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## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 0, -1191828872, 15836364428016])

gp: E = ellinit([1, 1, 0, -1191828872, 15836364428016])

magma: E := EllipticCurve([1, 1, 0, -1191828872, 15836364428016]);

$$y^2+xy=x^3+x^2-1191828872x+15836364428016$$ ## Mordell-Weil group structure

$$\Z\times \Z/{2}\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(\frac{118441240}{5929}, \frac{593692548}{456533}\right)$$ $$\hat{h}(P)$$ ≈ $15.192889726107699551853114871$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-39864, 19932\right)$$ ## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-39864, 19932\right)$$ ## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$10830$$ = $$2 \cdot 3 \cdot 5 \cdot 19^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-69225939256229080243440$$ = $$-1 \cdot 2^{4} \cdot 3 \cdot 5 \cdot 19^{16}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{3979640234041473454886161}{1471455901872240}$$ = $$-1 \cdot 2^{-4} \cdot 3^{-1} \cdot 5^{-1} \cdot 19^{-10} \cdot 269^{3} \cdot 589109^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$3.7344750203152855571061513509\dots$$ Stable Faltings height: $$2.2622555307320653271016376350\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$15.192889726107699551853114871\dots$$ sage: E.period_lattice().omega()  gp: E.omega  magma: RealPeriod(E); Real period: $$0.088815779865959138253221382299\dots$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$8$$  = $$2\cdot1\cdot1\cdot2^{2}$$ sage: E.torsion_order()  gp: elltors(E)  magma: Order(TorsionSubgroup(E)); Torsion order: $$2$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## Modular invariants

Modular form 10830.2.a.g

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

$$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - 2q^{7} - q^{8} + q^{9} - q^{10} + 2q^{11} - q^{12} - 4q^{13} + 2q^{14} - q^{15} + q^{16} - 2q^{17} - q^{18} + O(q^{20})$$ sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 4320000 $$\Gamma_0(N)$$-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/factorial(ar)

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

$$L'(E,1)$$ ≈ $$2.6987366988835473367844588550283422430$$

## Local data

This elliptic curve is not semistable. There are 4 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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$3$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$5$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$19$$ $$4$$ $$I_{10}^{*}$$ Additive -1 2 16 10

## Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X6.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right)$ and has index 3.

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
$$2$$ B
$$5$$ B.4.2

## $p$-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.

## Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 nonsplit nonsplit split ordinary ordinary ordinary ordinary add ordinary ss ordinary ordinary ordinary ordinary ordinary 2 3 2 1 1 1 1 - 1 1,1 1 1 1 1 1 0 0 0 0 0 0 0 - 0 0,0 0 0 0 0 0

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 $$d=$$ 2, 5 and 10.
Its isogeny class 10830g consists of 4 curves linked by isogenies of degrees dividing 10.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-15})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $4$ 4.2.86640.1 $$\Z/4\Z$$ Not in database $4$ 4.4.45125.1 $$\Z/10\Z$$ Not in database $8$ 8.0.1688960160000.20 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.0.23751002250000.11 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/6\Z$$ Not in database $8$ 8.0.164937515625.2 $$\Z/2\Z \times \Z/10\Z$$ Not in database $10$ 10.0.15864927284179687500000000.1 $$\Z/10\Z$$ Not in database $16$ Deg 16 $$\Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/20\Z$$ Not in database $20$ 20.0.56631581489769541852211952209472656250000000000000000.3 $$\Z/2\Z \times \Z/10\Z$$ 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.