# Properties

 Label 825.c1 Conductor $825$ Discriminant $-1276171875$ j-invariant $$-\frac{56197120}{3267}$$ CM no Rank $1$ Torsion structure $$\Z/{3}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2+y=x^3+x^2-583x+5494$$ y^2+y=x^3+x^2-583x+5494 (homogenize, simplify) $$y^2z+yz^2=x^3+x^2z-583xz^2+5494z^3$$ y^2z+yz^2=x^3+x^2z-583xz^2+5494z^3 (dehomogenize, simplify) $$y^2=x^3-756000x+265410000$$ y^2=x^3-756000x+265410000 (homogenize, minimize)

sage: E = EllipticCurve([0, 1, 1, -583, 5494])

gp: E = ellinit([0, 1, 1, -583, 5494])

magma: E := EllipticCurve([0, 1, 1, -583, 5494]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(14, 16\right)$$ (14, 16) $\hat{h}(P)$ ≈ $0.75080242449875493421233634991$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(8, 37\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-28, 25\right)$$, $$\left(-28, -26\right)$$, $$\left(8, 37\right)$$, $$\left(8, -38\right)$$, $$\left(14, 16\right)$$, $$\left(14, -17\right)$$, $$\left(58, 412\right)$$, $$\left(58, -413\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$825$$ = $3 \cdot 5^{2} \cdot 11$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-1276171875$ = $-1 \cdot 3^{3} \cdot 5^{8} \cdot 11^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{56197120}{3267}$$ = $-1 \cdot 2^{15} \cdot 3^{-3} \cdot 5 \cdot 7^{3} \cdot 11^{-2}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.50350814692454616825465761032\dots$ Stable Faltings height: $-0.56945046136485408147918194516\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.75080242449875493421233634991\dots$ 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: $1.5092459477680003368989645301\dots$ 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: $18$  = $3\cdot3\cdot2$ 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: $L'(E,1)$ ≈ $2.2662910334982718119316468866$

## 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);

$$q + q^{3} - 2 q^{4} - q^{7} + q^{9} - q^{11} - 2 q^{12} - q^{13} + 4 q^{16} - 6 q^{17} - 7 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 360 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## 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$(j)_{-}$
$3$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$5$ $3$ $IV^{*}$ Additive -1 2 8 0
$11$ $2$ $I_{2}$ 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
$3$ 3B.1.1 3.8.0.1

The image of the adelic Galois representation has level $6$, index $16$, and genus $0$.

## $p$-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 ss split add ord nonsplit ord ord ord ord ord ord ord ord ord ss 1,2 2 - 1 1 3 1 1 1 1 1 1 1 1 1,1 0,0 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 non-trivial cyclic isogenies of degree $d$ for $d=$ 3.
Its isogeny class 825.c consists of 2 curves linked by isogenies of degree 3.

## 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 $K$ $3$ 3.1.300.1 $$\Z/6\Z$$ Not in database $6$ 6.0.270000.1 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $6$ 6.0.247066875.1 $$\Z/3\Z \oplus \Z/3\Z$$ Not in database $9$ 9.3.3282536677921875.3 $$\Z/9\Z$$ Not in database $12$ Deg 12 $$\Z/12\Z$$ Not in database $18$ 18.0.61773685738999443000000000000.1 $$\Z/6\Z \oplus \Z/6\Z$$ 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.