Properties

 Label 25071a1 Conductor 25071 Discriminant 676917 j-invariant $$\frac{1593413632}{676917}$$ CM no Rank 3 Torsion Structure $$\mathrm{Trivial}$$

Related objects

Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([0, -1, 1, -24, -16]); // or
magma: E := EllipticCurve("25071a1");
sage: E = EllipticCurve([0, -1, 1, -24, -16]) # or
sage: E = EllipticCurve("25071a1")
gp: E = ellinit([0, -1, 1, -24, -16]) \\ or
gp: E = ellinit("25071a1")

$$y^2 + y = x^{3} - x^{2} - 24 x - 16$$

Mordell-Weil group structure

$$\Z^3$$

Infinite order Mordell-Weil generators and heights

magma: Generators(E);
sage: E.gens()

 $$P$$ = $$\left(-4, 0\right)$$ $$\left(-3, 4\right)$$ $$\left(-2, 4\right)$$ $$\hat{h}(P)$$ ≈ 1.73560234243 0.580483490775 1.61743941483

Integral points

magma: IntegralPoints(E);
sage: E.integral_points()

$$\left(-4, 0\right)$$, $$\left(-3, 4\right)$$, $$\left(-2, 4\right)$$, $$\left(-1, 2\right)$$, $$\left(6, 4\right)$$, $$\left(7, 10\right)$$, $$\left(8, 15\right)$$, $$\left(11, 30\right)$$, $$\left(24, 112\right)$$, $$\left(33, 184\right)$$, $$\left(53, 380\right)$$, $$\left(87, 805\right)$$, $$\left(213, 3100\right)$$, $$\left(662, 17019\right)$$, $$\left(1033, 33184\right)$$, $$\left(2341491, 3582932665\right)$$

Note: only one of each pair $\pm P$ is listed.

Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] Conductor: $$25071$$ = $$3 \cdot 61 \cdot 137$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$676917$$ = $$3^{4} \cdot 61 \cdot 137$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$\frac{1593413632}{676917}$$ = $$2^{12} \cdot 3^{-4} \cdot 61^{-1} \cdot 73^{3} \cdot 137^{-1}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

 magma: Rank(E); sage: E.rank() Rank: $$3$$ magma: Regulator(E); sage: E.regulator() Regulator: $$0.880641968445$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega[1] Real period: $$2.23311988021$$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] Tamagawa product: $$2$$  = $$2\cdot1\cdot1$$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E)[1] Torsion order: $$1$$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (rounded)

Modular invariants

Modular form 25071.2.a.a

magma: ModularForm(E);
sage: E.q_eigenform(20)
gp: xy = elltaniyama(E);
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

$$q - 2q^{2} - q^{3} + 2q^{4} - 2q^{5} + 2q^{6} - 4q^{7} + q^{9} + 4q^{10} - 6q^{11} - 2q^{12} - 4q^{13} + 8q^{14} + 2q^{15} - 4q^{16} - 6q^{17} - 2q^{18} - 5q^{19} + O(q^{20})$$

 magma: ModularDegree(E); sage: E.modular_degree() Modular degree: 8960 $$\Gamma_0(N)$$-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[2]/factorial(ar[1])

$$L^{(3)}(E,1)/3!$$ ≈ $$3.93315817416$$

Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
sage: E.local_data()
gp: ellglobalred(E)[5]
prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$3$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$61$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$137$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 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]

Note: $$p$$-adic regulator data only exists for primes $$p\ge5$$ 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 61 137 ss nonsplit ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss ordinary ordinary ordinary ordinary split split 7,4 3 3 3 3 3 3 3 3 3 3,3 3 3 3 3 4 4 0,0 0 0 0 0 0 0 0 0 0 0,0 0 0 0 0 0 0

Isogenies

This curve has no rational isogenies. Its isogeny class 25071a 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 3.3.33428.1 $$\Z/2\Z$$ Not in database
6 6.6.9338372404688.1 $$\Z/2\Z \times \Z/2\Z$$ 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.