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Properties

Label 2704a1
Conductor $2704$
Discriminant $-128508962816$
j-invariant \( -\frac{235298}{13} \)
CM no
Rank $1$
Torsion structure trivial

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Show commands: Magma / Oscar / PariGP / SageMath

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2=x^3-x^2-2760x+59344\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z=x^3-x^2z-2760xz^2+59344z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-223587x+42591042\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([0, -1, 0, -2760, 59344])
 
gp: E = ellinit([0, -1, 0, -2760, 59344])
 
magma: E := EllipticCurve([0, -1, 0, -2760, 59344]);
 
oscar: E = elliptic_curve([0, -1, 0, -2760, 59344])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

\(\Z\)

magma: MordellWeilGroup(E);
 

Mordell-Weil generators

$P$$\hat{h}(P)$Order
$(-30, 338)$$0.38861802119655383980103765134$$\infty$

Integral points

\((-30,\pm 338)\), \((61,\pm 338)\) Copy content Toggle raw display

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

Invariants

Conductor: $N$  =  \( 2704 \) = $2^{4} \cdot 13^{2}$
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: $\Delta$  =  $-128508962816$ = $-1 \cdot 2^{11} \cdot 13^{7} $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: $j$  =  \( -\frac{235298}{13} \) = $-1 \cdot 2 \cdot 7^{6} \cdot 13^{-1}$
comment: j-invariant
 
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
oscar: j_invariant(E)
 
Endomorphism ring: $\mathrm{End}(E)$ = $\Z$
Geometric endomorphism ring: $\mathrm{End}(E_{\overline{\Q}})$  =  \(\Z\)    (no potential complex multiplication)
sage: E.has_cm()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{ST}(E)$ = $\mathrm{SU}(2)$
Faltings height: $h_{\mathrm{Faltings}}$ ≈ $0.88944307216241255821258378246$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $h_{\mathrm{stable}}$ ≈ $-1.0284165220816390101132893830$
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 
$abc$ quality: $Q$ ≈ $0.9655917770548856$
Szpiro ratio: $\sigma_{m}$ ≈ $4.488979877606868$

BSD invariants

Analytic rank: $r_{\mathrm{an}}$ = $ 1$
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Mordell-Weil rank: $r$ = $ 1$
comment: Rank
 
sage: E.rank()
 
gp: [lower,upper] = ellrank(E)
 
magma: Rank(E);
 
Regulator: $\mathrm{Reg}(E/\Q)$ ≈ $0.38861802119655383980103765134$
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: $\Omega$ ≈ $1.0285660578059661122494538583$
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: $\prod_{p}c_p$ = $ 8 $  = $ 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: $\#E(\Q)_{\mathrm{tor}}$ = $1$
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)$ ≈ $3.1977544484359580854898024994 $
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: Ш${}_{\mathrm{an}}$  ≈  $1$    (rounded)
comment: Order of Sha
 
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 

BSD formula

$\displaystyle 3.197754448 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.028566 \cdot 0.388618 \cdot 8}{1^2} \approx 3.197754448$

# 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   2704.2.a.d

\( q - q^{3} + q^{5} + 5 q^{7} - 2 q^{9} - 2 q^{11} - q^{15} - 3 q^{17} - 2 q^{19} + O(q^{20}) \) 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: 2688
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 at primes of bad reduction

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

$p$ Tamagawa number Kodaira symbol Reduction type Root number $\mathrm{ord}_p(N)$ $\mathrm{ord}_p(\Delta)$ $\mathrm{ord}_p(\mathrm{den}(j))$
$2$ $2$ $I_{3}^{*}$ additive 1 4 11 0
$13$ $4$ $I_{1}^{*}$ additive 1 2 7 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 = [[41, 2, 41, 3], [1, 2, 0, 1], [53, 2, 53, 3], [1, 0, 2, 1], [103, 2, 102, 3], [1, 1, 103, 0], [79, 2, 79, 3]]
 
GL(2,Integers(104)).subgroup(gens)
 
Gens := [[41, 2, 41, 3], [1, 2, 0, 1], [53, 2, 53, 3], [1, 0, 2, 1], [103, 2, 102, 3], [1, 1, 103, 0], [79, 2, 79, 3]];
 
sub<GL(2,Integers(104))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 104 = 2^{3} \cdot 13 \), index $2$, genus $0$, and generators

$\left(\begin{array}{rr} 41 & 2 \\ 41 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 53 & 2 \\ 53 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 103 & 2 \\ 102 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 103 & 0 \end{array}\right),\left(\begin{array}{rr} 79 & 2 \\ 79 & 3 \end{array}\right)$.

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

The torsion field $K:=\Q(E[104])$ is a degree-$20127744$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/104\Z)$.

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
$2$ additive $4$ \( 169 = 13^{2} \)
$13$ additive $98$ \( 16 = 2^{4} \)

Isogenies

gp: ellisomat(E)
 

This curve has no rational isogenies. Its isogeny class 2704a consists of this curve only.

Twists

The minimal quadratic twist of this elliptic curve is 104a1, its twist by $-52$.

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
$3$ 3.1.104.1 \(\Z/2\Z\) not in database
$6$ 6.0.1124864.1 \(\Z/2\Z \oplus \Z/2\Z\) not in database
$8$ 8.2.43238323335168.7 \(\Z/3\Z\) not in database
$12$ 12.2.8421963387109376.3 \(\Z/4\Z\) not in database

We only show fields where the torsion growth is primitive.

Iwasawa invariants

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

$p$-adic regulators

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