Properties

Label 438993.a1
Conductor $438993$
Discriminant $-31168503$
j-invariant \( -\frac{32312399988051}{1154389} \)
CM no
Rank $1$
Torsion structure trivial

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy+y=x^3-x^2-1991x-33690\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz+yz^2=x^3-x^2z-1991xz^2-33690z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-31851x-2187994\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([1, -1, 1, -1991, -33690])
 
gp: E = ellinit([1, -1, 1, -1991, -33690])
 
magma: E := EllipticCurve([1, -1, 1, -1991, -33690]);
 
oscar: E = EllipticCurve([1, -1, 1, -1991, -33690])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

\(\Z\)

magma: MordellWeilGroup(E);
 

Infinite order Mordell-Weil generator and height

$P$ =  \(\left(150, 1664\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $3.0363552787444970212583298665$

sage: E.gens()
 
magma: Generators(E);
 
gp: E.gen
 

Integral points

\( \left(150, 1664\right) \), \( \left(150, -1815\right) \) Copy content Toggle raw display

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

Invariants

Conductor: \( 438993 \)  =  $3^{3} \cdot 71 \cdot 229$
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: $-31168503 $  =  $-1 \cdot 3^{3} \cdot 71^{2} \cdot 229 $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: \( -\frac{32312399988051}{1154389} \)  =  $-1 \cdot 3^{6} \cdot 71^{-2} \cdot 229^{-1} \cdot 3539^{3}$
comment: j-invariant
 
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
oscar: j_invariant(E)
 
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
sage: E.has_cm()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $0.52699892202517137127083138608\dots$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $0.25234584985814394842202007685\dots$
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 

BSD invariants

Analytic rank: $1$
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Regulator: $3.0363552787444970212583298665\dots$
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: $0.35736980546958049250662731945\dots$
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: $ 2 $  = $ 1\cdot2\cdot1 $
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: $1$
comment: Torsion order
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
oscar: prod(torsion_structure(E)[1])
 
Analytic order of Ш: $1$ ( rounded)
comment: Order of Sha
 
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Special value: $ L'(E,1) $ ≈ $ 2.1702033906029095050562175767 $
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);
 

BSD formula

$\displaystyle 2.170203391 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.357370 \cdot 3.036355 \cdot 2}{1^2} \approx 2.170203391$

# 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 analyiic 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 438993.2.a.a

\( q - q^{2} - q^{4} - 2 q^{5} + 2 q^{7} + 3 q^{8} + 2 q^{10} - 6 q^{11} - 2 q^{13} - 2 q^{14} - q^{16} - 4 q^{17} + 7 q^{19} + O(q^{20}) \) 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: 310368
comment: Modular degree
 
sage: E.modular_degree()
 
gp: ellmoddegree(E)
 
magma: ModularDegree(E);
 
$ \Gamma_0(N) $-optimal: yes
Manin constant: 1
comment: Manin constant
 
magma: ManinConstant(E);
 

Local data

This elliptic curve is not semistable. There are 3 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$3$ $1$ $II$ Additive -1 3 3 0
$71$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$229$ $1$ $I_{1}$ Split multiplicative -1 1 1 1

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 $\ell$.

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)];
 

gens = [[1373, 2, 1372, 3], [235, 2, 235, 3], [1, 0, 2, 1], [1, 2, 0, 1], [917, 2, 917, 3], [1, 1, 1373, 0]]
 
GL(2,Integers(1374)).subgroup(gens)
 
Gens := [[1373, 2, 1372, 3], [235, 2, 235, 3], [1, 0, 2, 1], [1, 2, 0, 1], [917, 2, 917, 3], [1, 1, 1373, 0]];
 
sub<GL(2,Integers(1374))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1374 = 2 \cdot 3 \cdot 229 \), index $2$, genus $0$, and generators

$\left(\begin{array}{rr} 1373 & 2 \\ 1372 & 3 \end{array}\right),\left(\begin{array}{rr} 235 & 2 \\ 235 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 917 & 2 \\ 917 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 1373 & 0 \end{array}\right)$.

Input positive integer $m$ to see the generators of the reduction of $H$ to $\mathrm{GL}_2(\Z/m\Z)$:

The torsion field $K:=\Q(E[1374])$ is a degree-$394271608320$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1374\Z)$.

Isogenies

gp: ellisomat(E)
 

This curve has no rational isogenies. Its isogeny class 438993.a consists of this curve only.

Twists

This elliptic curve is its own minimal quadratic twist.

Iwasawa invariants

No Iwasawa invariant data is available for this curve.

$p$-adic regulators

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.