# Properties

 Label 424830.s1 Conductor $424830$ Discriminant $-6.973\times 10^{21}$ j-invariant $$-\frac{12242088317612041}{600}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 0, -10875981078, -436571405239668])

gp: E = ellinit([1, 1, 0, -10875981078, -436571405239668])

magma: E := EllipticCurve([1, 1, 0, -10875981078, -436571405239668]);

$$y^2+xy=x^3+x^2-10875981078x-436571405239668$$

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(\frac{8243155377459577448466637684705411130049390553029143391522486604101973839189200669177075677661510806665506434723494738329}{67779665206853747420752620106350821796510249501084892202334518415058695811966510138765205777500547653195009157357824}, \frac{3474972445126811645515999011006740684523282546462424174632324948759784613202184862660557680763697946119626814792108665256429574366738002122505069647793414320476983482316697266012309}{558019183117110812100850606751119663427151108560402571131710775707401606375724757941659829142425888767413640648630415827656353156599696307039101626081157513775881705337720832}\right)$$ $$\hat{h}(P)$$ ≈ $278.01706057723995353078604678$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);



## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$424830$$ = $$2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 17^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-6973082191981377389400$$ = $$-1 \cdot 2^{3} \cdot 3 \cdot 5^{2} \cdot 7^{8} \cdot 17^{10}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{12242088317612041}{600}$$ = $$-1 \cdot 2^{-3} \cdot 3^{-1} \cdot 5^{-2} \cdot 7 \cdot 17^{2} \cdot 18223^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$4.1150544465835169997242874572\dots$$ Stable Faltings height: $$0.45676989383312806277944011334\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$278.01706057723995353078604678\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.0073918381341951943363642454680\dots$$ 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: $$2$$  = $$1\cdot1\cdot2\cdot1\cdot1$$ 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)

## Modular invariants

Modular form 424830.2.a.s

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);

$$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - q^{8} + q^{9} + q^{10} + 3q^{11} - q^{12} + 2q^{13} + q^{15} + q^{16} - q^{18} - 4q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 349780032 $$\Gamma_0(N)$$-optimal: no Manin constant: 1

#### Special L-value

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);

$$L'(E,1)$$ ≈ $$4.1101142206633953932291296546543169472$$

## 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$$(j)_{-}$$
$$2$$ $$1$$ $$I_{3}$$ Non-split multiplicative 1 1 3 3
$$3$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$5$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$7$$ $$1$$ $$IV^{*}$$ Additive 1 2 8 0
$$17$$ $$1$$ $$II^{*}$$ Additive 1 2 10 0

## Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

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 mod $$p$$ Galois representation has maximal image $$\GL(2,\F_p)$$ for all primes $$p$$ except those listed.

prime Image of Galois representation
$$3$$ B

## $p$-adic data

### $p$-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 non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 424830.s consists of 2 curves linked by isogenies of degree 3.