# Properties

 Label 277200.fl3 Conductor $277200$ Discriminant $7.531\times 10^{18}$ j-invariant $$\frac{2177286259681}{161417025}$$ CM no Rank $2$ Torsion structure $$\Z/{2}\Z \times \Z/{2}\Z$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -972075, 344452250]) # or

sage: E = EllipticCurve("277200fl2")

gp: E = ellinit([0, 0, 0, -972075, 344452250]) \\ or

gp: E = ellinit("277200fl2")

magma: E := EllipticCurve([0, 0, 0, -972075, 344452250]); // or

magma: E := EllipticCurve("277200fl2");

$$y^2 = x^{3} - 972075 x + 344452250$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(2335, -103950\right)$$ $$\left(-41, 19602\right)$$ $$\hat{h}(P)$$ ≈ $1.0740602801785843$ $2.174965581963359$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(685, 0\right)$$, $$\left(445, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-1130, 0\right)$$, $$(-899,\pm 22176)$$, $$(-305,\pm 24750)$$, $$(-41,\pm 19602)$$, $$(85,\pm 16200)$$, $$(245,\pm 11000)$$, $$(391,\pm 4914)$$, $$(410,\pm 3850)$$, $$\left(445, 0\right)$$, $$\left(685, 0\right)$$, $$(806,\pm 9196)$$, $$(949,\pm 16632)$$, $$(1495,\pm 47250)$$, $$(2335,\pm 103950)$$, $$(4285,\pm 273600)$$, $$(7445,\pm 637000)$$, $$(11245,\pm 1188000)$$, $$(28570,\pm 4826250)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$277200$$ = $$2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 11$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$7531072718400000000$$ = $$2^{12} \cdot 3^{8} \cdot 5^{8} \cdot 7^{2} \cdot 11^{4}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{2177286259681}{161417025}$$ = $$3^{-2} \cdot 5^{-2} \cdot 7^{-2} \cdot 11^{-4} \cdot 13^{3} \cdot 997^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$2$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$2.00210245123$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.229790053241$$ 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: $$512$$  = $$2^{2}\cdot2^{2}\cdot2^{2}\cdot2\cdot2^{2}$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$4$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (rounded)

## Modular invariants

Modular form 277200.2.a.fl

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 6291456 $$\Gamma_0(N)$$-optimal: no Manin constant: 1

#### Special L-value

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);

$$L^{(2)}(E,1)/2!$$ ≈ $$14.7220233236$$

## Local data

This elliptic curve is not semistable.

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$$(j)_{-}$$
$$2$$ $$4$$ $$I_4^{*}$$ Additive -1 4 12 0
$$3$$ $$4$$ $$I_2^{*}$$ Additive -1 2 8 2
$$5$$ $$4$$ $$I_2^{*}$$ Additive 1 2 8 2
$$7$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$11$$ $$4$$ $$I_{4}$$ Split multiplicative -1 1 4 4

## 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 X25k.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 0 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right)$ and has index 24.

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 mod $$p$$ Galois representation has maximal image $$\GL(2,\F_p)$$ for all primes $$p$$ except those listed.

prime Image of Galois representation
$$2$$ Cs

## $p$-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 and 4.
Its isogeny class 277200.fl 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ $$\Q(\sqrt{15})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$4$ $$\Q(\sqrt{-7}, \sqrt{-15})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$4$ $$\Q(\sqrt{7}, \sqrt{-15})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$4$ $$\Q(\sqrt{11}, \sqrt{15})$$ $$\Z/2\Z \times \Z/8\Z$$ 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.