Properties

Label 2304.a2
Conductor 23042304
Discriminant 373248-373248
j-invariant 1728 1728
CM yes (D=4D=-4)
Rank 11
Torsion structure Z/2Z\Z/{2}\Z

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Show commands: Magma / Oscar / PariGP / SageMath

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

y2=x3+18xy^2=x^3+18x Copy content Toggle raw display (homogenize, simplify)
y2z=x3+18xz2y^2z=x^3+18xz^2 Copy content Toggle raw display (dehomogenize, simplify)
y2=x3+18xy^2=x^3+18x Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
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)
 

Mordell-Weil group structure

ZZ/2Z\Z \oplus \Z/{2}\Z

magma: MordellWeilGroup(E);
 

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(3,9)(3, 9)0.714741348683870351184389083010.71474134868387035118438908301\infty
(0,0)(0, 0)0022

Integral points

(0,0) \left(0, 0\right) , (3,±9)(3,\pm 9), (6,±18)(6,\pm 18), (72,±612)(72,\pm 612) Copy content Toggle raw display

comment: Integral points
 
sage: E.integral_points()
 
magma: IntegralPoints(E);
 

Invariants

Conductor: NN  =  2304 2304  = 28322^{8} \cdot 3^{2}
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: Δ\Delta  =  373248-373248 = 12936-1 \cdot 2^{9} \cdot 3^{6}
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: jj  =  1728 1728  = 26332^{6} \cdot 3^{3}
comment: j-invariant
 
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
oscar: j_invariant(E)
 
Endomorphism ring: End(E)\mathrm{End}(E) = Z\Z
Geometric endomorphism ring: End(EQ)\mathrm{End}(E_{\overline{\Q}})  =  Z[1]\Z[\sqrt{-1}]    (potential complex multiplication)
sage: E.has_cm()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: ST(E)\mathrm{ST}(E) = N(U(1))N(\mathrm{U}(1))
Faltings height: hFaltingsh_{\mathrm{Faltings}} ≈ 0.24136639615749569049172837375-0.24136639615749569049172837375
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: hstableh_{\mathrm{stable}} ≈ 1.3105329259115095182522750833-1.3105329259115095182522750833
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 
abcabc quality: QQ ≈ 
Szpiro ratio: σm\sigma_{m} ≈ 2.6199515666140662.619951566614066

BSD invariants

Analytic rank: ranr_{\mathrm{an}} = 1 1
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Mordell-Weil rank: rr = 1 1
comment: Rank
 
sage: E.rank()
 
gp: [lower,upper] = ellrank(E)
 
magma: Rank(E);
 
Regulator: Reg(E/Q)\mathrm{Reg}(E/\Q) ≈ 0.714741348683870351184389083010.71474134868387035118438908301
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: Ω\Omega ≈ 1.80027599992145399124071907661.8002759999214539912407190766
comment: Real Period
 
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: pcp\prod_{p}c_p = 8 8  = 222 2\cdot2^{2}
comment: Tamagawa numbers
 
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)tor\#E(\Q)_{\mathrm{tor}} = 22
comment: Torsion order
 
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) ≈ 2.57346339237412660022896739902.5734633923741266002289673990
comment: Special L-value
 
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: Шan{}_{\mathrm{an}}  ≈  11    (rounded)
comment: Order of Sha
 
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 

BSD formula

2.573463392L(E,1)=#Ш(E/Q)ΩEReg(E/Q)pcp#E(Q)tor211.8002760.7147418222.573463392\displaystyle 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

# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)
 
E = EllipticCurve(%s); 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)))
 
/* self-contained Magma code snippet for the BSD formula (checks rank, computes analytic sha) */
 
E := EllipticCurve(%s); 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 invariants

Modular form   2304.2.a.a

q4q5+4q13+2q17+O(q20) q - 4 q^{5} + 4 q^{13} + 2 q^{17} + O(q^{20}) Copy content Toggle raw display

comment: q-expansion of modular form
 
sage: E.q_eigenform(20)
 
\\ 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.

Modular degree: 256
comment: Modular degree
 
sage: E.modular_degree()
 
gp: ellmoddegree(E)
 
magma: ModularDegree(E);
 
Γ0(N) \Gamma_0(N) -optimal: yes
Manin constant: 1
comment: Manin constant
 
magma: ManinConstant(E);
 

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 2 primes pp of bad reduction:

pp Tamagawa number Kodaira symbol Reduction type Root number ordp(N)\mathrm{ord}_p(N) ordp(Δ)\mathrm{ord}_p(\Delta) ordp(den(j))\mathrm{ord}_p(\mathrm{den}(j))
22 22 IIIIII additive -1 8 9 0
33 44 I0I_0^{*} additive -1 2 6 0

comment: Local data
 
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)]
 

Galois representations

The \ell-adic Galois representation has maximal image for all primes \ell except those listed in the table below.

prime \ell mod-\ell image \ell-adic image
22 2B 16.192.9.145

comment: mod p Galois image
 
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 \ell for which the Serre invariants associated to the mod-\ell Galois representation are exceptional.

\ell Reduction type Serre weight Serre conductor
22 additive 44 9=32 9 = 3^{2}
33 additive 66 256=28 256 = 2^{8}

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree dd for d=d= 2.
Its isogeny class 2304.a consists of 2 curves linked by isogenies of degree 2.

Twists

The minimal quadratic twist of this elliptic curve is 256.b2, its twist by 2424.

The minimal quartic twist of this elliptic curve is 32.a3, its quartic twist by 72-72.

Growth of torsion in number fields

The number fields KK of degree less than 24 such that E(K)torsE(K)_{\rm tors} is strictly larger than E(Q)torsE(\Q)_{\rm tors} Z/2Z\cong \Z/{2}\Z are as follows:

[K:Q][K:\Q] KK E(K)torsE(K)_{\rm tors} Base change curve
22 Q(2)\Q(\sqrt{-2}) Z/2ZZ/2Z\Z/2\Z \oplus \Z/2\Z not in database
44 4.2.18432.3 Z/4Z\Z/4\Z not in database
88 8.0.1358954496.9 Z/4ZZ/4Z\Z/4\Z \oplus \Z/4\Z not in database
88 8.4.5435817984.2 Z/8Z\Z/8\Z not in database
88 8.0.5435817984.7 Z/8Z\Z/8\Z not in database
88 8.2.36691771392.4 Z/6Z\Z/6\Z not in database
88 8.0.169869312000.8 Z/10Z\Z/10\Z not in database
1616 16.0.118192468620711297024.15 Z/4ZZ/8Z\Z/4\Z \oplus \Z/8\Z not in database
1616 16.0.1346286087882789617664.2 Z/3ZZ/6Z\Z/3\Z \oplus \Z/6\Z not in database
1616 deg 16 Z/10Z\Z/10\Z not in database
1616 16.0.5385144351531158470656.26 Z/2ZZ/6Z\Z/2\Z \oplus \Z/6\Z not in database
1616 deg 16 Z/2ZZ/10Z\Z/2\Z \oplus \Z/10\Z not in database

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.

Iwasawa invariants

pp 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-invariant(s) - - 3 5,1 1,1 1 1 1,1 1,1 1 1,1 1 1 1,1 1,1
μ\mu-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.

pp-adic regulators

Note: pp-adic regulator data only exists for primes p5p\ge 5 of good ordinary reduction.