Properties

 Label 348082.v1 Conductor $348082$ Discriminant $305263317240668236423364608$ j-invariant $\frac{267635230617574617998257}{2062089938478825472}$ 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, -710224302, -7236834692117]); // or
magma: E := EllipticCurve("348082v1");
sage: E = EllipticCurve([1, 1, 1, -710224302, -7236834692117]) # or
sage: E = EllipticCurve("348082v1")
gp: E = ellinit([1, 1, 1, -710224302, -7236834692117]) \\ or
gp: E = ellinit("348082v1")

$y^2 + x y + y = x^{3} + x^{2} - 710224302 x - 7236834692117$

Trivial

Integral points

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

Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] $N$ = $348082$ = $2 \cdot 7 \cdot 23^{2} \cdot 47$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc $\Delta$ = $305263317240668236423364608$ = $2^{20} \cdot 7^{7} \cdot 23^{7} \cdot 47^{3}$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j $j$ = $\frac{267635230617574617998257}{2062089938478825472}$ = $2^{-20} \cdot 7^{-7} \cdot 23^{-1} \cdot 47^{-3} \cdot 97^{3} \cdot 664369^{3}$ $\text{End} (E)$ = $\Z$ (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] $\Omega$ ≈ $0.0292585250701$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] $\prod_p c_p$ = $120$  = $( 2^{2} \cdot 5 )\cdot1\cdot2\cdot3$ 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 348082.2.1.v

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)

$q + q^{2} - q^{3} + q^{4} + 2q^{5} - q^{6} - q^{7} + q^{8} - 2q^{9} + 2q^{10} + 6q^{11} - q^{12} - q^{13} - q^{14} - 2q^{15} + q^{16} - 7q^{17} - 2q^{18} - q^{19} + O(q^{20})$

Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
223534080 : 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])

$L(E,1)$ ≈ $3.51102300842$

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$(j)_{-}$
$2$ $20$ $I_{20}$ Split multiplicative -1 1 20 20
$7$ $1$ $I_{7}$ Non-split multiplicative 1 1 7 7
$23$ $2$ $I_1^{*}$ Additive -1 2 7 1
$47$ $3$ $I_{3}$ 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 $0$.

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has no rational isogenies. Its isogeny class 348082.v consists of this curve only.