# Properties

 Label 75c2 Conductor $75$ Discriminant $-29296875$ j-invariant $$-\frac{102400}{3}$$ CM no Rank $0$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2+y=x^3+x^2-208x-1256$$ y^2+y=x^3+x^2-208x-1256 (homogenize, simplify) $$y^2z+yz^2=x^3+x^2z-208xz^2-1256z^3$$ y^2z+yz^2=x^3+x^2z-208xz^2-1256z^3 (dehomogenize, simplify) $$y^2=x^3-270000x-55350000$$ y^2=x^3-270000x-55350000 (homogenize, minimize)

sage: E = EllipticCurve([0, 1, 1, -208, -1256])

gp: E = ellinit([0, 1, 1, -208, -1256])

magma: E := EllipticCurve([0, 1, 1, -208, -1256]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

trivial

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

None

## 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: $-29296875$ = $-1 \cdot 3 \cdot 5^{10}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{102400}{3}$$ = $-1 \cdot 2^{12} \cdot 3^{-1} \cdot 5^{2}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.21162062212237737261518773535\dots$ Stable Faltings height: $-1.1295776382393729395521117090\dots$

## 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: $0.62723492949639230259155163689\dots$ 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: $1$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $1$ 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: 30 $\Gamma_0(N)$-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$(j)_{-}$
$3$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$5$ $1$ $II^{*}$ Additive 1 2 10 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.1.2 5.24.0.3

## $p$-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 75c 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $3$ 3.1.300.1 $$\Z/2\Z$$ Not in database $4$ $$\Q(\zeta_{5})$$ $$\Z/5\Z$$ Not in database $5$ 5.1.253125.1 $$\Z/5\Z$$ Not in database $6$ 6.0.270000.1 $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $8$ 8.2.2767921875.1 $$\Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database $12$ 12.0.1012500000000.1 $$\Z/10\Z$$ Not in database $15$ 15.1.1245564843750000000000.1 $$\Z/10\Z$$ Not in database $20$ 20.0.513156902790069580078125.1 $$\Z/5\Z \oplus \Z/5\Z$$ 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.