# Properties

 Label 25857.a1 Conductor $25857$ Discriminant $-18051780123$ j-invariant $$-\frac{692224}{867}$$ CM no Rank $2$ Torsion structure trivial

# Learn more

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 1, -507, 7816])

gp: E = ellinit([0, 0, 1, -507, 7816])

magma: E := EllipticCurve([0, 0, 1, -507, 7816]);

$$y^2+y=x^3-507x+7816$$

## Mordell-Weil group structure

$$\Z^2$$

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(13, 58\right)$$ $$\left(-13, 110\right)$$ $$\hat{h}(P)$$ ≈ $0.23298487991875034143383904560$ $0.46489496872072562153618036302$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-26, 58\right)$$, $$\left(-26, -59\right)$$, $$\left(-23, 85\right)$$, $$\left(-23, -86\right)$$, $$\left(-13, 110\right)$$, $$\left(-13, -111\right)$$, $$\left(1, 85\right)$$, $$\left(1, -86\right)$$, $$\left(4, 76\right)$$, $$\left(4, -77\right)$$, $$\left(13, 58\right)$$, $$\left(13, -59\right)$$, $$\left(22, 85\right)$$, $$\left(22, -86\right)$$, $$\left(26, 110\right)$$, $$\left(26, -111\right)$$, $$\left(55, 382\right)$$, $$\left(55, -383\right)$$, $$\left(130, 1462\right)$$, $$\left(130, -1463\right)$$, $$\left(208, 2983\right)$$, $$\left(208, -2984\right)$$, $$\left(221, 3269\right)$$, $$\left(221, -3270\right)$$, $$\left(2353, 114133\right)$$, $$\left(2353, -114134\right)$$, $$\left(2911, 157054\right)$$, $$\left(2911, -157055\right)$$, $$\left(230932, 110975260\right)$$, $$\left(230932, -110975261\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$25857$$ = $$3^{2} \cdot 13^{2} \cdot 17$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-18051780123$$ = $$-1 \cdot 3^{7} \cdot 13^{4} \cdot 17^{2}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{692224}{867}$$ = $$-1 \cdot 2^{12} \cdot 3^{-1} \cdot 13^{2} \cdot 17^{-2}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$0.66141249173157693004473737420\dots$$ Stable Faltings height: $$-0.74287677175632349433738105812\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$2$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.10463691624080360514426459466\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$1.1091177577325926209530431185\dots$$ 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: $$24$$  = $$2^{2}\cdot3\cdot2$$ 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: $$1$$ (rounded)

## Modular invariants

Modular form 25857.2.a.a

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);

$$q - 2q^{2} + 2q^{4} - 4q^{5} - 3q^{7} + 8q^{10} - 4q^{11} + 6q^{14} - 4q^{16} - q^{17} - 4q^{19} + O(q^{20})$$

For more coefficients, see the Downloads section to the right.

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 41856 $$\Gamma_0(N)$$-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[2]/factorial(ar[1])

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

$$L^{(2)}(E,1)/2!$$ ≈ $$2.7853118860092767791779543746040930453$$

## Local data

This elliptic curve is not semistable. There are 3 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$$(j)_{-}$$
$$3$$ $$4$$ $$I_1^{*}$$ Additive -1 2 7 1
$$13$$ $$3$$ $$IV$$ Additive 1 2 4 0
$$17$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2

## 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]

Note: $$p$$-adic regulator data only exists for primes $$p\ge 5$$ of good ordinary reduction.

## 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 add ordinary ordinary ordinary add nonsplit ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary 5,12 - 2 2 2 - 2 2 2 2 2 2 2 2 2 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 25857.a 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 $K$ $3$ 3.1.2028.1 $$\Z/2\Z$$ Not in database $6$ 6.0.12338352.2 $$\Z/2\Z \times \Z/2\Z$$ Not in database $8$ 8.2.422574120899307.4 $$\Z/3\Z$$ 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.