}a^{8}-\frac{58\!\cdots\!03}{70\!\cdots\!95}a^{7}+\frac{14\!\cdots\!24}{14\!\cdots\!39}a^{6}+\frac{10\!\cdots\!82}{70\!\cdots\!95}a^{5}+\frac{19\!\cdots\!38}{70\!\cdots\!95}a^{4}-\frac{40\!\cdots\!87}{70\!\cdots\!95}a^{3}-\frac{13\!\cdots\!67}{14\!\cdots\!39}a^{2}-\frac{98\!\cdots\!59}{70\!\cdots\!95}a+\frac{44\!\cdots\!18}{10\!\cdots\!85}$, $\frac{322815302439019}{70\!\cdots\!95}a^{13}+\frac{567727444977848}{70\!\cdots\!95}a^{12}-\frac{57\!\cdots\!01}{70\!\cdots\!95}a^{11}-\frac{14\!\cdots\!74}{70\!\cdots\!95}a^{10}+\frac{933253932788153}{10\!\cdots\!85}a^{9}+\frac{92\!\cdots\!96}{70\!\cdots\!95}a^{8}+\frac{18\!\cdots\!81}{70\!\cdots\!95}a^{7}+\frac{50\!\cdots\!40}{14\!\cdots\!39}a^{6}-\frac{28\!\cdots\!79}{70\!\cdots\!95}a^{5}-\frac{46\!\cdots\!41}{70\!\cdots\!95}a^{4}-\frac{37\!\cdots\!51}{70\!\cdots\!95}a^{3}+\frac{36\!\cdots\!26}{14\!\cdots\!39}a^{2}+\frac{75\!\cdots\!58}{70\!\cdots\!95}a+\frac{45\!\cdots\!48}{70\!\cdots\!95}$, $\frac{330064322436055}{14\!\cdots\!39}a^{13}-\frac{190512884937170}{14\!\cdots\!39}a^{12}+\frac{59\!\cdots\!90}{14\!\cdots\!39}a^{11}+\frac{77\!\cdots\!43}{14\!\cdots\!39}a^{10}-\frac{17\!\cdots\!08}{200197659270377}a^{9}-\frac{78\!\cdots\!96}{14\!\cdots\!39}a^{8}-\frac{10\!\cdots\!52}{14\!\cdots\!39}a^{7}+\frac{67\!\cdots\!87}{14\!\cdots\!39}a^{6}+\frac{18\!\cdots\!34}{14\!\cdots\!39}a^{5}+\frac{32\!\cdots\!92}{14\!\cdots\!39}a^{4}+\frac{50\!\cdots\!13}{14\!\cdots\!39}a^{3}-\frac{11\!\cdots\!68}{14\!\cdots\!39}a^{2}-\frac{20\!\cdots\!64}{14\!\cdots\!39}a+\frac{49\!\cdots\!93}{14\!\cdots\!39}$, $\frac{50\!\cdots\!84}{70\!\cdots\!95}a^{13}+\frac{534095711454518}{70\!\cdots\!95}a^{12}-\frac{91\!\cdots\!11}{70\!\cdots\!95}a^{11}-\frac{75\!\cdots\!44}{70\!\cdots\!95}a^{10}+\frac{22\!\cdots\!81}{70\!\cdots\!95}a^{9}+\frac{10\!\cdots\!66}{70\!\cdots\!95}a^{8}+\frac{11\!\cdots\!11}{70\!\cdots\!95}a^{7}-\frac{30\!\cdots\!29}{14\!\cdots\!39}a^{6}-\frac{21\!\cdots\!84}{70\!\cdots\!95}a^{5}-\frac{40\!\cdots\!21}{70\!\cdots\!95}a^{4}+\frac{10\!\cdots\!54}{70\!\cdots\!95}a^{3}+\frac{23\!\cdots\!82}{14\!\cdots\!39}a^{2}-\frac{59\!\cdots\!97}{70\!\cdots\!95}a-\frac{32\!\cdots\!47}{70\!\cdots\!95}$, $\frac{26259158605966}{10\!\cdots\!85}a^{13}-\frac{252724888783324}{70\!\cdots\!95}a^{12}+\frac{33\!\cdots\!98}{70\!\cdots\!95}a^{11}+\frac{69\!\cdots\!42}{70\!\cdots\!95}a^{10}-\frac{60\!\cdots\!88}{70\!\cdots\!95}a^{9}-\frac{50\!\cdots\!73}{70\!\cdots\!95}a^{8}-\frac{89\!\cdots\!48}{70\!\cdots\!95}a^{7}+\frac{35\!\cdots\!48}{14\!\cdots\!39}a^{6}+\frac{16\!\cdots\!22}{70\!\cdots\!95}a^{5}+\frac{22\!\cdots\!58}{70\!\cdots\!95}a^{4}+\frac{12\!\cdots\!28}{70\!\cdots\!95}a^{3}-\frac{33\!\cdots\!19}{14\!\cdots\!39}a^{2}-\frac{48\!