Properties

Label 493680.cf4
Conductor $493680$
Discriminant $1.045\times 10^{28}$
j-invariant \( \frac{1696892787277117093383481}{1440538624914939000} \)
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

\(y^2=x^3-x^2-4810780960x-128335641252608\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z=x^3-x^2z-4810780960xz^2-128335641252608z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-389673257787x-93557851492924566\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([0, -1, 0, -4810780960, -128335641252608])
 
gp: E = ellinit([0, -1, 0, -4810780960, -128335641252608])
 
magma: E := EllipticCurve([0, -1, 0, -4810780960, -128335641252608]);
 
oscar: E = EllipticCurve([0, -1, 0, -4810780960, -128335641252608])
 
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

$P$ =  \(\left(-39406, 213330\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $7.8191492402136548075538469876$

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

Torsion generators

\( \left(-40927, 0\right) \) 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

\( \left(-40927, 0\right) \), \((-39406,\pm 213330)\) Copy content Toggle raw display

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

Invariants

Conductor: \( 493680 \)  =  $2^{4} \cdot 3 \cdot 5 \cdot 11^{2} \cdot 17$
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: $10453000384073458687094784000 $  =  $2^{15} \cdot 3^{3} \cdot 5^{3} \cdot 11^{18} \cdot 17 $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: \( \frac{1696892787277117093383481}{1440538624914939000} \)  =  $2^{-3} \cdot 3^{-3} \cdot 5^{-3} \cdot 11^{-12} \cdot 17^{-1} \cdot 181^{3} \cdot 227^{3} \cdot 2903^{3}$
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: $4.3035761215371378145650002670\dots$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $2.4114813045780072331167963566\dots$
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 

BSD invariants

Analytic rank: $1$
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Regulator: $7.8191492402136548075538469876\dots$
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: $0.018128747152679201410256860407\dots$
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: $ 24 $  = $ 2\cdot1\cdot3\cdot2^{2}\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: $4$ = $2^2$ ( rounded)
comment: Order of Sha
 
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Special value: $ L'(E,1) $ ≈ $ 3.4020331085975288557179572295 $
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

$\displaystyle 3.402033109 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{4 \cdot 0.018129 \cdot 7.819149 \cdot 24}{2^2} \approx 3.402033109$

# 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 493680.2.a.cf

\( q - q^{3} + q^{5} - 4 q^{7} + q^{9} - 2 q^{13} - q^{15} + q^{17} - 4 q^{19} + O(q^{20}) \) 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: 637009920
comment: Modular degree
 
sage: E.modular_degree()
 
gp: ellmoddegree(E)
 
magma: ModularDegree(E);
 
$ \Gamma_0(N) $-optimal: no
Manin constant: 1 (conditional*)
comment: Manin constant
 
magma: ManinConstant(E);
 
* The Manin constant is correct provided that curve 493680.cf8 is optimal.

Local data

This elliptic curve is not semistable. There are 5 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $2$ $I_{7}^{*}$ Additive -1 4 15 3
$3$ $1$ $I_{3}$ Non-split multiplicative 1 1 3 3
$5$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$11$ $4$ $I_{12}^{*}$ Additive -1 2 18 12
$17$ $1$ $I_{1}$ 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 $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 8.12.0.6
$3$ 3B 3.4.0.1

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 = [[7936, 21, 12915, 22066], [1, 12, 12, 145], [4079, 22416, 4068, 22151], [8992, 21, 22155, 22066], [16, 11223, 3281, 7394], [15, 106, 21134, 13211], [22417, 24, 22416, 25], [1, 24, 0, 1], [16826, 22437, 16785, 22406], [17766, 21529, 12155, 15896], [1, 0, 24, 1]]
 
GL(2,Integers(22440)).subgroup(gens)
 
Gens := [[7936, 21, 12915, 22066], [1, 12, 12, 145], [4079, 22416, 4068, 22151], [8992, 21, 22155, 22066], [16, 11223, 3281, 7394], [15, 106, 21134, 13211], [22417, 24, 22416, 25], [1, 24, 0, 1], [16826, 22437, 16785, 22406], [17766, 21529, 12155, 15896], [1, 0, 24, 1]];
 
sub<GL(2,Integers(22440))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 22440 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 17 \), index $384$, genus $5$, and generators

$\left(\begin{array}{rr} 7936 & 21 \\ 12915 & 22066 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 4079 & 22416 \\ 4068 & 22151 \end{array}\right),\left(\begin{array}{rr} 8992 & 21 \\ 22155 & 22066 \end{array}\right),\left(\begin{array}{rr} 16 & 11223 \\ 3281 & 7394 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 21134 & 13211 \end{array}\right),\left(\begin{array}{rr} 22417 & 24 \\ 22416 & 25 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 16826 & 22437 \\ 16785 & 22406 \end{array}\right),\left(\begin{array}{rr} 17766 & 21529 \\ 12155 & 15896 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right)$.

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

The torsion field $K:=\Q(E[22440])$ is a degree-$95296684032000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/22440\Z)$.

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3, 4, 6 and 12.
Its isogeny class 493680.cf consists of 8 curves linked by isogenies of degrees dividing 12.

Twists

The minimal quadratic twist of this elliptic curve is 5610.q4, its twist by $44$.

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.