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Properties

Label 5054b1
Conductor $5054$
Discriminant $-4.476\times 10^{17}$
j-invariant \( \frac{53261199}{26353376} \)
CM no
Rank $1$
Torsion structure trivial

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, 20148, 32163887])
 
gp: E = ellinit([1, -1, 1, 20148, 32163887])
 
magma: E := EllipticCurve([1, -1, 1, 20148, 32163887]);
 

\(y^2+xy+y=x^3-x^2+20148x+32163887\) Copy content Toggle raw display

Mordell-Weil group structure

$\Z$

Infinite order Mordell-Weil generator and height

sage: E.gens()
 
magma: Generators(E);
 

$P$ =  \(\left(271, 7445\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $0.23428542293602147057813604464$

Integral points

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

\( \left(145, 6101\right) \), \( \left(145, -6247\right) \), \( \left(271, 7445\right) \), \( \left(271, -7717\right) \) Copy content Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 5054 \)  =  $2 \cdot 7 \cdot 19^{2}$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $-447574222639176416 $  =  $-1 \cdot 2^{5} \cdot 7^{7} \cdot 19^{8} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{53261199}{26353376} \)  =  $2^{-5} \cdot 3^{3} \cdot 7^{-7} \cdot 19 \cdot 47^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $2.0659636738345777293519765893\dots$
Stable Faltings height: $0.10300435439028408934595830138\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $0.23428542293602147057813604464\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.23104263642920630508424412789\dots$
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: $ 105 $  = $ 5\cdot7\cdot3 $
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: $1$ (exact)
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);
 
Special value: $ L'(E,1) $ ≈ $ 5.6836417881673542659860103321 $

Modular invariants

Modular form   5054.2.a.b

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);
 

\( q + q^{2} + q^{4} + q^{5} + q^{7} + q^{8} - 3 q^{9} + q^{10} - 2 q^{11} - 5 q^{13} + q^{14} + q^{16} - 3 q^{18} + O(q^{20}) \) Copy content Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 47880
$ \Gamma_0(N) $-optimal: yes
Manin constant: 1

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$(j)_{-}$
$2$ $5$ $I_{5}$ Split multiplicative -1 1 5 5
$7$ $7$ $I_{7}$ Split multiplicative -1 1 7 7
$19$ $3$ $IV^{*}$ Additive 1 2 8 0

Galois representations

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 $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$7$ 7B.2.1 7.16.0.1

$p$-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

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type split ss ord split ord ord ss add ord ord ord ord ord ord ss
$\lambda$-invariant(s) 2 3,3 3 2 1 1 1,1 - 1 1 1 1 1 1 1,1
$\mu$-invariant(s) 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 non-trivial cyclic isogenies of degree $d$ for $d=$ 7.
Its isogeny class 5054b consists of 2 curves linked by isogenies of degree 7.

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:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$3$ 3.1.20216.1 \(\Z/2\Z\) Not in database
$3$ 3.3.361.1 \(\Z/7\Z\) Not in database
$6$ 6.0.22886452736.1 \(\Z/2\Z \times \Z/2\Z\) Not in database
$8$ 8.2.10949022029232.4 \(\Z/3\Z\) Not in database
$9$ 9.3.8262009437696.1 \(\Z/14\Z\) Not in database
$12$ Deg 12 \(\Z/4\Z\) Not in database
$18$ 18.0.11987688643769434377362138464256.1 \(\Z/2\Z \times \Z/14\Z\) 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.