# Properties

 Label 2304.o4 Conductor $2304$ Discriminant $-3359232$ j-invariant $$\frac{64}{9}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2=x^3+6x-88$$ y^2=x^3+6x-88 (homogenize, simplify) $$y^2z=x^3+6xz^2-88z^3$$ y^2z=x^3+6xz^2-88z^3 (dehomogenize, simplify) $$y^2=x^3+6x-88$$ y^2=x^3+6x-88 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([0, 0, 0, 6, -88])

gp: E = ellinit([0, 0, 0, 6, -88])

magma: E := EllipticCurve([0, 0, 0, 6, -88]);

oscar: E = elliptic_curve([0, 0, 0, 6, -88])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

$$\Z/{2}\Z$$

magma: MordellWeilGroup(E);

## Torsion generators

$$\left(4, 0\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

$$\left(4, 0\right)$$

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$2304$$ = $2^{8} \cdot 3^{2}$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $-3359232$ = $-1 \cdot 2^{9} \cdot 3^{8}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$\frac{64}{9}$$ = $2^{6} \cdot 3^{-2}$ 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: $-0.068409221203428861993528224773\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $-1.1375757509574426897540749343\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E) $abc$ quality: $1.311101219926227\dots$ Szpiro ratio: $2.90321002490296\dots$

## BSD invariants

 Analytic rank: $0$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $1$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $1.1869637097891439356122242537\dots$ 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: $8$  = $2\cdot2^{2}$ 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: $2$ 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: $L(E,1)$ ≈ $2.3739274195782878712244485075$ 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

$\displaystyle 2.373927420 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.186964 \cdot 1.000000 \cdot 8}{2^2} \approx 2.373927420$

# 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

$$q + 2 q^{5} + 2 q^{7} - 4 q^{13} + 2 q^{17} + 4 q^{19} + O(q^{20})$$

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: 256
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 2 primes $p$ of bad reduction:

$p$ Tamagawa number Kodaira symbol Reduction type Root number $v_p(N)$ $v_p(\Delta)$ $v_p(\mathrm{den}(j))$
$2$ $2$ $III$ additive 1 8 9 0
$3$ $4$ $I_{2}^{*}$ additive -1 2 8 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 $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 8.6.0.5
$5$ 5B 5.6.0.1

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 = [[51, 20, 220, 107], [79, 220, 70, 39], [116, 225, 185, 146], [221, 20, 220, 21], [1, 20, 0, 1], [138, 5, 115, 82], [1, 0, 20, 1], [11, 16, 0, 131], [1, 10, 10, 101]]

GL(2,Integers(240)).subgroup(gens)

Gens := [[51, 20, 220, 107], [79, 220, 70, 39], [116, 225, 185, 146], [221, 20, 220, 21], [1, 20, 0, 1], [138, 5, 115, 82], [1, 0, 20, 1], [11, 16, 0, 131], [1, 10, 10, 101]];

sub<GL(2,Integers(240))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$240 = 2^{4} \cdot 3 \cdot 5$$, index $288$, genus $5$, and generators

$\left(\begin{array}{rr} 51 & 20 \\ 220 & 107 \end{array}\right),\left(\begin{array}{rr} 79 & 220 \\ 70 & 39 \end{array}\right),\left(\begin{array}{rr} 116 & 225 \\ 185 & 146 \end{array}\right),\left(\begin{array}{rr} 221 & 20 \\ 220 & 21 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 138 & 5 \\ 115 & 82 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 16 \\ 0 & 131 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 10 & 101 \end{array}\right)$.

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

The torsion field $K:=\Q(E[240])$ is a degree-$1966080$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/240\Z)$.

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 $4$ $$9 = 3^{2}$$
$3$ additive $8$ $$256 = 2^{8}$$

## Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 5 and 10.
Its isogeny class 2304.o consists of 4 curves linked by isogenies of degrees dividing 10.

## Twists

The minimal quadratic twist of this elliptic curve is 768.a4, its twist by $-3$.

## 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:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-2})$$ $$\Z/2\Z \oplus \Z/2\Z$$ not in database $4$ 4.2.18432.1 $$\Z/4\Z$$ not in database $4$ 4.0.18432.2 $$\Z/10\Z$$ not in database $8$ 8.0.339738624.9 $$\Z/2\Z \oplus \Z/4\Z$$ not in database $8$ 8.0.5435817984.4 $$\Z/2\Z \oplus \Z/4\Z$$ not in database $8$ 8.2.743008370688.21 $$\Z/6\Z$$ not in database $8$ 8.0.1358954496.4 $$\Z/2\Z \oplus \Z/10\Z$$ not in database $16$ 16.4.38294359833110460235776.6 $$\Z/8\Z$$ not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/6\Z$$ not in database $16$ 16.0.29548117155177824256.5 $$\Z/20\Z$$ not in database $20$ 20.4.425973332494163902464000000000000000.2 $$\Z/10\Z$$ 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) 2 3 5 add add ord - - 0 - - 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.

An entry - indicates that the invariants are not computed because the reduction is additive.

## $p$-adic regulators

All $p$-adic regulators are identically $1$ since the rank is $0$.