# Properties

 Label 450840.o1 Conductor $450840$ Discriminant $-7.439\times 10^{24}$ j-invariant $$\frac{3103032160078}{1801814625}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 0, 48748424, 7605648076])

gp: E = ellinit([0, -1, 0, 48748424, 7605648076])

magma: E := EllipticCurve([0, -1, 0, 48748424, 7605648076]);

$$y^2=x^3-x^2+48748424x+7605648076$$

## Mordell-Weil group structure

$\Z$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(\frac{4758915379956874282138068992207653}{2231810917400635778931794354596}, \frac{1160952448207186321274275519038226781743393485494949}{3334157386611448286243262070691294689509535144}\right)$$ $\hat{h}(P)$ ≈ $76.693434228852267789367040996$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

None

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$450840$$ = $2^{3} \cdot 3 \cdot 5 \cdot 13 \cdot 17^{2}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-7439252057579115042048000$ = $-1 \cdot 2^{11} \cdot 3^{8} \cdot 5^{3} \cdot 13^{3} \cdot 17^{10}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{3103032160078}{1801814625}$$ = $2 \cdot 3^{-8} \cdot 5^{-3} \cdot 13^{-3} \cdot 17^{5} \cdot 103^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $3.4618505971430317166460948290\dots$ Stable Faltings height: $0.46545456158290178280568653610\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $76.693434228852267789367040996\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.044707687182261296691473426424\dots$ 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: $2$  = $1\cdot2\cdot1\cdot1\cdot1$ 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: $L'(E,1)$ ≈ $6.8575721328737166556801005344119297743$

## Modular invariants

Modular form 450840.2.a.o

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);

$$q - q^{3} - q^{5} + 4q^{7} + q^{9} - 4q^{11} - q^{13} + q^{15} + 8q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 141709824 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## Local data

This elliptic curve is not semistable. There are 5 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$(j)_{-}$
$2$ $1$ $II^{*}$ Additive 1 3 11 0
$3$ $2$ $I_{8}$ Non-split multiplicative 1 1 8 8
$5$ $1$ $I_{3}$ Non-split multiplicative 1 1 3 3
$13$ $1$ $I_{3}$ Non-split multiplicative 1 1 3 3
$17$ $1$ $II^{*}$ 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 $\ell$.

## $p$-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.

No Iwasawa invariant data is available for this curve.

## Isogenies

This curve has no rational isogenies. Its isogeny class 450840.o consists of this curve only.