# Properties

 Label 546d2 Conductor $546$ Discriminant $-10417365504$ j-invariant $$-\frac{198461344537}{10417365504}$$ CM no Rank $0$ Torsion structure $$\Z/{3}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2+xy+y=x^3-122x-4948$$ y^2+xy+y=x^3-122x-4948 (homogenize, simplify) $$y^2z+xyz+yz^2=x^3-122xz^2-4948z^3$$ y^2z+xyz+yz^2=x^3-122xz^2-4948z^3 (dehomogenize, simplify) $$y^2=x^3-157491x-230369778$$ y^2=x^3-157491x-230369778 (homogenize, minimize)

sage: E = EllipticCurve([1, 0, 1, -122, -4948])

gp: E = ellinit([1, 0, 1, -122, -4948])

magma: E := EllipticCurve([1, 0, 1, -122, -4948]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

$$\Z/{3}\Z$$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(30, 121\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(30, 121\right)$$, $$\left(30, -152\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$546$$ = $2 \cdot 3 \cdot 7 \cdot 13$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-10417365504$ = $-1 \cdot 2^{9} \cdot 3^{3} \cdot 7^{3} \cdot 13^{3}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{198461344537}{10417365504}$$ = $-1 \cdot 2^{-9} \cdot 3^{-3} \cdot 7^{-3} \cdot 13^{-3} \cdot 19^{3} \cdot 307^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.60215319124588882820324886666\dots$ Stable Faltings height: $0.60215319124588882820324886666\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $0$ sage: E.regulator()  magma: Regulator(E); Regulator: $1$ sage: E.period_lattice().omega()  gp: if(E.disc>0,2,1)*E.omega[1]  magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E); Real period: $0.56365221639777019227436627275\dots$ 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: $27$  = $1\cdot3\cdot3\cdot3$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $3$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $1$ (exact) 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: $L(E,1)$ ≈ $1.6909566491933105768230988183$

## 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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 648 $\Gamma_0(N)$-optimal: no Manin constant: 1

## 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$(j)_{-}$
$2$ $1$ $I_{9}$ Non-split multiplicative 1 1 9 9
$3$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$7$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$13$ $3$ $I_{3}$ Split multiplicative -1 1 3 3

## 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 for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$3$ 3Cs.1.1 3.24.0.1
sage: gens = [[10, 9, 3267, 6544], [1, 18, 0, 1], [3278, 4923, 1593, 5986], [2812, 9, 1863, 6532], [1, 9, 9, 82], [1, 6, 6, 37], [6064, 9, 3663, 76], [1, 0, 18, 1], [1, 12, 0, 1], [3286, 9, 4905, 6544], [6535, 18, 6534, 19]]

sage: GL(2,Integers(6552)).subgroup(gens)

magma: Gens := [[10, 9, 3267, 6544], [1, 18, 0, 1], [3278, 4923, 1593, 5986], [2812, 9, 1863, 6532], [1, 9, 9, 82], [1, 6, 6, 37], [6064, 9, 3663, 76], [1, 0, 18, 1], [1, 12, 0, 1], [3286, 9, 4905, 6544], [6535, 18, 6534, 19]];

magma: sub<GL(2,Integers(6552))|Gens>;

The image of the adelic Galois representation has level $6552$, index $144$, genus $3$, and generators

$\left(\begin{array}{rr} 10 & 9 \\ 3267 & 6544 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3278 & 4923 \\ 1593 & 5986 \end{array}\right),\left(\begin{array}{rr} 2812 & 9 \\ 1863 & 6532 \end{array}\right),\left(\begin{array}{rr} 1 & 9 \\ 9 & 82 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 6064 & 9 \\ 3663 & 76 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3286 & 9 \\ 4905 & 6544 \end{array}\right),\left(\begin{array}{rr} 6535 & 18 \\ 6534 & 19 \end{array}\right)$

## $p$-adic regulators

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

All $p$-adic regulators are identically $1$ since the rank is $0$.

## Iwasawa invariants

$p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 7 13 nonsplit split split split 2 9 1 1 0 1 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3.
Its isogeny class 546d consists of 3 curves linked by isogenies of degrees dividing 9.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-3})$$ $$\Z/3\Z \oplus \Z/3\Z$$ 2.0.3.1-99372.5-k3 $3$ 3.1.2184.1 $$\Z/6\Z$$ Not in database $6$ 6.0.10417365504.1 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $6$ 6.0.14309568.3 $$\Z/3\Z \oplus \Z/6\Z$$ Not in database $9$ 9.3.8148303085695869187.16 $$\Z/9\Z$$ Not in database $12$ deg 12 $$\Z/12\Z$$ Not in database $12$ deg 12 $$\Z/6\Z \oplus \Z/6\Z$$ Not in database $18$ 18.0.3156819565980502545707572443.1 $$\Z/3\Z \oplus \Z/9\Z$$ Not in database $18$ 18.0.2459068265791141604143011767237631147.18 $$\Z/3\Z \oplus \Z/9\Z$$ Not in database $18$ 18.0.1792660765761742229420255578316233106163.1 $$\Z/3\Z \oplus \Z/9\Z$$ 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.