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Properties

Label 32487.e6
Conductor $32487$
Discriminant $-1.278\times 10^{19}$
j-invariant \( \frac{158346567380527343}{108665074944153} \)
CM no
Rank $1$
Torsion structure \(\Z/{2}\Z\)

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Show commands: Magma / Oscar / PariGP / SageMath

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy=x^3+552278x+68146637\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz=x^3+552278xz^2+68146637z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3+715752261x+3177302239062\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([1, 0, 0, 552278, 68146637])
 
gp: E = ellinit([1, 0, 0, 552278, 68146637])
 
magma: E := EllipticCurve([1, 0, 0, 552278, 68146637]);
 
oscar: E = EllipticCurve([1, 0, 0, 552278, 68146637])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

\(\Z \oplus \Z/{2}\Z\)

magma: MordellWeilGroup(E);
 

Infinite order Mordell-Weil generator and height

$P$ =  \(\left(1061, 42467\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $2.0642626470206162648123084785$

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

Torsion generators

\( \left(-\frac{481}{4}, \frac{481}{8}\right) \) Copy content Toggle raw display

comment: Torsion subgroup
 
sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 
magma: TorsionSubgroup(E);
 
oscar: torsion_structure(E)
 

Integral points

\( \left(1061, 42467\right) \), \( \left(1061, -43528\right) \) 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: $-12784337402104656297 $  =  $-1 \cdot 3^{8} \cdot 7^{14} \cdot 13^{2} \cdot 17 $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: \( \frac{158346567380527343}{108665074944153} \)  =  $3^{-8} \cdot 7^{-8} \cdot 13^{-2} \cdot 17^{-1} \cdot 541007^{3}$
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: $2.3548417573043155541679270421\dots$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $1.3818866827766589016152506704\dots$
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 

BSD invariants

Analytic rank: $1$
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Regulator: $2.0642626470206162648123084785\dots$
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: $0.14169147262354586618715938905\dots$
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: $ 64 $  = $ 2^{3}\cdot2^{2}\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: $2$
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: $1$ (exact)
comment: Order of Sha
 
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Special value: $ L'(E,1) $ ≈ $ 4.6798146294100795709226626654 $
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

$\displaystyle 4.679814629 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.141691 \cdot 2.064263 \cdot 64}{2^2} \approx 4.679814629$

# 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.e

\( q - q^{2} + q^{3} - q^{4} + 2 q^{5} - q^{6} + 3 q^{8} + q^{9} - 2 q^{10} - 4 q^{11} - q^{12} - q^{13} + 2 q^{15} - q^{16} - q^{17} - q^{18} + 4 q^{19} + O(q^{20}) \) 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: 589824
comment: Modular degree
 
sage: E.modular_degree()
 
gp: ellmoddegree(E)
 
magma: ModularDegree(E);
 
$ \Gamma_0(N) $-optimal: no
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$(j)_{-}$
$3$ $8$ $I_{8}$ Split multiplicative -1 1 8 8
$7$ $4$ $I_{8}^{*}$ Additive -1 2 14 8
$13$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$17$ $1$ $I_{1}$ Non-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 $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 8.12.0.5

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 = [[8752, 5, 21795, 74242], [13, 16, 18308, 18249], [1, 16, 0, 1], [49505, 16, 24760, 129], [42431, 74240, 42424, 74127], [74241, 16, 74240, 17], [64982, 18569, 9285, 37130], [15, 2, 74158, 74243], [1, 0, 16, 1], [13, 16, 34016, 73941], [5, 4, 74252, 74253]]
 
GL(2,Integers(74256)).subgroup(gens)
 
Gens := [[8752, 5, 21795, 74242], [13, 16, 18308, 18249], [1, 16, 0, 1], [49505, 16, 24760, 129], [42431, 74240, 42424, 74127], [74241, 16, 74240, 17], [64982, 18569, 9285, 37130], [15, 2, 74158, 74243], [1, 0, 16, 1], [13, 16, 34016, 73941], [5, 4, 74252, 74253]];
 
sub<GL(2,Integers(74256))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 74256 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \cdot 17 \), index $192$, genus $1$, and generators

$\left(\begin{array}{rr} 8752 & 5 \\ 21795 & 74242 \end{array}\right),\left(\begin{array}{rr} 13 & 16 \\ 18308 & 18249 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 49505 & 16 \\ 24760 & 129 \end{array}\right),\left(\begin{array}{rr} 42431 & 74240 \\ 42424 & 74127 \end{array}\right),\left(\begin{array}{rr} 74241 & 16 \\ 74240 & 17 \end{array}\right),\left(\begin{array}{rr} 64982 & 18569 \\ 9285 & 37130 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 74158 & 74243 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 13 & 16 \\ 34016 & 73941 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 74252 & 74253 \end{array}\right)$.

The torsion field $K:=\Q(E[74256])$ is a degree-$25429452313853952$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/74256\Z)$.

$p$-adic regulators

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

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type ord split ord add ord nonsplit nonsplit ord ss ord ss ord ord ord ss
$\lambda$-invariant(s) 8 6 3 - 1 1 1 1 1,1 1 1,1 1 1 1 1,1
$\mu$-invariant(s) 1 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.

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 4 and 8.
Its isogeny class 32487.e consists of 6 curves linked by isogenies of degrees dividing 8.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{-17}) \) \(\Z/2\Z \oplus \Z/2\Z\) Not in database
$2$ \(\Q(\sqrt{7}) \) \(\Z/4\Z\) Not in database
$2$ \(\Q(\sqrt{-119}) \) \(\Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{7}, \sqrt{-17})\) \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{7}, \sqrt{442})\) \(\Z/8\Z\) Not in database
$4$ \(\Q(\sqrt{7}, \sqrt{-26})\) \(\Z/8\Z\) Not in database
$8$ deg 8 \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$8$ 8.0.927268850704.3 \(\Z/8\Z\) Not in database
$8$ deg 8 \(\Z/2\Z \oplus \Z/8\Z\) Not in database
$8$ deg 8 \(\Z/6\Z\) Not in database
$16$ deg 16 \(\Z/4\Z \oplus \Z/4\Z\) Not in database
$16$ deg 16 \(\Z/2\Z \oplus \Z/8\Z\) Not in database
$16$ deg 16 \(\Z/16\Z\) Not in database
$16$ deg 16 \(\Z/16\Z\) Not in database
$16$ deg 16 \(\Z/2\Z \oplus \Z/6\Z\) Not in database
$16$ deg 16 \(\Z/12\Z\) Not in database
$16$ deg 16 \(\Z/12\Z\) 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.