↑

Properties

Label 32487.a1
Conductor $32487$
Discriminant $-4.706\times 10^{12}$
j-invariant \( -\frac{325660672}{40000779} \)
CM no
Rank $2$
Torsion structure trivial

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+y=x^3-x^2-702x+104852\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+yz^2=x^3-x^2z-702xz^2+104852z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-910224x+4881065616\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([0, -1, 1, -702, 104852])
 
gp: E = ellinit([0, -1, 1, -702, 104852])
 
magma: E := EllipticCurve([0, -1, 1, -702, 104852]);
 
oscar: E = EllipticCurve([0, -1, 1, -702, 104852])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

\(\Z \oplus \Z\)

magma: MordellWeilGroup(E);
 

Infinite order Mordell-Weil generators and heights

$P$ =  \(\left(96, 955\right)\) Copy content Toggle raw display \(\left(47, 416\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $0.16363152227611104561417852688$$0.62143516255980567073602138201$

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

Integral points

\( \left(-51, 73\right) \), \( \left(-51, -74\right) \), \( \left(-44, 220\right) \), \( \left(-44, -221\right) \), \( \left(-21, 331\right) \), \( \left(-21, -332\right) \), \( \left(-8, 331\right) \), \( \left(-8, -332\right) \), \( \left(5, 318\right) \), \( \left(5, -319\right) \), \( \left(30, 331\right) \), \( \left(30, -332\right) \), \( \left(44, 396\right) \), \( \left(44, -397\right) \), \( \left(47, 416\right) \), \( \left(47, -417\right) \), \( \left(96, 955\right) \), \( \left(96, -956\right) \), \( \left(161, 2034\right) \), \( \left(161, -2035\right) \), \( \left(537, 12421\right) \), \( \left(537, -12422\right) \), \( \left(642, 16243\right) \), \( \left(642, -16244\right) \), \( \left(1860, 80188\right) \), \( \left(1860, -80189\right) \), \( \left(2504, 125268\right) \), \( \left(2504, -125269\right) \), \( \left(2631, 134920\right) \), \( \left(2631, -134921\right) \), \( \left(73351, 19865800\right) \), \( \left(73351, -19865801\right) \) Copy content Toggle raw display

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

Invariants

Conductor: \( 32487 \)  =  $3 \cdot 7^{2} \cdot 13 \cdot 17$
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: $-4706051648571 $  =  $-1 \cdot 3^{2} \cdot 7^{7} \cdot 13^{3} \cdot 17^{2} $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: \( -\frac{325660672}{40000779} \)  =  $-1 \cdot 2^{12} \cdot 3^{-2} \cdot 7^{-1} \cdot 13^{-3} \cdot 17^{-2} \cdot 43^{3}$
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: $1.1111144493925204616629758621\dots$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $0.13815937486486380911029949038\dots$
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 

BSD invariants

Analytic rank: $2$
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Regulator: $0.099161187770099366668695682584\dots$
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: $0.63291503084419577836096063563\dots$
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: $ 48 $  = $ 2\cdot2^{2}\cdot3\cdot2 $
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^{(2)}(E,1)/2! $ ≈ $ 3.0125090983708574328419968010 $
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

$\displaystyle 3.012509098 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.632915 \cdot 0.099161 \cdot 48}{1^2} \approx 3.012509098$

# 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("anayltic 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 32487.2.a.a

\( q - 2 q^{2} - q^{3} + 2 q^{4} - 3 q^{5} + 2 q^{6} + q^{9} + 6 q^{10} - 6 q^{11} - 2 q^{12} + q^{13} + 3 q^{15} - 4 q^{16} + q^{17} - 2 q^{18} + q^{19} + O(q^{20}) \) 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: 105984
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 4 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$3$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$7$ $4$ $I_{1}^{*}$ Additive -1 2 7 1
$13$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$17$ $2$ $I_{2}$ Split multiplicative -1 1 2 2

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 = [[181, 2, 180, 3], [1, 1, 181, 0], [1, 2, 0, 1], [1, 0, 2, 1], [15, 2, 15, 3], [157, 2, 157, 3]]
 
GL(2,Integers(182)).subgroup(gens)
 
Gens := [[181, 2, 180, 3], [1, 1, 181, 0], [1, 2, 0, 1], [1, 0, 2, 1], [15, 2, 15, 3], [157, 2, 157, 3]];
 
sub<GL(2,Integers(182))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 182 = 2 \cdot 7 \cdot 13 \), index $2$, genus $0$, and generators

$\left(\begin{array}{rr} 181 & 2 \\ 180 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 181 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 15 & 3 \end{array}\right),\left(\begin{array}{rr} 157 & 2 \\ 157 & 3 \end{array}\right)$.

The torsion field $K:=\Q(E[182])$ is a degree-$13288$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/182\Z)$.

$p$-adic regulators

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

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type ss nonsplit ord add ord split split ord ord ord ord ord ord ord ord
$\lambda$-invariant(s) 23,8 2 2 - 2 3 3 2 2 2 2 2 2 2 2
$\mu$-invariant(s) 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.

Isogenies

gp: ellisomat(E)
 

This curve has no rational isogenies. Its isogeny class 32487.a consists of this curve only.

Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$3$ 3.1.364.1 \(\Z/2\Z\) Not in database
$6$ 6.0.12057136.1 \(\Z/2\Z \oplus \Z/2\Z\) Not in database
$8$ 8.2.1740675142665963.6 \(\Z/3\Z\) Not in database
$12$ deg 12 \(\Z/4\Z\) 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.