# Properties

 Label 336e4 Conductor $336$ Discriminant $88510464$ j-invariant $$\frac{13027640977}{21609}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z \oplus \Z/{4}\Z$$

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2=x^3-x^2-784x+8704$$ y^2=x^3-x^2-784x+8704 (homogenize, simplify) $$y^2z=x^3-x^2z-784xz^2+8704z^3$$ y^2z=x^3-x^2z-784xz^2+8704z^3 (dehomogenize, simplify) $$y^2=x^3-63531x+6154650$$ y^2=x^3-63531x+6154650 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([0, -1, 0, -784, 8704])

gp: E = ellinit([0, -1, 0, -784, 8704])

magma: E := EllipticCurve([0, -1, 0, -784, 8704]);

oscar: E = elliptic_curve([0, -1, 0, -784, 8704])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

$$\Z \oplus \Z/{2}\Z \oplus \Z/{4}\Z$$

magma: MordellWeilGroup(E);

### Infinite order Mordell-Weil generator and height

 $P$ = $$\left(18, 10\right)$$ (18, 10) $\hat{h}(P)$ ≈ $1.4925984173890274178188752073$

sage: E.gens()

magma: Generators(E);

gp: E.gen

## Torsion generators

$$\left(16, 0\right)$$, $$\left(24, 56\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

$$\left(-32, 0\right)$$, $$(-11,\pm 126)$$, $$(-8,\pm 120)$$, $$(10,\pm 42)$$, $$\left(16, 0\right)$$, $$\left(17, 0\right)$$, $$(18,\pm 10)$$, $$(24,\pm 56)$$, $$(66,\pm 490)$$, $$(80,\pm 672)$$, $$(6744,\pm 553784)$$

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$336$$ = $2^{4} \cdot 3 \cdot 7$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $88510464$ = $2^{12} \cdot 3^{2} \cdot 7^{4}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$\frac{13027640977}{21609}$$ = $3^{-2} \cdot 7^{-4} \cdot 13^{3} \cdot 181^{3}$ 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: $0.42077897367091242546688191540\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $-0.27236820688903288395035020606\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E) $abc$ quality: $1.0814902740568197\dots$ Szpiro ratio: $5.433642999715578\dots$

## BSD invariants

 Analytic rank: $1$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $1.4925984173890274178188752073\dots$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $1.9109897807518291965531482188\dots$ 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: $32$  = $2^{2}\cdot2\cdot2^{2}$ 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: $8$ 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: $L'(E,1)$ ≈ $1.4261701611983923742568767354$ 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

$\displaystyle 1.426170161 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.910990 \cdot 1.492598 \cdot 32}{8^2} \approx 1.426170161$

# 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

$$q - q^{3} - 2 q^{5} + q^{7} + q^{9} - 4 q^{11} - 2 q^{13} + 2 q^{15} - 6 q^{17} - 4 q^{19} + O(q^{20})$$

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);

Modular degree: 128
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 3 primes $p$ of bad reduction:

$p$ Tamagawa number Kodaira symbol Reduction type Root number $v_p(N)$ $v_p(\Delta)$ $v_p(\mathrm{den}(j))$
$2$ $4$ $I_{4}^{*}$ additive -1 4 12 0
$3$ $2$ $I_{2}$ nonsplit multiplicative 1 1 2 2
$7$ $4$ $I_{4}$ split multiplicative -1 1 4 4

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
$2$ 2Cs 8.48.0.25

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 = [[89, 44, 64, 93], [119, 2, 150, 163], [73, 8, 124, 33], [5, 4, 164, 165], [161, 8, 160, 9], [1, 0, 8, 1], [161, 120, 6, 47], [1, 8, 0, 1]]

GL(2,Integers(168)).subgroup(gens)

Gens := [[89, 44, 64, 93], [119, 2, 150, 163], [73, 8, 124, 33], [5, 4, 164, 165], [161, 8, 160, 9], [1, 0, 8, 1], [161, 120, 6, 47], [1, 8, 0, 1]];

sub<GL(2,Integers(168))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$168 = 2^{3} \cdot 3 \cdot 7$$, index $192$, genus $1$, and generators

$\left(\begin{array}{rr} 89 & 44 \\ 64 & 93 \end{array}\right),\left(\begin{array}{rr} 119 & 2 \\ 150 & 163 \end{array}\right),\left(\begin{array}{rr} 73 & 8 \\ 124 & 33 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 164 & 165 \end{array}\right),\left(\begin{array}{rr} 161 & 8 \\ 160 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 161 & 120 \\ 6 & 47 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right)$.

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

The torsion field $K:=\Q(E[168])$ is a degree-$774144$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/168\Z)$.

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 $2$ $$1$$
$3$ nonsplit multiplicative $4$ $$112 = 2^{4} \cdot 7$$
$7$ split multiplicative $8$ $$48 = 2^{4} \cdot 3$$

## Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 336e consists of 6 curves linked by isogenies of degrees dividing 8.

## Twists

The minimal quadratic twist of this elliptic curve is 21a2, its twist by $-4$.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{4}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $4$ $$\Q(\zeta_{12})$$ $$\Z/4\Z \oplus \Z/4\Z$$ not in database $4$ $$\Q(\sqrt{-6}, \sqrt{-7})$$ $$\Z/2\Z \oplus \Z/8\Z$$ not in database $4$ $$\Q(\sqrt{2}, \sqrt{7})$$ $$\Z/2\Z \oplus \Z/8\Z$$ not in database $8$ 8.0.2985984.1 $$\Z/4\Z \oplus \Z/8\Z$$ not in database $8$ 8.2.108884466432.1 $$\Z/2\Z \oplus \Z/12\Z$$ not in database $16$ 16.0.162447943996702457856.1 $$\Z/4\Z \oplus \Z/8\Z$$ not in database $16$ deg 16 $$\Z/4\Z \oplus \Z/8\Z$$ not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/16\Z$$ not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/16\Z$$ 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.

## Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 add nonsplit ord split ord ord ord ord ss ord ss ord ord ord ss - 1 1 2 1 1 1 1 1,1 1 1,1 1 1 1 1,1 - 0 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.

## $p$-adic regulators

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