# Properties

 Label 75.a2 Conductor $75$ Discriminant $-6075$ j-invariant $$\frac{20480}{243}$$ CM no Rank $0$ Torsion structure $$\Z/{5}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2+y=x^3+x^2+2x+4$$ y^2+y=x^3+x^2+2x+4 (homogenize, simplify) $$y^2z+yz^2=x^3+x^2z+2xz^2+4z^3$$ y^2z+yz^2=x^3+x^2z+2xz^2+4z^3 (dehomogenize, simplify) $$y^2=x^3+2160x+170640$$ y^2=x^3+2160x+170640 (homogenize, minimize)

sage: E = EllipticCurve([0, 1, 1, 2, 4])

gp: E = ellinit([0, 1, 1, 2, 4])

magma: E := EllipticCurve([0, 1, 1, 2, 4]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

$$\Z/{5}\Z$$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(2, 4\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-1, 1\right)$$, $$\left(-1, -2\right)$$, $$\left(2, 4\right)$$, $$\left(2, -5\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$75$$ = $3 \cdot 5^{2}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-6075$ = $-1 \cdot 3^{5} \cdot 5^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{20480}{243}$$ = $2^{12} \cdot 3^{-5} \cdot 5$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.59309833409467281468519193126\dots$ Stable Faltings height: $-0.86133798616702287711865182013\dots$

## 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: E.omega[1]  magma: RealPeriod(E); Real period: $3.1361746474819615129577581845\dots$ 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: $5$  = $5\cdot1$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $5$ 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: $L(E,1)$ ≈ $0.62723492949639230259155163689$

## 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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 6 $\Gamma_0(N)$-optimal: yes 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$(j)_{-}$
$3$ $5$ $I_{5}$ Split multiplicative -1 1 5 5
$5$ $1$ $II$ Additive 1 2 2 0

## 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
$5$ 5B.1.1 5.24.0.1

## $p$-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) 2 3 5 ss split add 0,1 1 - 0,0 0 -

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.

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 $d=$ 5.
Its isogeny class 75.a consists of 2 curves linked by isogenies of degree 5.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{5}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $3$ 3.1.300.1 $$\Z/10\Z$$ Not in database $6$ 6.0.270000.1 $$\Z/2\Z \oplus \Z/10\Z$$ Not in database $8$ 8.2.2767921875.1 $$\Z/15\Z$$ Not in database $12$ Deg 12 $$\Z/20\Z$$ Not in database $20$ 20.0.4656612873077392578125.1 $$\Z/5\Z \oplus \Z/5\Z$$ 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.