# Properties

 Label 2475j4 Conductor $2475$ Discriminant $833850703125$ j-invariant $$\frac{2749884201}{73205}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2+xy=x^3-x^2-6567x+201716$$ y^2+xy=x^3-x^2-6567x+201716 (homogenize, simplify) $$y^2z+xyz=x^3-x^2z-6567xz^2+201716z^3$$ y^2z+xyz=x^3-x^2z-6567xz^2+201716z^3 (dehomogenize, simplify) $$y^2=x^3-105075x+12804750$$ y^2=x^3-105075x+12804750 (homogenize, minimize)

sage: E = EllipticCurve([1, -1, 0, -6567, 201716])

gp: E = ellinit([1, -1, 0, -6567, 201716])

magma: E := EllipticCurve([1, -1, 0, -6567, 201716]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(28, 184\right)$$ (28, 184) $\hat{h}(P)$ ≈ $1.0787475094281050123631454675$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(\frac{211}{4}, -\frac{211}{8}\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-16, 558\right)$$, $$\left(-16, -542\right)$$, $$\left(28, 184\right)$$, $$\left(28, -212\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$2475$$ = $3^{2} \cdot 5^{2} \cdot 11$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $833850703125$ = $3^{6} \cdot 5^{7} \cdot 11^{4}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{2749884201}{73205}$$ = $3^{3} \cdot 5^{-1} \cdot 11^{-4} \cdot 467^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.0683596205763461323350148754\dots$ Stable Faltings height: $-0.28566547997475890066298740967\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $1.0787475094281050123631454675\dots$ 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: $0.88886178683078034013809041399\dots$ 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: $16$  = $2\cdot2\cdot2^{2}$ 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: $L'(E,1)$ ≈ $3.8354297550780779307618261547$

## 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);

$$q + q^{2} - q^{4} - 3 q^{8} + q^{11} - 2 q^{13} - q^{16} + 6 q^{17} - 4 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 3072 $\Gamma_0(N)$-optimal: no 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$(j)_{-}$
$3$ $2$ $I_0^{*}$ Additive -1 2 6 0
$5$ $2$ $I_{1}^{*}$ Additive 1 2 7 1
$11$ $4$ $I_{4}$ Split multiplicative -1 1 4 4

## 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 4.6.0.1
sage: gens = [[1, 0, 8, 1], [1, 8, 0, 1], [1, 4, 4, 17], [499, 498, 618, 835], [1052, 879, 1209, 434], [1313, 8, 1312, 9], [1201, 888, 1284, 913], [499, 936, 738, 61], [439, 0, 0, 1319], [7, 6, 1314, 1315]]

sage: GL(2,Integers(1320)).subgroup(gens)

magma: Gens := [[1, 0, 8, 1], [1, 8, 0, 1], [1, 4, 4, 17], [499, 498, 618, 835], [1052, 879, 1209, 434], [1313, 8, 1312, 9], [1201, 888, 1284, 913], [499, 936, 738, 61], [439, 0, 0, 1319], [7, 6, 1314, 1315]];

magma: sub<GL(2,Integers(1320))|Gens>;

The image of the adelic Galois representation has level $1320$, index $48$, genus $0$, and generators

$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 499 & 498 \\ 618 & 835 \end{array}\right),\left(\begin{array}{rr} 1052 & 879 \\ 1209 & 434 \end{array}\right),\left(\begin{array}{rr} 1313 & 8 \\ 1312 & 9 \end{array}\right),\left(\begin{array}{rr} 1201 & 888 \\ 1284 & 913 \end{array}\right),\left(\begin{array}{rr} 499 & 936 \\ 738 & 61 \end{array}\right),\left(\begin{array}{rr} 439 & 0 \\ 0 & 1319 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 1314 & 1315 \end{array}\right)$

## $p$-adic regulators

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

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.

## 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 ord add add ss split ord ord ord ord ord ord ord ord ord ord 3 - - 1,1 2 1 1 1 1 3 1 1 1 1 1 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 $d=$ 2 and 4.
Its isogeny class 2475j 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 $K$ $2$ $$\Q(\sqrt{5})$$ $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{15})$$ $$\Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{3})$$ $$\Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{3}, \sqrt{5})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.324000000.4 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.8.48575324160000.14 $$\Z/8\Z$$ Not in database $8$ 8.0.1214383104000000.4 $$\Z/8\Z$$ Not in database $8$ 8.2.500310421875.2 $$\Z/6\Z$$ Not in database $16$ 16.0.26873856000000000000.6 $$\Z/4\Z \oplus \Z/4\Z$$ 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.