Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 0, -41721799, -103737357815]); // or

magma: E := EllipticCurve("25886c1");

sage: E = EllipticCurve([1, 0, 0, -41721799, -103737357815]) # or

sage: E = EllipticCurve("25886c1")

gp: E = ellinit([1, 0, 0, -41721799, -103737357815]) \\ or

gp: E = ellinit("25886c1")

$$y^2 + x y = x^{3} - 41721799 x - 103737357815$$

Trivial

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

None

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E) Conductor: $$25886$$ = $$2 \cdot 7 \cdot 43^{2}$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$-598796585779245744128$$ = $$-1 \cdot 2^{17} \cdot 7^{5} \cdot 43^{7}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$-\frac{1270580128269753673}{94725865472}$$ = $$-1 \cdot 2^{-17} \cdot 7^{-5} \cdot 41^{3} \cdot 43^{-1} \cdot 26417^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$0$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$1$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega Real period: $$0.0297013958364$$ magma: TamagawaNumbers(E);  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]] Tamagawa product: $$170$$  = $$17\cdot5\cdot2$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E) Torsion order: $$1$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form 25886.2.a.c

magma: ModularForm(E);

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

$$q + q^{2} + q^{3} + q^{4} - 2q^{5} + q^{6} + q^{7} + q^{8} - 2q^{9} - 2q^{10} + 5q^{11} + q^{12} + 2q^{13} + q^{14} - 2q^{15} + q^{16} - 2q^{18} + 3q^{19} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 2513280 $$\Gamma_0(N)$$-optimal: yes Manin constant: 1

#### Special L-value

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

sage: r = E.rank();

sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()

gp: ar = ellanalyticrank(E);

gp: ar/factorial(ar)

$$L(E,1)$$ ≈ $$5.04923729219$$

## Local data

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

sage: E.local_data()

gp: ellglobalred(E)

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$17$$ $$I_{17}$$ Split multiplicative -1 1 17 17
$$7$$ $$5$$ $$I_{5}$$ Split multiplicative -1 1 5 5
$$43$$ $$2$$ $$I_1^{*}$$ Additive -1 2 7 1

## Galois representations

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

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

sage: rho = E.galois_representation();

sage: [rho.image_type(p) for p in rho.non_surjective()]

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(3,20) 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 split ordinary ordinary split ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ss add ordinary 2 4 6 3 0 0 4,4 0 0 0 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 are not computed because the reduction is additive.

## Isogenies

This curve has no rational isogenies. Its isogeny class 25886c consists of this curve only.

## Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
3 3.1.2408.1 $$\Z/2\Z$$ Not in database
6 6.0.13962701312.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.