# Properties

 Label 111600.de3 Conductor $111600$ Discriminant $-2.388\times 10^{24}$ j-invariant $$-\frac{749011598724977281}{51173462246460}$$ CM no Rank $2$ Torsion structure $$\Z/{4}\Z$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -68112075, -228779279750]) # or

sage: E = EllipticCurve("111600.de3")

gp: E = ellinit([0, 0, 0, -68112075, -228779279750]) \\ or

gp: E = ellinit("111600.de3")

magma: E := EllipticCurve([0, 0, 0, -68112075, -228779279750]); // or

magma: E := EllipticCurve("111600.de3");

$$y^2=x^3-68112075x-228779279750$$

## Mordell-Weil group structure

$$\Z^2 \times \Z/{4}\Z$$

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(10055, 320850\right)$$ $$\left(2422661, 3770825616\right)$$ $$\hat{h}(P)$$ ≈ $2.5247930649201248203583968363$ $5.4321543200085436943937267588$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(24005, 3459600\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(9590, 0\right)$$, $$(10055,\pm 320850)$$, $$(16565,\pm 1785600)$$, $$(24005,\pm 3459600)$$, $$(31199,\pm 5292882)$$, $$(39381,\pm 7626496)$$, $$(222005,\pm 104529600)$$, $$(456455,\pm 308336850)$$, $$(2422661,\pm 3770825616)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$111600$$ = $$2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 31$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-2387549054570837760000000$$ = $$-1 \cdot 2^{14} \cdot 3^{7} \cdot 5^{7} \cdot 31^{8}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{749011598724977281}{51173462246460}$$ = $$-1 \cdot 2^{-2} \cdot 3^{-1} \cdot 5^{-1} \cdot 31^{-8} \cdot 71^{3} \cdot 12791^{3}$$ 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); Analytic rank: $$2$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$13.577059281864077791297080748$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.026173125079378563878245344769$$ 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: $$512$$  = $$2^{2}\cdot2^{2}\cdot2^{2}\cdot2^{3}$$ 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 111600.2.a.de

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);

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

For more coefficients, see the Downloads section to the right.

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 18874368 $$\Gamma_0(N)$$-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);

$$L^{(2)}(E,1)/2!$$ ≈ $$11.371330265419718672816447132996509412$$

## Local data

This elliptic curve is semistable. There are 4 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$$(j)_{-}$$
$$2$$ $$4$$ $$I_6^{*}$$ Additive -1 4 14 2
$$3$$ $$4$$ $$I_1^{*}$$ Additive -1 2 7 1
$$5$$ $$4$$ $$I_1^{*}$$ Additive 1 2 7 1
$$31$$ $$8$$ $$I_{8}$$ Split multiplicative -1 1 8 8

## 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 X85p.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 3 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 1 \end{array}\right)$ and has index 48.

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
$$2$$ 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.

## 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 add add add ss ordinary ordinary ordinary ordinary ordinary ordinary split ordinary ordinary ordinary ss - - - 2,2 2 2 2 2 2 2 3 2 2 2 2,2 - - - 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, 4 and 8.
Its isogeny class 111600.de consists of 4 curves linked by isogenies of degrees dividing 8.

## Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-15})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ 4.2.54000.2 $$\Z/8\Z$$ Not in database $4$ $$\Q(\sqrt{2}, \sqrt{-15})$$ $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ 8.0.746496000000.4 $$\Z/4\Z \times \Z/4\Z$$ Not in database $8$ 8.0.2916000000.2 $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ Deg 8 $$\Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/4\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/4\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/12\Z$$ Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.