# Properties

 Label 53361bm1 Conductor $53361$ Discriminant $-1.256\times 10^{12}$ j-invariant $$-121$$ CM no Rank $2$ Torsion structure trivial

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, -1112, -55492])

gp: E = ellinit([1, -1, 1, -1112, -55492])

magma: E := EllipticCurve([1, -1, 1, -1112, -55492]);

$$y^2+xy+y=x^3-x^2-1112x-55492$$

## Mordell-Weil group structure

$$\Z^2$$

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(58, 240\right)$$ $$\left(597, 14254\right)$$ $$\hat{h}(P)$$ ≈ $0.90336827061961534309571528724$ $1.7912477935888324968358801339$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(48, -20\right)$$, $$\left(48, -29\right)$$, $$\left(58, 240\right)$$, $$\left(58, -299\right)$$, $$\left(102, 889\right)$$, $$\left(102, -992\right)$$, $$\left(157, 1824\right)$$, $$\left(157, -1982\right)$$, $$\left(597, 14254\right)$$, $$\left(597, -14852\right)$$, $$\left(405037, 257572984\right)$$, $$\left(405037, -257978022\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$53361$$ = $$3^{2} \cdot 7^{2} \cdot 11^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-1255701777561$$ = $$-1 \cdot 3^{6} \cdot 7^{6} \cdot 11^{4}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-121$$ = $$-1 \cdot 11^{2}$$ 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: $$1.5470453485844442395067601164$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.36358524797052414255707168895$$ 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: $$12$$  = $$2\cdot2\cdot3$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$1$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (rounded)

## Modular invariants

Modular form 53361.2.a.r

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);

$$q - q^{2} - q^{4} + q^{5} + 3q^{8} - q^{10} - q^{13} - q^{16} - 5q^{17} - 6q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 51840 $$\Gamma_0(N)$$-optimal: yes 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);

$$L^{(2)}(E,1)/2!$$ ≈ $$6.7497944002406534353273963246450983859$$

## Local data

This elliptic curve is not semistable. There are 3 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$$(j)_{-}$$
$$3$$ $$2$$ $$I_0^{*}$$ Additive -1 2 6 0
$$7$$ $$2$$ $$I_0^{*}$$ Additive -1 2 6 0
$$11$$ $$3$$ $$IV$$ Additive -1 2 4 0

## 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 X3.

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

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
$$11$$ B.10.4

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]

Note: $$p$$-adic regulator data only exists for primes $$p\ge 5$$ 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 ordinary add ordinary add add ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss ordinary ? - 2 - - 2 4 2 2 2 2 2 2 2,2 2 ? - 0 - - 0 0 0 0 0 0 0 0 0,0 0

An entry ? indicates that the invariants have not yet been computed.

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 $$d=$$ 11.
Its isogeny class 53361bm consists of 2 curves linked by isogenies of degree 11.

## 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $3$ 3.1.484.1 $$\Z/2\Z$$ Not in database $6$ 6.0.937024.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $8$ 8.2.76879700667.4 $$\Z/3\Z$$ Not in database $10$ 10.0.9630096522760791.1 $$\Z/11\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ 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.