# Properties

 Label 2898.p1 Conductor $2898$ Discriminant $1.280\times 10^{12}$ j-invariant $$\frac{24553362849625}{1755162752}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, -5450, -143607])

gp: E = ellinit([1, -1, 1, -5450, -143607])

magma: E := EllipticCurve([1, -1, 1, -5450, -143607]);

$$y^2+xy+y=x^3-x^2-5450x-143607$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(-39, 111\right)$$ $$\hat{h}(P)$$ ≈ $0.30438561983503594292367688071$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-\frac{133}{4}, \frac{129}{8}\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-49, 87\right)$$, $$\left(-49, -39\right)$$, $$\left(-39, 111\right)$$, $$\left(-39, -73\right)$$, $$\left(99, 479\right)$$, $$\left(99, -579\right)$$, $$\left(309, 5099\right)$$, $$\left(309, -5409\right)$$, $$\left(329, 5631\right)$$, $$\left(329, -5961\right)$$, $$\left(304619, 167973711\right)$$, $$\left(304619, -168278331\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$2898$$ = $$2 \cdot 3^{2} \cdot 7 \cdot 23$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$1279513646208$$ = $$2^{7} \cdot 3^{6} \cdot 7^{2} \cdot 23^{4}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{24553362849625}{1755162752}$$ = $$2^{-7} \cdot 5^{3} \cdot 7^{-2} \cdot 23^{-4} \cdot 5813^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$1.0700597066161294723187075671\dots$$ Stable Faltings height: $$0.52075356228207462662108494864\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.30438561983503594292367688071\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.55816149994345755919820798984\dots$$ 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: $$112$$  = $$7\cdot2\cdot2\cdot2^{2}$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$2$$ 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[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

magma: ModularForm(E);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 5376 $$\Gamma_0(N)$$-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[2]/factorial(ar[1])

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

$$L'(E,1)$$ ≈ $$4.7570973555935958398411283956433472970$$

## Local data

This elliptic curve is not semistable. There are 4 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$$(j)_{-}$$
$$2$$ $$7$$ $$I_{7}$$ Split multiplicative -1 1 7 7
$$3$$ $$2$$ $$I_0^{*}$$ Additive -1 2 6 0
$$7$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$23$$ $$4$$ $$I_{4}$$ Split multiplicative -1 1 4 4

## 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 X17.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 0 & 5 \end{array}\right)$ and has index 6.

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

## $p$-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 split add ss split ordinary ss ordinary ordinary split ordinary ordinary ordinary ordinary ordinary ordinary 4 - 1,1 2 3 1,1 1 1 2 1 1 1 1 1 1 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 non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2.
Its isogeny class 2898.p consists of 2 curves linked by isogenies of degree 2.

## 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 $K$ $2$ $$\Q(\sqrt{2})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $4$ 4.0.14112.1 $$\Z/4\Z$$ Not in database $8$ 8.4.16647192576.5 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.0.12745506816.13 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.2.23511063249072.8 $$\Z/6\Z$$ 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

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.