# Properties

 Label 4410.bi7 Conductor 4410 Discriminant 518713499808000 j-invariant $$\frac{13619385906841}{6048000}$$ CM no Rank 1 Torsion Structure $$\Z/{4}\Z$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 1, -219407, 39596631]); // or
magma: E := EllipticCurve("4410bk1");
sage: E = EllipticCurve([1, -1, 1, -219407, 39596631]) # or
sage: E = EllipticCurve("4410bk1")
gp: E = ellinit([1, -1, 1, -219407, 39596631]) \\ or
gp: E = ellinit("4410bk1")

$$y^2 + x y + y = x^{3} - x^{2} - 219407 x + 39596631$$

## Mordell-Weil group structure

$$\Z\times \Z/{4}\Z$$

### Infinite order Mordell-Weil generator and height

magma: Generators(E);
sage: E.gens()

 $$P$$ = $$\left(-229, 8934\right)$$ $$\hat{h}(P)$$ ≈ 0.45390173426

## Torsion generators

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

$$\left(359, 2466\right)$$

## Integral points

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

$$\left(-451, 7056\right)$$, $$\left(-229, 8934\right)$$, $$\left(-19, 6624\right)$$, $$\left(251, 414\right)$$, $$\left(261, 114\right)$$, $$\left(275, -138\right)$$, $$\left(311, 1014\right)$$, $$\left(359, 2466\right)$$, $$\left(471, 6134\right)$$, $$\left(5651, 420534\right)$$

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

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] Conductor: $$4410$$ = $$2 \cdot 3^{2} \cdot 5 \cdot 7^{2}$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$518713499808000$$ = $$2^{8} \cdot 3^{9} \cdot 5^{3} \cdot 7^{7}$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$\frac{13619385906841}{6048000}$$ = $$2^{-8} \cdot 3^{-3} \cdot 5^{-3} \cdot 7^{-1} \cdot 11^{3} \cdot 13^{3} \cdot 167^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E); sage: E.rank() Rank: $$1$$ magma: Regulator(E); sage: E.regulator() Regulator: $$0.45390173426$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega[1] Real period: $$0.513537867334$$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] Tamagawa product: $$384$$  = $$2^{3}\cdot2^{2}\cdot3\cdot2^{2}$$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E)[1] Torsion order: $$4$$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form4410.2.a.bi

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

$$q + q^{2} + q^{4} + q^{5} + q^{8} + q^{10} - 2q^{13} + q^{16} - 6q^{17} - 8q^{19} + O(q^{20})$$

 magma: ModularDegree(E); sage: E.modular_degree() Modular degree: 36864 $$\Gamma_0(N)$$-optimal: yes Manin constant: 1

#### Special L-value

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])

$$L'(E,1)$$ ≈ $$5.59429748619$$

## 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$$(j)_{-}$$
$$2$$ $$8$$ $$I_{8}$$ Split multiplicative -1 1 8 8
$$3$$ $$4$$ $$I_3^{*}$$ Additive -1 2 9 3
$$5$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3
$$7$$ $$4$$ $$I_1^{*}$$ Additive -1 2 7 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 X13h.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 0 & 3 \end{array}\right)$ 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
$$3$$ B

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

Note: $$p$$-adic regulator data only exists for primes $$p\ge5$$ of good ordinary reduction.

## 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 split add split add ss ordinary ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ss 3 - 2 - 1,1 1 1 1 1,1 1 1 1 1 1 1,1 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 non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3, 4, 6 and 12.
Its isogeny class 4410.bi consists of 8 curves linked by isogenies of degrees dividing 12.

## Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{105})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$$\Q(\sqrt{21})$$ $$\Z/12\Z$$ 2.2.21.1-2100.1-n8
4 4.4.46305.1 $$\Z/24\Z$$ Not in database
$$\Q(\sqrt{5}, \sqrt{21})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
6 6.0.2420208.1 $$\Z/12\Z$$ Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.