# Properties

 Label 115608.a1 Conductor $115608$ Discriminant $-343361146407936$ j-invariant $-\frac{46001912569921444}{335313619539}$ CM no Rank $0$ Torsion Structure $\mathrm{Trivial}$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([0, -1, 0, -75240, -7968516]); // or
magma: E := EllipticCurve("115608a1");
sage: E = EllipticCurve([0, -1, 0, -75240, -7968516]) # or
sage: E = EllipticCurve("115608a1")
gp: E = ellinit([0, -1, 0, -75240, -7968516]) \\ or
gp: E = ellinit("115608a1")

$y^2 = x^{3} - x^{2} - 75240 x - 7968516$

Trivial

## Integral points

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

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] $N$ = $115608$ = $2^{3} \cdot 3 \cdot 4817$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc $\Delta$ = $-343361146407936$ = $-1 \cdot 2^{10} \cdot 3 \cdot 4817^{3}$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j $j$ = $-\frac{46001912569921444}{335313619539}$ = $-1 \cdot 2^{2} \cdot 3^{-1} \cdot 4817^{-3} \cdot 225721^{3}$ $\text{End} (E)$ = $\Z$ (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] $\Omega$ ≈ $0.144068095164$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] $\prod_p c_p$ = $2$  = $2\cdot1\cdot1$ 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 115608.2.1.a

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

$q - q^{3} + q^{5} + q^{9} - 5q^{11} + 4q^{13} - q^{15} - 7q^{17} + 8q^{19} + O(q^{20})$

### Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
714144 : 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])

$L(E,1)$ ≈ $0.288136190328$

## 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$(j)_{-}$
$2$ $2$ $III^{*}$ Additive -1 3 10 0
$3$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$4817$ $1$ $I_{3}$ Non-split multiplicative 1 1 3 3

## 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 $0$.

## 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 4817 add nonsplit ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary nonsplit - 0 2 0,0 0 0 0 0 0 0 0 0 0 2 0 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.

## Isogenies

This curve has no rational isogenies. Its isogeny class 115608.a consists of this curve only.