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## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -22581888, -24991420608])

gp: E = ellinit([1, 0, 0, -22581888, -24991420608])

magma: E := EllipticCurve([1, 0, 0, -22581888, -24991420608]);

$$y^2+xy=x^3-22581888x-24991420608$$ ## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(-3444, 110964\right)$$ $$\hat{h}(P)$$ ≈ $0.95609986755989842964611431841$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-3444, 110964\right)$$, $$\left(-3444, -107520\right)$$, $$\left(10428, 929412\right)$$, $$\left(10428, -939840\right)$$ ## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$303450$$ = $$2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 17^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$467214434156176875000000$$ = $$2^{6} \cdot 3^{7} \cdot 5^{10} \cdot 7^{2} \cdot 17^{8}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{18694465225}{6858432}$$ = $$2^{-6} \cdot 3^{-7} \cdot 5^{2} \cdot 7^{-2} \cdot 17 \cdot 353^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$3.2420049936175520484048995380\dots$$ Stable Faltings height: $$0.011997837218324349404577015064\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.95609986755989842964611431841\dots$$ sage: E.period_lattice().omega()  gp: E.omega  magma: RealPeriod(E); Real period: $$0.071388427705647313075908426731\dots$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$252$$  = $$( 2 \cdot 3 )\cdot7\cdot1\cdot2\cdot3$$ sage: E.torsion_order()  gp: elltors(E)  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 303450.2.a.hb

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

$$q + q^{2} + q^{3} + q^{4} + q^{6} + q^{7} + q^{8} + q^{9} + 4q^{11} + q^{12} - 3q^{13} + q^{14} + q^{16} + q^{18} - 2q^{19} + O(q^{20})$$ sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 67858560 $$\Gamma_0(N)$$-optimal: yes 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/factorial(ar)

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

$$L'(E,1)$$ ≈ $$17.200125501219052496516285503696608795$$

## Local data

This elliptic curve is not semistable. There are 5 primes of bad reduction:

sage: E.local_data()

gp: ellglobalred(E)

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$6$$ $$I_{6}$$ Split multiplicative -1 1 6 6
$$3$$ $$7$$ $$I_{7}$$ Split multiplicative -1 1 7 7
$$5$$ $$1$$ $$II^{*}$$ Additive 1 2 10 0
$$7$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$17$$ $$3$$ $$IV^{*}$$ Additive -1 2 8 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 $$p$$ .

## $p$-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 no rational isogenies. Its isogeny class 303450.hb consists of this curve only.

## 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 $K$ $3$ 3.3.86700.1 $$\Z/2\Z$$ Not in database $6$ 6.6.90202680000.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $8$ Deg 8 $$\Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ 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.