Properties

 Label 362992.c4 Conductor $362992$ Discriminant $-11463709506244968448$ j-invariant $\frac{250404380127}{23789043712}$ CM no Rank $0$ Torsion Structure $\Z/{2}\Z$

Related objects

Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([0, 0, 0, 102949, 162402954]); // or
magma: E := EllipticCurve("362992c1");
sage: E = EllipticCurve([0, 0, 0, 102949, 162402954]) # or
sage: E = EllipticCurve("362992c1")
gp: E = ellinit([0, 0, 0, 102949, 162402954]) \\ or
gp: E = ellinit("362992c1")

$y^2 = x^{3} + 102949 x + 162402954$

Mordell-Weil group structure

$\Z/{2}\Z$

Torsion generators

magma: TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp: elltors(E)

$\left(-483, 0\right)$

Integral points

magma: IntegralPoints(E);
sage: E.integral_points()

$\left(-483, 0\right)$

Note: only one of each pair $\pm P$ is listed.

Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] $N$ = $362992$ = $2^{4} \cdot 7^{2} \cdot 463$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc $\Delta$ = $-11463709506244968448$ = $-1 \cdot 2^{32} \cdot 7^{8} \cdot 463$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j $j$ = $\frac{250404380127}{23789043712}$ = $2^{-20} \cdot 3^{3} \cdot 7^{-2} \cdot 11^{3} \cdot 191^{3} \cdot 463^{-1}$ $\text{End} (E)$ = $\Z$ (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] $\Omega$ ≈ $0.173638039181$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] $\prod_p c_p$ = $8$  = $2^{2}\cdot2\cdot1$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E)[1] $\#E_{\text{tor}}$ = $2$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Ш$_{\text{an}}$ = $4$ (exact)

Modular invariants

Modular form 362992.2.1.c

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)

$q - 2q^{5} - 3q^{9} + 6q^{13} - 2q^{17} - 8q^{19} + O(q^{20})$

Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
4792320 : 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])

$L(E,1)$ ≈ $1.38910431345$

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$(j)_{-}$
$2$ $4$ $I_24^{*}$ Additive -1 4 32 20
$7$ $2$ $I_2^{*}$ Additive -1 2 8 2
$463$ $1$ $I_{1}$ Split multiplicative -1 1 1 1

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X34.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 5 & 0 \\ 4 & 3 \end{array}\right),\left(\begin{array}{rr} 7 & 7 \\ 4 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 4 & 3 \end{array}\right)$ and has index 12.

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 $p$ except those listed.

prime Image of Galois representation
$2$ B

$p$-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 non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 362992.c consists of 4 curves linked by isogenies of degrees dividing 4.