Properties

Label 2394.c3
Conductor $2394$
Discriminant $-2.159\times 10^{15}$
j-invariant \( -\frac{11108001800138902875}{79947274872976} \)
CM no
Rank $1$
Torsion structure \(\Z/{6}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 0, -139452, 20203200])
 
gp: E = ellinit([1, -1, 0, -139452, 20203200])
 
magma: E := EllipticCurve([1, -1, 0, -139452, 20203200]);
 

Minimal equation

Minimal equation

Simplified equation

\(y^2+xy=x^3-x^2-139452x+20203200\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz=x^3-x^2z-139452xz^2+20203200z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-2231235x+1290773566\) Copy content Toggle raw display (homogenize, minimize)

Mordell-Weil group structure

\(\Z \oplus \Z/{6}\Z\)

Infinite order Mordell-Weil generator and height

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

$P$ =  \(\left(204, 360\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $1.9155091243036073061665050065$

Torsion generators

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

\( \left(267, 1263\right) \) Copy content Toggle raw display

Integral points

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

\( \left(169, 1116\right) \), \( \left(169, -1285\right) \), \( \left(204, 360\right) \), \( \left(204, -564\right) \), \( \left(267, 1263\right) \), \( \left(267, -1530\right) \), \( \left(400, 5120\right) \), \( \left(400, -5520\right) \), \( \left(1884, 79320\right) \), \( \left(1884, -81204\right) \) Copy content Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 2394 \)  =  $2 \cdot 3^{2} \cdot 7 \cdot 19$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $-2158576421570352 $  =  $-1 \cdot 2^{4} \cdot 3^{3} \cdot 7^{12} \cdot 19^{2} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{11108001800138902875}{79947274872976} \)  =  $-1 \cdot 2^{-4} \cdot 3^{6} \cdot 5^{3} \cdot 7^{-12} \cdot 19^{-2} \cdot 179^{3} \cdot 277^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $1.7746176554310721921943948507\dots$
Stable Faltings height: $1.4999645832640447693455835415\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $1.9155091243036073061665050065\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.46562914308316946128471437787\dots$
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: $ 96 $  = $ 2\cdot2\cdot( 2^{2} \cdot 3 )\cdot2 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $6$
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) $ ≈ $ 2.3784449923132826766216916003 $

Modular invariants

Modular form   2394.2.a.c

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 18432
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

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$(j)_{-}$
$2$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$3$ $2$ $III$ Additive 1 2 3 0
$7$ $12$ $I_{12}$ Split multiplicative -1 1 12 12
$19$ $2$ $I_{2}$ Split multiplicative -1 1 2 2

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
$2$ 2B 2.3.0.1
$3$ 3B.1.1 3.8.0.1

$p$-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

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type nonsplit add ss split ord ord ss split ord ord ord ord ord ord ord
$\lambda$-invariant(s) 7 - 1,3 2 1 3 1,1 2 1 1 1 1 1 1 1
$\mu$-invariant(s) 1 - 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, 3 and 6.
Its isogeny class 2394.c consists of 4 curves linked by isogenies of degrees dividing 6.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{-3}) \) \(\Z/2\Z \oplus \Z/6\Z\) Not in database
$4$ 4.2.155952.2 \(\Z/12\Z\) Not in database
$6$ 6.0.4560192432.1 \(\Z/6\Z \oplus \Z/6\Z\) Not in database
$8$ 8.0.67371264.1 \(\Z/2\Z \oplus \Z/12\Z\) Not in database
$8$ 8.0.24321026304.2 \(\Z/2\Z \oplus \Z/12\Z\) Not in database
$9$ 9.3.3519990143155766448.2 \(\Z/18\Z\) Not in database
$16$ Deg 16 \(\Z/24\Z\) Not in database
$18$ 18.0.37170991823741259516493521942967610112.3 \(\Z/2\Z \oplus \Z/18\Z\) 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.