# Properties

 Label 60984ca1 Conductor $60984$ Discriminant $-143775024624$ j-invariant $$-\frac{91625216}{9261}$$ CM no Rank $2$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -2343, 47311])

gp: E = ellinit([0, 0, 0, -2343, 47311])

magma: E := EllipticCurve([0, 0, 0, -2343, 47311]);

$$y^2=x^3-2343x+47311$$

## Mordell-Weil group structure

$\Z^2$

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(5, 189\right)$$ $$\left(33, 77\right)$$ $\hat{h}(P)$ ≈ $0.27845603830067074593209368893$ $0.73569822977002245840062557250$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(-55,\pm 99)$$, $$(-30,\pm 301)$$, $$(-22,\pm 297)$$, $$(5,\pm 189)$$, $$(26,\pm 63)$$, $$(33,\pm 77)$$, $$(113,\pm 1107)$$, $$(221,\pm 3213)$$, $$(341,\pm 6237)$$, $$(1266,\pm 45013)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$60984$$ = $2^{3} \cdot 3^{2} \cdot 7 \cdot 11^{2}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-143775024624$ = $-1 \cdot 2^{4} \cdot 3^{9} \cdot 7^{3} \cdot 11^{3}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{91625216}{9261}$$ = $-1 \cdot 2^{8} \cdot 3^{-3} \cdot 7^{-3} \cdot 71^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.88095372881737140983242559575\dots$ Stable Faltings height: $-0.49887529390292450835309362436\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $2$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.20461900218626478404177070107\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $1.0066920720883843034915876727\dots$ 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: $48$  = $2\cdot2^{2}\cdot3\cdot2$ 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) 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); Special value: $L^{(2)}(E,1)/2!$ ≈ $9.8874397103783296020969287904658423917$

## Modular invariants

Modular form 60984.2.a.ba

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);

$$q - q^{5} + q^{7} - q^{13} - 8q^{17} - 5q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 55296 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## Local data

This elliptic curve is not 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$(j)_{-}$
$2$ $2$ $III$ Additive -1 3 4 0
$3$ $4$ $I_3^{*}$ Additive -1 2 9 3
$7$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$11$ $2$ $III$ Additive 1 2 3 0

## Galois representations

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 $\ell$-adic Galois representation has maximal image $\GL(2,\Z_\ell)$ for all primes $\ell$.

## $p$-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 add add ordinary split add ordinary ordinary ordinary ss ordinary ordinary ordinary ss ordinary ordinary - - 4 3 - 2 2 2 2,2 2 2 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 no rational isogenies. Its isogeny class 60984ca consists of this curve only.

## 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 $K$ $3$ 3.1.231.1 $$\Z/2\Z$$ Not in database $6$ 6.0.12326391.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $8$ 8.2.3967389600768.3 $$\Z/3\Z$$ 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.