# Properties

 Label 191466.t1 Conductor $191466$ Discriminant $12659686629866496$ j-invariant $\frac{30154864531691593}{17365825281024}$ CM no Rank $0$ Torsion Structure $\mathrm{Trivial}$

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Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 1, -58361, 393113]); // or
magma: E := EllipticCurve("191466j1");
sage: E = EllipticCurve([1, -1, 1, -58361, 393113]) # or
sage: E = EllipticCurve("191466j1")
gp: E = ellinit([1, -1, 1, -58361, 393113]) \\ or
gp: E = ellinit("191466j1")

$y^2 + x y + y = x^{3} - x^{2} - 58361 x + 393113$

Trivial

## Integral points

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

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] $N$ = $191466$ = $2 \cdot 3^{2} \cdot 11 \cdot 967$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc $\Delta$ = $12659686629866496$ = $2^{10} \cdot 3^{19} \cdot 11 \cdot 967$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j $j$ = $\frac{30154864531691593}{17365825281024}$ = $2^{-10} \cdot 3^{-13} \cdot 11^{-1} \cdot 29^{3} \cdot 967^{-1} \cdot 10733^{3}$ $\text{End} (E)$ = $\Z$ (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] $\Omega$ ≈ $0.340621805271$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] $\prod_p c_p$ = $20$  = $( 2 \cdot 5 )\cdot2\cdot1\cdot1$ 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 191466.2.1.t

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)

$q + q^{2} + q^{4} + 3q^{5} - 2q^{7} + q^{8} + 3q^{10} + q^{11} - 3q^{13} - 2q^{14} + q^{16} + 3q^{17} + O(q^{20})$

For more coefficients, see the Downloads section to the right.

### Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
1497600 : 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])

$L(E,1)$ ≈ $6.81243610541$

## 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
$3$ $2$ $I_13^{*}$ Additive -1 2 19 13
$11$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$967$ $1$ $I_{1}$ Non-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 is surjective for all primes $p$.

## p-adic data

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 $0$.