# Properties

 Label 329742.j1 Conductor $329742$ Discriminant $24817701888$ j-invariant $\frac{34508408842474480159875}{919174144}$ CM no Rank $0$ Torsion Structure $\mathrm{Trivial}$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 1, -2034800, -1116690477]); // or
magma: E := EllipticCurve("329742j1");
sage: E = EllipticCurve([1, -1, 1, -2034800, -1116690477]) # or
sage: E = EllipticCurve("329742j1")
gp: E = ellinit([1, -1, 1, -2034800, -1116690477]) \\ or
gp: E = ellinit("329742j1")

$y^2 + x y + y = x^{3} - x^{2} - 2034800 x - 1116690477$

Trivial

## Integral points

magma: IntegralPoints(E);
sage: E.integral_points()
None

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] $N$ = $329742$ = $2 \cdot 3^{2} \cdot 7 \cdot 2617$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc $\Delta$ = $24817701888$ = $2^{10} \cdot 3^{3} \cdot 7^{3} \cdot 2617$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j $j$ = $\frac{34508408842474480159875}{919174144}$ = $2^{-10} \cdot 3^{3} \cdot 5^{3} \cdot 7^{-3} \cdot 2617^{-1} \cdot 2170453^{3}$ $\text{End} (E)$ = $\Z$ (no Complex Multiplication) $\text{ST} (E)$ = $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E); sage: E.rank() $r$ = $0$ magma: Regulator(E); sage: E.regulator() $\text{Reg}$ = $1$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega[1] $\Omega$ ≈ $0.126406422158$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] $\prod_p c_p$ = $60$  = $( 2 \cdot 5 )\cdot2\cdot3\cdot1$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E)[1] $\#E_{\text{tor}}$ = $1$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Ш$_{\text{an}}$ = $1$ (exact)

## Modular invariants

### Modular form 329742.2.1.j

magma: ModularForm(E);
sage: E.q_eigenform(20)
gp: xy = elltaniyama(E);
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

$q + q^{2} + q^{4} + q^{7} + q^{8} + 3q^{11} + 6q^{13} + q^{14} + q^{16} + 3q^{17} + 4q^{19} + O(q^{20})$

### Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
3223680 : curve is $\Gamma_0(N)$-optimal

### Special L-value attached to the curve

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
sage: r = E.rank();
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
gp: ar = ellanalyticrank(E);
gp: ar[2]/factorial(ar[1])

$L(E,1)$ ≈ $7.58438532951$

## Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
sage: E.local_data()
gp: ellglobalred(E)[5]
prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $10$ $I_{10}$ Split multiplicative -1 1 10 10
$3$ $2$ $III$ Additive 1 2 3 0
$7$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$2617$ $1$ $I_{1}$ Split multiplicative -1 1 1 1

## Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
sage: rho = E.galois_representation();
sage: [rho.image_type(p) for p in rho.non_surjective()]

The mod $p$ Galois representation has maximal image $\GL(2,\F_p)$ for all primes $p$ .

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

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

No Iwasawa invariant data is available for this curve.

## Isogenies

This curve has no rational isogenies. Its isogeny class 329742.j consists of this curve only.

## Growth of torsion in number fields

The number fields $K$ of degree up to 7 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 3.3.54957.1 $\Z/2\Z$ Not in database
6 6.6.165985080005493.1 $\Z/2\Z \times \Z/2\Z$ Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.