sage: E = EllipticCurve([0, 0, 0, 18, 0])
gp: E = ellinit([0, 0, 0, 18, 0])
magma: E := EllipticCurve([0, 0, 0, 18, 0]);
oscar: E = elliptic_curve([0, 0, 0, 18, 0])
sage: E.short_weierstrass_model()
magma: WeierstrassModel(E);
oscar: short_weierstrass_model(E)
Z ⊕ Z / 2 Z \Z \oplus \Z/{2}\Z Z ⊕ Z / 2 Z
magma: MordellWeilGroup(E);
P P P h ^ ( P ) \hat{h}(P) h ^ ( P ) Order
( 3 , 9 ) (3, 9) ( 3 , 9 ) 0.71474134868387035118438908301 0.71474134868387035118438908301 0 . 7 1 4 7 4 1 3 4 8 6 8 3 8 7 0 3 5 1 1 8 4 3 8 9 0 8 3 0 1 ∞ \infty ∞
( 0 , 0 ) (0, 0) ( 0 , 0 ) 0 0 0 2 2 2
( 0 , 0 ) \left(0, 0\right) ( 0 , 0 ) , ( 3 , ± 9 ) (3,\pm 9) ( 3 , ± 9 ) , ( 6 , ± 18 ) (6,\pm 18) ( 6 , ± 1 8 ) , ( 72 , ± 612 ) (72,\pm 612) ( 7 2 , ± 6 1 2 )
sage: E.integral_points()
magma: IntegralPoints(E);
Invariants
Conductor :
N N N
=
2304 2304 2 3 0 4 = 2 8 ⋅ 3 2 2^{8} \cdot 3^{2} 2 8 ⋅ 3 2
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
Discriminant :
Δ \Delta Δ
=
− 373248 -373248 − 3 7 3 2 4 8 = − 1 ⋅ 2 9 ⋅ 3 6 -1 \cdot 2^{9} \cdot 3^{6} − 1 ⋅ 2 9 ⋅ 3 6
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
j-invariant :
j j j
=
1728 1728 1 7 2 8 = 2 6 ⋅ 3 3 2^{6} \cdot 3^{3} 2 6 ⋅ 3 3
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
Endomorphism ring :
E n d ( E ) \mathrm{End}(E) E n d ( E ) = Z \Z Z
Geometric endomorphism ring :
E n d ( E Q ‾ ) \mathrm{End}(E_{\overline{\Q}}) E n d ( E Q )
=
Z [ − 1 ] \Z[\sqrt{-1}] Z [ − 1 ]
(potential complex multiplication )
sage: E.has_cm()
magma: HasComplexMultiplication(E);
Sato-Tate group :
S T ( E ) \mathrm{ST}(E) S T ( E ) = N ( U ( 1 ) ) N(\mathrm{U}(1)) N ( U ( 1 ) )
Faltings height :
h F a l t i n g s h_{\mathrm{Faltings}} h F a l t i n g s ≈ − 0.24136639615749569049172837375 -0.24136639615749569049172837375 − 0 . 2 4 1 3 6 6 3 9 6 1 5 7 4 9 5 6 9 0 4 9 1 7 2 8 3 7 3 7 5
gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
Stable Faltings height :
h s t a b l e h_{\mathrm{stable}} h s t a b l e ≈ − 1.3105329259115095182522750833 -1.3105329259115095182522750833 − 1 . 3 1 0 5 3 2 9 2 5 9 1 1 5 0 9 5 1 8 2 5 2 2 7 5 0 8 3 3
magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
a b c abc a b c quality :
Q Q Q ≈
Szpiro ratio :
σ m \sigma_{m} σ m ≈ 2.619951566614066 2.619951566614066 2 . 6 1 9 9 5 1 5 6 6 6 1 4 0 6 6
Analytic rank :
r a n r_{\mathrm{an}} r a n = 1 1 1
sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
Mordell-Weil rank :
r r r = 1 1 1
sage: E.rank()
gp: [lower,upper] = ellrank(E)
magma: Rank(E);
Regulator :
R e g ( E / Q ) \mathrm{Reg}(E/\Q) R e g ( E / Q ) ≈ 0.71474134868387035118438908301 0.71474134868387035118438908301 0 . 7 1 4 7 4 1 3 4 8 6 8 3 8 7 0 3 5 1 1 8 4 3 8 9 0 8 3 0 1
sage: E.regulator()
gp: G = E.gen \\ if available
matdet(ellheightmatrix(E,G))
magma: Regulator(E);
Real period :
Ω \Omega Ω ≈ 1.8002759999214539912407190766 1.8002759999214539912407190766 1 . 8 0 0 2 7 5 9 9 9 9 2 1 4 5 3 9 9 1 2 4 0 7 1 9 0 7 6 6
sage: E.period_lattice().omega()
gp: if(E.disc>0,2,1)*E.omega[1]
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
Tamagawa product :
∏ p c p \prod_{p}c_p ∏ p c p = 8 8 8
= 2 ⋅ 2 2 2\cdot2^{2} 2 ⋅ 2 2
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
oscar: tamagawa_numbers(E)
Torsion order :
# E ( Q ) t o r \#E(\Q)_{\mathrm{tor}} # E ( Q ) t o r = 2 2 2
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
Special value :
L ′ ( E , 1 ) L'(E,1) L ′ ( E , 1 ) ≈ 2.5734633923741266002289673990 2.5734633923741266002289673990 2 . 5 7 3 4 6 3 3 9 2 3 7 4 1 2 6 6 0 0 2 2 8 9 6 7 3 9 9 0
sage: r = E.rank();
E.lseries().dokchitser().derivative(1,r)/r.factorial()
