Properties

Label 369600eg2
Conductor $369600$
Discriminant $2.490\times 10^{21}$
j-invariant \( \frac{474334834335054841}{607815140625} \)
CM no
Rank $0$
Torsion structure \(\Z/{2}\Z \oplus \Z/{2}\Z\)

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2=x^3-x^2-25996033x-50951096063\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z=x^3-x^2z-25996033xz^2-50951096063z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-2105678700x-37149666066000\) Copy content Toggle raw display (homogenize, minimize)

sage: E = EllipticCurve([0, -1, 0, -25996033, -50951096063])
 
gp: E = ellinit([0, -1, 0, -25996033, -50951096063])
 
magma: E := EllipticCurve([0, -1, 0, -25996033, -50951096063]);
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 

Mordell-Weil group structure

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

Torsion generators

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

\( \left(-3023, 0\right) \), \( \left(5887, 0\right) \) Copy content Toggle raw display

Integral points

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

\( \left(-3023, 0\right) \), \( \left(-2863, 0\right) \), \( \left(5887, 0\right) \) Copy content Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 369600 \)  =  $2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $2489610816000000000000 $  =  $2^{18} \cdot 3^{8} \cdot 5^{12} \cdot 7^{2} \cdot 11^{2} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{474334834335054841}{607815140625} \)  =  $3^{-8} \cdot 5^{-6} \cdot 7^{-2} \cdot 11^{-2} \cdot 199^{3} \cdot 3919^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $3.0133429912309112984064846411\dots$
Stable Faltings height: $1.1689032641739431469802567923\dots$

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: if(E.disc>0,2,1)*E.omega[1]
 
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
 
Real period: $0.066866131939895260970728834905\dots$
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: $ 128 $  = $ 2^{2}\cdot2\cdot2^{2}\cdot2\cdot2 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $4$
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) $ ≈ $ 0.53492905551916208776583067924 $

Modular invariants

Modular form 369600.2.a.eg

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);
 

\( q - q^{3} - q^{7} + q^{9} + q^{11} - 2 q^{13} + 6 q^{17} - 4 q^{19} + O(q^{20}) \) Copy content Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 28311552
$ \Gamma_0(N) $-optimal: no
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$(j)_{-}$
$2$ $4$ $I_{8}^{*}$ Additive 1 6 18 0
$3$ $2$ $I_{8}$ Non-split multiplicative 1 1 8 8
$5$ $4$ $I_{6}^{*}$ Additive 1 2 12 6
$7$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$11$ $2$ $I_{2}$ 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
$2$ 2Cs 2.6.0.1

$p$-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 $0$.

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 369600eg 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$4$ \(\Q(\sqrt{-10}, \sqrt{11})\) \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{14}, \sqrt{110})\) \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{10}, \sqrt{-14})\) \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$8$ Deg 8 \(\Z/2\Z \oplus \Z/6\Z\) 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 \(\Z/2\Z \oplus \Z/8\Z\) 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.