Properties

Label 2394.c1
Conductor $2394$
Discriminant $1.110\times 10^{17}$
j-invariant \( \frac{19804628171203875}{5638671302656} \)
CM no
Rank $1$
Torsion structure \(\Z/{2}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 0, -152187, 16325477])
 
gp: E = ellinit([1, -1, 0, -152187, 16325477])
 
magma: E := EllipticCurve([1, -1, 0, -152187, 16325477]);
 

Minimal equation

Minimal equation

Simplified equation

\(y^2+xy=x^3-x^2-152187x+16325477\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz=x^3-x^2z-152187xz^2+16325477z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-2434995x+1042395534\) Copy content Toggle raw display (homogenize, minimize)

Mordell-Weil group structure

\(\Z \oplus \Z/{2}\Z\)

Infinite order Mordell-Weil generator and height

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

$P$ =  \(\left(91, 1750\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $1.2770060828690715374443366710$

Torsion generators

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

\( \left(118, -59\right) \) Copy content Toggle raw display

Integral points

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

\( \left(91, 1750\right) \), \( \left(91, -1841\right) \), \( \left(118, -59\right) \), \( \left(1079, 32739\right) \), \( \left(1079, -33818\right) \), \( \left(4214, 270277\right) \), \( \left(4214, -274491\right) \) 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: $110985967250178048 $  =  $2^{24} \cdot 3^{9} \cdot 7^{2} \cdot 19^{3} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{19804628171203875}{5638671302656} \)  =  $2^{-24} \cdot 3^{3} \cdot 5^{3} \cdot 7^{-2} \cdot 17^{3} \cdot 19^{-3} \cdot 1061^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $1.9773502094851543831834014084\dots$
Stable Faltings height: $1.1533909929840721146369674807\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $1.2770060828690715374443366710\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.31041942872211297418980958525\dots$
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: $ 24 $  = $ 2\cdot2\cdot2\cdot3 $
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)
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: 27648
$ \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$(j)_{-}$
$2$ $2$ $I_{24}$ Non-split multiplicative 1 1 24 24
$3$ $2$ $III^{*}$ Additive 1 2 9 0
$7$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$19$ $3$ $I_{3}$ Split multiplicative -1 1 3 3

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.1.2 3.8.0.2

$p$-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) 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, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{57}) \) \(\Z/2\Z \oplus \Z/2\Z\) Not in database
$2$ \(\Q(\sqrt{-3}) \) \(\Z/6\Z\) Not in database
$3$ 3.1.1323.1 \(\Z/6\Z\) Not in database
$4$ 4.0.513.1 \(\Z/12\Z\) Not in database
$4$ \(\Q(\sqrt{-3}, \sqrt{-19})\) \(\Z/2\Z \oplus \Z/6\Z\) Not in database
$6$ 6.0.5250987.1 \(\Z/3\Z \oplus \Z/6\Z\) Not in database
$6$ 6.2.36016519833.1 \(\Z/2\Z \oplus \Z/6\Z\) Not in database
$8$ 8.0.95004009.1 \(\Z/2\Z \oplus \Z/12\Z\) Not in database
$8$ 8.4.21080517080281344.14 \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$12$ Deg 12 \(\Z/6\Z \oplus \Z/6\Z\) Not in database
$12$ Deg 12 \(\Z/3\Z \oplus \Z/12\Z\) Not in database
$16$ Deg 16 \(\Z/24\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \oplus \Z/12\Z\) Not in database
$18$ 18.0.32280955105118098668547786909650858258432.1 \(\Z/18\Z\) 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.