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Properties

Label 257400ej1
Conductor $257400$
Discriminant $-4.741\times 10^{23}$
j-invariant \( -\frac{329381898333928144}{162600887109375} \)
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([0, 0, 0, -20555175, -48828311750])
 
gp: E = ellinit([0, 0, 0, -20555175, -48828311750])
 
magma: E := EllipticCurve([0, 0, 0, -20555175, -48828311750]);
 

Minimal equation

Minimal equation

Simplified equation

\(y^2=x^3-20555175x-48828311750\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z=x^3-20555175xz^2-48828311750z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-20555175x-48828311750\) 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);
 

$P$ =  \(\left(\frac{6870714695}{237169}, \frac{561915043885800}{115501303}\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $17.542511280248774924365824777$

Torsion generators

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

\( \left(5435, 0\right) \) Copy content Toggle raw display

Integral points

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

\( \left(5435, 0\right) \) Copy content Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 257400 \)  =  $2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 11 \cdot 13$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $-474144186810937500000000 $  =  $-1 \cdot 2^{8} \cdot 3^{13} \cdot 5^{14} \cdot 11^{4} \cdot 13 $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{329381898333928144}{162600887109375} \)  =  $-1 \cdot 2^{4} \cdot 3^{-7} \cdot 5^{-8} \cdot 11^{-4} \cdot 13^{-1} \cdot 274069^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $3.2482240539029559560506297256\dots$
Stable Faltings height: $1.4321008329785540501078060262\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $17.542511280248774924365824777\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.034639062907678743995492908685\dots$
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: $ 64 $  = $ 2\cdot2\cdot2^{2}\cdot2^{2}\cdot1 $
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) $ ≈ $ 9.7224984287232207265940391327 $

Modular invariants

Modular form 257400.2.a.ej

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);
 

\( q + 4 q^{7} + q^{11} - q^{13} - 2 q^{17} + 4 q^{19} + O(q^{20}) \) Copy content Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 33030144
$ \Gamma_0(N) $-optimal: yes
Manin constant: 1

Local data

This elliptic curve is not semistable. There are 5 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$(j)_{-}$
$2$ $2$ $I_{1}^{*}$ Additive 1 3 8 0
$3$ $2$ $I_{7}^{*}$ Additive -1 2 13 7
$5$ $4$ $I_{8}^{*}$ Additive 1 2 14 8
$11$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$13$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1

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 4.6.0.1

$p$-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.

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 257400ej consists of 4 curves linked by isogenies of degrees dividing 4.

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
$2$ \(\Q(\sqrt{-39}) \) \(\Z/2\Z \oplus \Z/2\Z\) Not in database
$2$ \(\Q(\sqrt{5}) \) \(\Z/4\Z\) Not in database
$2$ \(\Q(\sqrt{-195}) \) \(\Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{5}, \sqrt{-39})\) \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$8$ 8.0.562999001760000.35 \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$8$ Deg 8 \(\Z/8\Z\) Not in database
$8$ Deg 8 \(\Z/8\Z\) Not in database
$8$ Deg 8 \(\Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/4\Z \oplus \Z/4\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \oplus \Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \oplus \Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \oplus \Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/12\Z\) Not in database
$16$ Deg 16 \(\Z/12\Z\) 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.