# Properties

 Label 546.g2 Conductor $546$ Discriminant $298116$ j-invariant $$\frac{2181825073}{298116}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z \oplus \Z/{2}\Z$$

# Learn more

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2+xy=x^3-27x+45$$ y^2+xy=x^3-27x+45 (homogenize, simplify) $$y^2z+xyz=x^3-27xz^2+45z^3$$ y^2z+xyz=x^3-27xz^2+45z^3 (dehomogenize, simplify) $$y^2=x^3-35019x+2204550$$ y^2=x^3-35019x+2204550 (homogenize, minimize)

sage: E = EllipticCurve([1, 0, 0, -27, 45])

gp: E = ellinit([1, 0, 0, -27, 45])

magma: E := EllipticCurve([1, 0, 0, -27, 45]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

$$\Z/{2}\Z \oplus \Z/{2}\Z$$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-6, 3\right)$$, $$\left(2, -1\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-6, 3\right)$$, $$\left(2, -1\right)$$

## 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: $298116$ = $2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{2181825073}{298116}$$ = $2^{-2} \cdot 3^{-2} \cdot 7^{-2} \cdot 13^{-2} \cdot 1297^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.22260093612944015531022418482\dots$ Stable Faltings height: $-0.22260093612944015531022418482\dots$

## 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: $2.9547312603904991530188126999\dots$ 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: $16$  = $2\cdot2\cdot2\cdot2$ 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: $1$ (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: $L(E,1)$ ≈ $2.9547312603904991530188126999$

## 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);

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

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 96 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$3$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$7$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$13$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2

## 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
$2$ 2Cs 8.12.0.1
sage: gens = [[2017, 4, 1850, 9], [1563, 2, 1870, 2183], [1, 4, 0, 1], [1457, 4, 730, 9], [1, 0, 4, 1], [2181, 4, 2180, 5], [1093, 2, 0, 1], [1639, 4, 0, 1]]

sage: GL(2,Integers(2184)).subgroup(gens)

magma: Gens := [[2017, 4, 1850, 9], [1563, 2, 1870, 2183], [1, 4, 0, 1], [1457, 4, 730, 9], [1, 0, 4, 1], [2181, 4, 2180, 5], [1093, 2, 0, 1], [1639, 4, 0, 1]];

magma: sub<GL(2,Integers(2184))|Gens>;

The image of the adelic Galois representation has level $2184$, index $48$, genus $0$, and generators

$\left(\begin{array}{rr} 2017 & 4 \\ 1850 & 9 \end{array}\right),\left(\begin{array}{rr} 1563 & 2 \\ 1870 & 2183 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1457 & 4 \\ 730 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 2181 & 4 \\ 2180 & 5 \end{array}\right),\left(\begin{array}{rr} 1093 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1639 & 4 \\ 0 & 1 \end{array}\right)$

## $p$-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 split split split nonsplit 2 1 1 0 0 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 546.g consists of 4 curves linked by isogenies of degrees dividing 4.

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

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $4$ $$\Q(\sqrt{7}, \sqrt{39})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{2}, \sqrt{-7})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-2}, \sqrt{-39})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.2.194365577860272.8 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $16$ 16.0.132513778481397717569346994176.17 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ 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.