# Properties

 Label 277200.gy4 Conductor 277200 Discriminant 75442752000000 j-invariant $$\frac{4354703137}{1617}$$ CM no Rank 2 Torsion Structure $$\Z/{2}\Z$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([0, 0, 0, -122475, 16492250]); // or

magma: E := EllipticCurve("277200gy1");

sage: E = EllipticCurve([0, 0, 0, -122475, 16492250]) # or

sage: E = EllipticCurve("277200gy1")

gp: E = ellinit([0, 0, 0, -122475, 16492250]) \\ or

gp: E = ellinit("277200gy1")

$$y^2 = x^{3} - 122475 x + 16492250$$

## Mordell-Weil group structure

$$\Z^2 \times \Z/{2}\Z$$

### Infinite order Mordell-Weil generators and heights

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(-89, 5166\right)$$ $$\left(181, 504\right)$$ $$\hat{h}(P)$$ ≈ 3.83061558003 0.822817110506

## Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(205, 0\right)$$

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$(-395,\pm 1800)$$, $$(-89,\pm 5166)$$, $$(55,\pm 3150)$$, $$(181,\pm 504)$$, $$(199,\pm 18)$$, $$\left(205, 0\right)$$, $$(214,\pm 288)$$, $$(230,\pm 700)$$, $$(349,\pm 4032)$$, $$(461,\pm 7616)$$, $$(605,\pm 12800)$$, $$(10006,\pm 1000296)$$

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$277200$$ = $$2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 11$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$75442752000000$$ = $$2^{12} \cdot 3^{7} \cdot 5^{6} \cdot 7^{2} \cdot 11$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{4354703137}{1617}$$ = $$3^{-1} \cdot 7^{-2} \cdot 11^{-1} \cdot 23^{3} \cdot 71^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$2$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$1.52957839985$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.601385350686$$ 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: $$64$$  = $$2^{2}\cdot2^{2}\cdot2\cdot2\cdot1$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E)[1] Torsion order: $$2$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (rounded)

## Modular invariants

#### Modular form 277200.2.a.gy

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)

$$q + q^{7} - q^{11} - 6q^{13} + 2q^{17} - 4q^{19} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 1310720 $$\Gamma_0(N)$$-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])

$$L^{(2)}(E,1)/2!$$ ≈ $$14.7178566783$$

## 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$$(j)_{-}$$
$$2$$ $$4$$ $$I_4^{*}$$ Additive -1 4 12 0
$$3$$ $$4$$ $$I_1^{*}$$ Additive -1 2 7 1
$$5$$ $$2$$ $$I_0^{*}$$ Additive 1 2 6 0
$$7$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$11$$ $$1$$ $$I_{1}$$ Non-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 X36.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right)$ 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

## $p$-adic data

### $p$-adic regulators

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

$$p$$-adic regulators are not yet computed for curves that are not $$\Gamma_0$$-optimal.

No Iwasawa invariant data is available for this curve.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 4 and 8.
Its isogeny class 277200.gy consists of 6 curves linked by isogenies of degrees dividing 8.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{33})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
$$\Q(\sqrt{15})$$ $$\Z/4\Z$$ Not in database
$$\Q(\sqrt{55})$$ $$\Z/4\Z$$ Not in database
4 $$\Q(\sqrt{15}, \sqrt{33})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$$\Q(\sqrt{15}, \sqrt{21})$$ $$\Z/8\Z$$ Not in database
$$\Q(\sqrt{15}, \sqrt{77})$$ $$\Z/8\Z$$ 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.