Properties

Label 2898m1
Conductor $2898$
Discriminant $-75352080384$
j-invariant \( -\frac{5999796014211}{2790817792} \)
CM no
Rank $0$
Torsion structure \(\Z/{3}\Z\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, -1136, -19501])
 
gp: E = ellinit([1, -1, 1, -1136, -19501])
 
magma: E := EllipticCurve([1, -1, 1, -1136, -19501]);
 

\(y^2+xy+y=x^3-x^2-1136x-19501\)  Toggle raw display

Mordell-Weil group structure

\(\Z/{3}\Z\)

Torsion generators

sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 
magma: TorsionSubgroup(E);
 

\( \left(61, 337\right) \)  Toggle raw display

Integral points

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

\( \left(61, 337\right) \), \( \left(61, -399\right) \)  Toggle raw display

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: \(-75352080384 \)  =  \(-1 \cdot 2^{15} \cdot 3^{3} \cdot 7 \cdot 23^{3} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{5999796014211}{2790817792} \)  =  \(-1 \cdot 2^{-15} \cdot 3^{9} \cdot 7^{-1} \cdot 23^{-3} \cdot 673^{3}\)
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: \(0.79332511476188698381814224174\dots\)
Stable Faltings height: \(0.51867204259485956096933093251\dots\)

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[1]
 
magma: RealPeriod(E);
 
Real period: \(0.40221101140525797558409803579\dots\)
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: \( 90 \)  = \( ( 3 \cdot 5 )\cdot2\cdot1\cdot3 \)
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: \(3\)
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form   2898.2.a.t

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);
 

\( q + q^{2} + q^{4} + 3q^{5} + q^{7} + q^{8} + 3q^{10} - 6q^{11} + 5q^{13} + q^{14} + q^{16} + 6q^{17} + 2q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 5280
\( \Gamma_0(N) \)-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);
 

\( L(E,1) \) ≈ \( 4.0221101140525797558409803578945731583 \)

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\((j)_{-}\)
\(2\) \(15\) \(I_{15}\) Split multiplicative -1 1 15 15
\(3\) \(2\) \(III\) Additive 1 2 3 0
\(7\) \(1\) \(I_{1}\) Split multiplicative -1 1 1 1
\(23\) \(3\) \(I_{3}\) Split multiplicative -1 1 3 3

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
\(3\) B.1.1

$p$-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

$p$ 2 3 5 7 23
Reduction type split add ordinary split split
$\lambda$-invariant(s) 2 - 6 1 1
$\mu$-invariant(s) 0 - 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.

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 \(d=\) 3.
Its isogeny class 2898m consists of 2 curves linked by isogenies of degree 3.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$3$ 3.1.3864.1 \(\Z/6\Z\) Not in database
$6$ 6.0.57691436544.1 \(\Z/2\Z \times \Z/6\Z\) Not in database
$6$ 6.0.5250987.1 \(\Z/3\Z \times \Z/3\Z\) Not in database
$9$ 9.3.326405877927673505472.2 \(\Z/9\Z\) Not in database
$12$ Deg 12 \(\Z/12\Z\) Not in database
$18$ 18.0.275312447559745684931594320211607552.1 \(\Z/3\Z \times \Z/6\Z\) 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.