# Properties

 Label 546.g1 Conductor $546$ Discriminant $187278$ j-invariant $$\frac{8020417344913}{187278}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z$$

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Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2+xy=x^3-417x+3243$$ y^2+xy=x^3-417x+3243 (homogenize, simplify) $$y^2z+xyz=x^3-417xz^2+3243z^3$$ y^2z+xyz=x^3-417xz^2+3243z^3 (dehomogenize, simplify) $$y^2=x^3-540459x+152926758$$ y^2=x^3-540459x+152926758 (homogenize, minimize)

sage: E = EllipticCurve([1, 0, 0, -417, 3243])

gp: E = ellinit([1, 0, 0, -417, 3243])

magma: E := EllipticCurve([1, 0, 0, -417, 3243]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

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

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(\frac{47}{4}, -\frac{47}{8}\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

None

## 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: $187278$ = $2 \cdot 3 \cdot 7^{4} \cdot 13$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{8020417344913}{187278}$$ = $2^{-1} \cdot 3^{-1} \cdot 7^{-4} \cdot 13^{-1} \cdot 37^{3} \cdot 541^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.12397265415053249939839187591\dots$ Stable Faltings height: $0.12397265415053249939839187591\dots$

## 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: $4$  = $1\cdot1\cdot2^{2}\cdot1$ 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) 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: 192 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$3$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$7$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$13$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1

## 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
$2$ 2B 8.12.0.8
sage: gens = [[2177, 8, 2176, 9], [1, 0, 8, 1], [1, 8, 0, 1], [1, 4, 4, 17], [736, 3, 733, 2], [1249, 8, 628, 33], [848, 3, 1349, 2], [1912, 827, 1915, 1944], [1368, 281, 811, 798], [7, 6, 2178, 2179]]

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

magma: Gens := [[2177, 8, 2176, 9], [1, 0, 8, 1], [1, 8, 0, 1], [1, 4, 4, 17], [736, 3, 733, 2], [1249, 8, 628, 33], [848, 3, 1349, 2], [1912, 827, 1915, 1944], [1368, 281, 811, 798], [7, 6, 2178, 2179]];

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

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

$\left(\begin{array}{rr} 2177 & 8 \\ 2176 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 736 & 3 \\ 733 & 2 \end{array}\right),\left(\begin{array}{rr} 1249 & 8 \\ 628 & 33 \end{array}\right),\left(\begin{array}{rr} 848 & 3 \\ 1349 & 2 \end{array}\right),\left(\begin{array}{rr} 1912 & 827 \\ 1915 & 1944 \end{array}\right),\left(\begin{array}{rr} 1368 & 281 \\ 811 & 798 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 2178 & 2179 \end{array}\right)$

## $p$-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 and 4.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{78})$$ $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{39})$$ $$\Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{2})$$ $$\Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{2}, \sqrt{39})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.14758681031737344.5 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.8.34605071442579456.6 $$\Z/8\Z$$ Not in database $8$ deg 8 $$\Z/8\Z$$ Not in database $8$ 8.2.194365577860272.8 $$\Z/6\Z$$ Not in database $16$ deg 16 $$\Z/4\Z \oplus \Z/4\Z$$ 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 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $16$ deg 16 $$\Z/12\Z$$ Not in database $16$ deg 16 $$\Z/12\Z$$ 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.