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## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 1, -31258, 3842394])

gp: E = ellinit([0, 1, 1, -31258, 3842394])

magma: E := EllipticCurve([0, 1, 1, -31258, 3842394]);

$$y^2+y=x^3+x^2-31258x+3842394$$ trivial

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E); ## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$3025$$ = $$5^{2} \cdot 11^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-4457994853296875$$ = $$-1 \cdot 5^{6} \cdot 11^{11}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{122023936}{161051}$$ = $$-1 \cdot 2^{12} \cdot 11^{-5} \cdot 31^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$1.6956567514978323946423371941\dots$$ Stable Faltings height: $$-0.30800984111840306468901426149\dots$$

## 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  magma: RealPeriod(E); Real period: $$0.39341358499961023967367211512\dots$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$2$$  = $$1\cdot2$$ 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

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

$$q - 2q^{2} + q^{3} + 2q^{4} - 2q^{6} - 2q^{7} - 2q^{9} + 2q^{12} + 4q^{13} + 4q^{14} - 4q^{16} - 2q^{17} + 4q^{18} + O(q^{20})$$ For more coefficients, see the Downloads section to the right.

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 16800 $$\Gamma_0(N)$$-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);

$$L(E,1)$$ ≈ $$0.78682716999922047934734423023914056002$$

## Local data

This elliptic curve is not semistable. There are 2 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$$(j)_{-}$$
$$5$$ $$1$$ $$I_0^{*}$$ Additive 1 2 6 0
$$11$$ $$2$$ $$I_5^{*}$$ Additive -1 2 11 5

## 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
$$5$$ Cs.4.1

## $p$-adic data

### $p$-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) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 ss ordinary add ordinary add ordinary ordinary ss ordinary ss ordinary ordinary ordinary ordinary ordinary ? 2 - 0 - 0 0 0,0 2 0,0 0 0 0 0 0 ? 0 - 0 - 0 0 0,0 0 0,0 0 0 0 0 0

An entry ? indicates that the invariants have not yet been computed.

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 3025g consists of 2 curves linked by isogenies of degrees dividing 25.

## 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 $K$ $2$ $$\Q(\sqrt{-55})$$ $$\Z/5\Z$$ Not in database $3$ 3.1.44.1 $$\Z/2\Z$$ Not in database $4$ 4.4.15125.1 $$\Z/5\Z$$ Not in database $6$ 6.0.21296.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $6$ 6.0.2662000.1 $$\Z/10\Z$$ Not in database $8$ 8.2.2421502441875.2 $$\Z/3\Z$$ Not in database $8$ 8.0.228765625.1 $$\Z/5\Z \times \Z/5\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database $12$ 12.4.885780500000000.1 $$\Z/10\Z$$ Not in database $12$ 12.0.7086244000000.1 $$\Z/2\Z \times \Z/10\Z$$ Not in database $16$ Deg 16 $$\Z/15\Z$$ 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.