# Properties

 Label 1122i2 Conductor $1122$ Discriminant $-264305213568$ j-invariant $$-\frac{2035346265217}{264305213568}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -264, 24768])

gp: E = ellinit([1, 0, 0, -264, 24768])

magma: E := EllipticCurve([1, 0, 0, -264, 24768]);

$$y^2+xy=x^3-264x+24768$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(6, 150\right)$$ $$\hat{h}(P)$$ ≈ $0.061869181235238456817779665213$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-\frac{129}{4}, \frac{129}{8}\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-28, 116\right)$$, $$\left(-28, -88\right)$$, $$\left(-24, 144\right)$$, $$\left(-24, -120\right)$$, $$\left(-12, 168\right)$$, $$\left(-12, -156\right)$$, $$\left(6, 150\right)$$, $$\left(6, -156\right)$$, $$\left(24, 168\right)$$, $$\left(24, -192\right)$$, $$\left(42, 276\right)$$, $$\left(42, -318\right)$$, $$\left(78, 654\right)$$, $$\left(78, -732\right)$$, $$\left(108, 1068\right)$$, $$\left(108, -1176\right)$$, $$\left(312, 5352\right)$$, $$\left(312, -5664\right)$$, $$\left(636, 15720\right)$$, $$\left(636, -16356\right)$$, $$\left(3474, 203028\right)$$, $$\left(3474, -206502\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$1122$$ = $$2 \cdot 3 \cdot 11 \cdot 17$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-264305213568$$ = $$-1 \cdot 2^{7} \cdot 3^{10} \cdot 11^{2} \cdot 17^{2}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{2035346265217}{264305213568}$$ = $$-1 \cdot 2^{-7} \cdot 3^{-10} \cdot 11^{-2} \cdot 17^{-2} \cdot 19^{3} \cdot 23^{3} \cdot 29^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$0.87113429906744046995289314110\dots$$ Stable Faltings height: $$0.87113429906744046995289314110\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.061869181235238456817779665213\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.80409727800559556494218245603\dots$$ 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: $$280$$  = $$7\cdot( 2 \cdot 5 )\cdot2\cdot2$$ 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)

## Modular invariants

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);

$$q + q^{2} + q^{3} + q^{4} - 2q^{5} + q^{6} - 4q^{7} + q^{8} + q^{9} - 2q^{10} - q^{11} + q^{12} - 4q^{13} - 4q^{14} - 2q^{15} + q^{16} - q^{17} + q^{18} - 8q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 2240 $$\Gamma_0(N)$$-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);

$$L'(E,1)$$ ≈ $$3.4824188156583079680733022639616856269$$

## 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$$(j)_{-}$$
$$2$$ $$7$$ $$I_{7}$$ Split multiplicative -1 1 7 7
$$3$$ $$10$$ $$I_{10}$$ Split multiplicative -1 1 10 10
$$11$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$17$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2

## 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 X19.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 0 & 5 \end{array}\right)$ and has index 6.

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(5,20) 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 split split ordinary ordinary nonsplit ordinary nonsplit ordinary ss ss ordinary ordinary ordinary ordinary ordinary 2 2 1 1 1 3 1 1 1,3 1,1 1 1 1 1 1 1 0 0 0 0 0 0 0 0,0 0,0 0 0 0 0 0

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2.
Its isogeny class 1122i consists of 2 curves linked by isogenies of degree 2.

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

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-2})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $4$ 4.2.10071072.4 $$\Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.0.6491295438667776.21 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.2.3465933379972272.3 $$\Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/6\Z$$ 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.