\cdots\!19}{70\!\cdots\!95}a+\frac{42\!\cdots\!26}{70\!\cdots\!95}$, $\frac{580616714261903}{70\!\cdots\!95}a^{13}-\frac{358086391915074}{70\!\cdots\!95}a^{12}-\frac{10\!\cdots\!32}{70\!\cdots\!95}a^{11}-\frac{160521174044099}{10\!\cdots\!85}a^{10}+\frac{30\!\cdots\!27}{70\!\cdots\!95}a^{9}+\frac{15\!\cdots\!61}{10\!\cdots\!85}a^{8}+\frac{45\!\cdots\!42}{70\!\cdots\!95}a^{7}-\frac{51\!\cdots\!66}{14\!\cdots\!39}a^{6}-\frac{10\!\cdots\!03}{70\!\cdots\!95}a^{5}-\frac{31\!\cdots\!27}{70\!\cdots\!95}a^{4}+\frac{62\!\cdots\!24}{10\!\cdots\!85}a^{3}-\frac{236969904443726}{200197659270377}a^{2}-\frac{11\!\cdots\!67}{10\!\cdots\!85}a+\frac{11\!\cdots\!01}{70\!\cdots\!95}$, $a$, $\frac{73077217794234}{10\!\cdots\!85}a^{13}+\frac{139486873526011}{70\!\cdots\!95}a^{12}-\frac{91\!\cdots\!82}{70\!\cdots\!95}a^{11}-\frac{92\!\cdots\!83}{70\!\cdots\!95}a^{10}+\frac{21\!\cdots\!42}{70\!\cdots\!95}a^{9}+\frac{11\!\cdots\!47}{70\!\cdots\!95}a^{8}+\frac{13\!\cdots\!07}{70\!\cdots\!95}a^{7}-\frac{27\!\cdots\!41}{14\!\cdots\!39}a^{6}-\frac{25\!\cdots\!33}{70\!\cdots\!95}a^{5}-\frac{45\!\cdots\!07}{70\!\cdots\!95}a^{4}+\frac{59\!\cdots\!33}{70\!\cdots\!95}a^{3}+\frac{31\!\cdots\!26}{14\!\cdots\!39}a^{2}+\frac{59\!\cdots\!16}{70\!\cdots\!95}a-\frac{16\!\cdots\!24}{70\!\cdots\!95}$, $\frac{12\!\cdots\!19}{70\!\cdots\!95}a^{13}-\frac{19\!\cdots\!73}{70\!\cdots\!95}a^{12}+\frac{21\!\cdots\!56}{70\!\cdots\!95}a^{11}+\frac{19\!\cdots\!44}{70\!\cdots\!95}a^{10}-\frac{75\!\cdots\!28}{10\!\cdots\!85}a^{9}-\frac{26\!\cdots\!86}{70\!\cdots\!95}a^{8}-\frac{29\!\cdots\!41}{70\!\cdots\!95}a^{7}+\frac{70\!\cdots\!38}{14\!\cdots\!39}a^{6}+\frac{52\!\cdots\!49}{70\!\cdots\!95}a^{5}+\frac{99\!\cdots\!51}{70\!\cdots\!95}a^{4}-\frac{20\!\cdots\!04}{70\!\cdots\!95}a^{3}-\frac{58\!\cdots\!39}{14\!\cdots\!39}a^{2}+\frac{21\!\cdots\!77}{70\!\cdots\!95}a+\frac{81\!\cdots\!72}{70\!\cdots\!95}$, $\frac{68\!\cdots\!92}{70\!\cdots\!95}a^{13}-\frac{14\!\cdots\!94}{70\!\cdots\!95}a^{12}+\frac{12\!\cdots\!38}{70\!\cdots\!95}a^{11}+\frac{11\!\cdots\!17}{70\!\cdots\!95}a^{10}-\frac{29\!\cdots\!23}{70\!\cdots\!95}a^{9}-\frac{15\!\cdots\!83}{70\!\cdots\!95}a^{8}-\frac{17\!\cdots\!08}{70\!\cdots\!95}a^{7}+\frac{37\!\cdots\!70}{14\!\cdots\!39}a^{6}+\frac{30\!\cdots\!37}{70\!\cdots\!95}a^{5}+\frac{57\!\cdots\!63}{70\!\cdots\!95}a^{4}-\frac{86\!\cdots\!57}{70\!\cdots\!95}a^{3}-\frac{35\!\cdots\!80}{14\!\cdots\!39}a^{2}-\frac{11\!\cdots\!64}{70\!\cdots\!95}a+\frac{58\!\cdots\!28}{10\!\cdots\!85}$, $\frac{48\!\cdots\!98}{70\!\cdots\!95}a^{13}-\frac{7955381337472}{10\!\cdots\!85}a^{12}-\frac{87\!\cdots\!87}{70\!\cdots\!95}a^{11}-\frac{62\!\cdots\!28}{70\!\cdots\!95}a^{10}+\frac{22\!