# Properties

 Label 275.a2 Conductor $275$ Discriminant $1143828125$ j-invariant $$\frac{2749884201}{73205}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Learn more

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2+xy+y=x^3-x^2-730x-7228$$ y^2+xy+y=x^3-x^2-730x-7228 (homogenize, simplify) $$y^2z+xyz+yz^2=x^3-x^2z-730xz^2-7228z^3$$ y^2z+xyz+yz^2=x^3-x^2z-730xz^2-7228z^3 (dehomogenize, simplify) $$y^2=x^3-11675x-474250$$ y^2=x^3-11675x-474250 (homogenize, minimize)

sage: E = EllipticCurve([1, -1, 1, -730, -7228])

gp: E = ellinit([1, -1, 1, -730, -7228])

magma: E := EllipticCurve([1, -1, 1, -730, -7228]);

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);

 $P$ = $$\left(-16, 20\right)$$ (-16, 20) $\hat{h}(P)$ ≈ $0.59612262310663940712038741071$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-\frac{69}{4}, \frac{65}{8}\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-16, 20\right)$$, $$\left(-16, -5\right)$$, $$\left(39, 130\right)$$, $$\left(39, -170\right)$$, $$\left(134, 1445\right)$$, $$\left(134, -1580\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$275$$ = $5^{2} \cdot 11$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $1143828125$ = $5^{7} \cdot 11^{4}$ 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: $0.51905347624229128663739225697\dots$ Stable Faltings height: $-0.28566547997475890066298740964\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.59612262310663940712038741071\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.92007028992774512545841083606\dots$ 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: $8$  = $2^{2}\cdot2$ 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: $L'(E,1)$ ≈ $1.0969494293484273097814192758$

## 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);

$$q - q^{2} - q^{4} + 3 q^{8} - 3 q^{9} - q^{11} - 2 q^{13} - q^{16} - 6 q^{17} + 3 q^{18} - 4 q^{19} + O(q^{20})$$

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 not semistable. There are 2 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$(j)_{-}$
$5$ $4$ $I_{1}^{*}$ Additive 1 2 7 1
$11$ $2$ $I_{4}$ Non-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 4.12.0.8

## $p$-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 ss add ss nonsplit ord ord ord ord ord ord ord ord ord ord 1 1,1 - 1,1 1 1 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

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 275.a 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 $K$ $2$ $$\Q(\sqrt{5})$$ $$\Z/2\Z \oplus \Z/2\Z$$ 2.2.5.1-605.1-c4 $2$ $$\Q(\sqrt{-1})$$ $$\Z/4\Z$$ 2.0.4.1-75625.3-a3 $2$ $$\Q(\sqrt{-5})$$ $$\Z/4\Z$$ Not in database $4$ 4.2.2000.1 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(i, \sqrt{5})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.64000000.3 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.14992384000000.10 $$\Z/8\Z$$ Not in database $8$ 8.0.599695360000.19 $$\Z/8\Z$$ Not in database $8$ 8.2.500310421875.1 $$\Z/6\Z$$ 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 $16$ 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.