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
Analytic order of Ш :
Шa n {}_{\mathrm{an}} a n
≈
1 1 1
(rounded )
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
2.573463392 ≈ L ′ ( E , 1 ) = # Ш ( E / Q ) ⋅ Ω E ⋅ R e g ( E / Q ) ⋅ ∏ p c p # E ( Q ) t o r 2 ≈ 1 ⋅ 1.800276 ⋅ 0.714741 ⋅ 8 2 2 ≈ 2.573463392 \begin{aligned} 2.573463392 \approx L'(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.800276 \cdot 0.714741 \cdot 8}{2^2} \\ & \approx 2.573463392\end{aligned} 2 . 5 7 3 4 6 3 3 9 2 ≈ L ′ ( E , 1 ) = # E ( Q ) t o r 2 # Ш ( E / Q ) ⋅ Ω E ⋅ R e g ( E / Q ) ⋅ ∏ p c p ≈ 2 2 1 ⋅ 1 . 8 0 0 2 7 6 ⋅ 0 . 7 1 4 7 4 1 ⋅ 8 ≈ 2 . 5 7 3 4 6 3 3 9 2
sage: # self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)
E = EllipticCurve([0, 0, 0, 18, 0]); r = E.rank(); ar = E.analytic_rank(); assert r == ar;
Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical();
omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order();
assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))
magma: /* self-contained Magma code snippet for the BSD formula (checks rank, computes analytic sha) */
E := EllipticCurve([0, 0, 0, 18, 0]); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar;
sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1);
reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E);
assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);
Modular form
2304.2.a.a
q − 4 q 5 + 4 q 13 + 2 q 17 + O ( q 20 ) q - 4 q^{5} + 4 q^{13} + 2 q^{17} + O(q^{20}) q − 4 q 5 + 4 q 1 3 + 2 q 1 7 + O ( q 2 0 )
sage: E.q_eigenform(20)
gp: \\ actual modular form, use for small N
[mf,F] = mffromell(E)
Ser(mfcoefs(mf,20),q)
\\ or just the series
Ser(ellan(E,20),q)*q
magma: ModularForm(E);
For more coefficients, see the Downloads section to the right.
This elliptic curve is not semistable .
There
are 2 primes p p p
of bad reduction :
sage: E.local_data()
gp: ellglobalred(E)[5]
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
oscar: [(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]
The ℓ \ell ℓ -adic Galois representation has maximal image
for all primes ℓ \ell ℓ except those listed in the table below.
sage: rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
The table below list all primes ℓ \ell ℓ for which the Serre invariants associated to the mod-ℓ \ell ℓ Galois representation are exceptional.
gp: ellisomat(E)
This curve has non-trivial cyclic isogenies of degree d d d for d = d= d =
2.
Its isogeny class 2304.a
consists of 2 curves linked by isogenies of
degree 2.
The minimal quadratic twist of this elliptic curve is
256.b2 , its twist by 24 24 2 4 .
The minimal quartic twist of this elliptic curve is
32.a3 ,
its quartic twist by − 72 -72 − 7 2 .
The number fields K K K of degree less than 24 such that
E ( K ) t o r s E(K)_{\rm tors} E ( K ) t o r s is strictly larger than E ( Q ) t o r s E(\Q)_{\rm tors} E ( Q ) t o r s
≅ Z / 2 Z \cong \Z/{2}\Z ≅ Z / 2 Z
are as follows:
We only show fields where the torsion growth is primitive .
For fields not in the database, click on the degree shown to reveal the defining polynomial.
p p p
2
3
5
7
11
13
17
19
23
29
31
37
41
43
47
Reduction type
add
add
ord
ss
ss
ord
ord
ss
ss
ord
ss
ord
ord
ss
ss
λ \lambda λ -invariant(s)
-
-
3
5,1
1,1
1
1
1,1
1,1
1
1,1
1
1
1,1
1,1
μ \mu μ -invariant(s)
-
-
0
0,0
0,0
0
0
0,0
0,0
0
0,0
0
0
0,0
0,0
An entry - indicates that the invariants are not computed because the reduction is additive.
p p p -adic regulators
Note: p p p -adic regulator data only exists for primes p ≥ 5 p\ge 5 p ≥ 5 of good ordinary
reduction.
Choose a prime...
11
19
23
37
43
47
61
71
83