\cdots\!72}{70\!\cdots\!95}a^{9}+\frac{10\!\cdots\!07}{70\!\cdots\!95}a^{8}+\frac{99\!\cdots\!67}{70\!\cdots\!95}a^{7}-\frac{31\!\cdots\!23}{14\!\cdots\!39}a^{6}-\frac{26\!\cdots\!84}{10\!\cdots\!85}a^{5}-\frac{36\!\cdots\!62}{70\!\cdots\!95}a^{4}+\frac{14\!\cdots\!73}{70\!\cdots\!95}a^{3}+\frac{19\!\cdots\!86}{14\!\cdots\!39}a^{2}-\frac{68\!\cdots\!14}{70\!\cdots\!95}a-\frac{27\!\cdots\!74}{70\!\cdots\!95}$
| 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: | \( 510717.70251821855 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{2}\cdot 510717.70251821855 \cdot 2}{2\cdot\sqrt{3598465942658427991657}}\cr\approx \mathstrut & 0.344177049048315 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^14 - 18*x^12 - 13*x^11 + 46*x^10 + 211*x^9 + 206*x^8 - 328*x^7 - 386*x^6 - 752*x^5 + 299*x^4 + 213*x^3 - 38*x^2 - 7*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^14 - 18*x^12 - 13*x^11 + 46*x^10 + 211*x^9 + 206*x^8 - 328*x^7 - 386*x^6 - 752*x^5 + 299*x^4 + 213*x^3 - 38*x^2 - 7*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^14 - 18*x^12 - 13*x^11 + 46*x^10 + 211*x^9 + 206*x^8 - 328*x^7 - 386*x^6 - 752*x^5 + 299*x^4 + 213*x^3 - 38*x^2 - 7*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^14 - 18*x^12 - 13*x^11 + 46*x^10 + 211*x^9 + 206*x^8 - 328*x^7 - 386*x^6 - 752*x^5 + 299*x^4 + 213*x^3 - 38*x^2 - 7*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
$C_2^6:D_7$ (as 14T27):
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 solvable group of order 896 |
| The 20 conjugacy class representatives for $C_2^6:D_7$ |
| Character table for $C_2^6:D_7$ |
Intermediate fields
| 7.7.192100033.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
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]
Sibling fields
| Degree 14 siblings: | data not computed |
| Degree 16 sibling: | data not computed |
| Degree 28 siblings: | data not computed |
| 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 | ${\href{/padicField/2.7.0.1}{7} }^{2}$ | ${\href{/padicField/3.7.0.1}{7} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{4}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}$ | ${\href{/padicField/11.7.0.1}{7} }^{2}$ | R | ${\href{/padicField/17.7.0.1}{7} }^{2}$ | ${\href{/padicField/19.7.0.1}{7} }^{2}$ | ${\href{/padicField/23.7.0.1}{7} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.7.0.1}{7} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.7.0.1}{7} }^{2}$ | ${\href{/padicField/59.7.0.1}{7} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(13\)
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
|
\(577\)
| 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 $4$ | $2$ | $2$ | $2$ |