# Properties

 Label 481338df1 Conductor $481338$ Discriminant $-7.153\times 10^{32}$ j-invariant $$\frac{79374649975090937760383}{553856914190911653543936}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, 975020881, -1286710394570697])

gp: E = ellinit([1, -1, 1, 975020881, -1286710394570697])

magma: E := EllipticCurve([1, -1, 1, 975020881, -1286710394570697]);

$$y^2+xy+y=x^3-x^2+975020881x-1286710394570697$$

## Mordell-Weil group structure

$\Z/{2}\Z$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(105779, -52890\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(105779, -52890\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$481338$$ = $2 \cdot 3^{2} \cdot 11^{2} \cdot 13 \cdot 17$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-715288464086743951460818678185984$ = $-1 \cdot 2^{36} \cdot 3^{7} \cdot 11^{10} \cdot 13^{3} \cdot 17^{4}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{79374649975090937760383}{553856914190911653543936}$$ = $2^{-36} \cdot 3^{-1} \cdot 11^{-4} \cdot 13^{-3} \cdot 17^{-4} \cdot 42976127^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $4.9831039518834965021787820586\dots$ Stable Faltings height: $3.2348501711502563844501876512\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $0$ sage: E.regulator()  magma: Regulator(E); Regulator: $1$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.0074262432057626710770496433725\dots$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $1152$  = $( 2^{2} \cdot 3^{2} )\cdot2^{2}\cdot2^{2}\cdot1\cdot2$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $2$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $1$ (exact) sage: r = E.rank(); sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()  gp: ar = ellanalyticrank(E); gp: ar[2]/factorial(ar[1])  magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12); Special value: $L(E,1)$ ≈ $2.1387580432596492701902972912724958834$

## Modular invariants

Modular form 481338.2.a.df

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

magma: ModularForm(E);

$$q + q^{2} + q^{4} + 2q^{5} + q^{8} + 2q^{10} - q^{13} + q^{16} - q^{17} - 8q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 2269347840 $\Gamma_0(N)$-optimal: not computed* (one of 3 curves in this isogeny class which might be optimal) Manin constant: 1 (conditional*)
* The optimal curve in each isogeny class has not been determined in all cases for conductors over 400000. The Manin constant is correct provided that this curve is optimal.

## Local data

This elliptic curve is not semistable. There are 5 primes of bad reduction:

sage: E.local_data()

gp: ellglobalred(E)[5]

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $36$ $I_{36}$ Split multiplicative -1 1 36 36
$3$ $4$ $I_1^{*}$ Additive -1 2 7 1
$11$ $4$ $I_4^{*}$ Additive -1 2 10 4
$13$ $1$ $I_{3}$ Non-split multiplicative 1 1 3 3
$17$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4

## Galois representations

sage: rho = E.galois_representation();

sage: [rho.image_type(p) for p in rho.non_surjective()]

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

The $\ell$-adic Galois representation has maximal image $\GL(2,\Z_\ell)$ for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 4.6.0.1

## $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) 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 481338df consists of 4 curves linked by isogenies of degrees dividing 4.