Properties

Label 381150je4
Conductor $381150$
Discriminant $1.373\times 10^{27}$
j-invariant \( \frac{260744057755293612689909}{8504954620259328} \)
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, 1, -7247053265, 237455026687937])
 
gp: E = ellinit([1, -1, 1, -7247053265, 237455026687937])
 
magma: E := EllipticCurve([1, -1, 1, -7247053265, 237455026687937]);
 

\(y^2+xy+y=x^3-x^2-7247053265x+237455026687937\)  Toggle raw display

Mordell-Weil group structure

$\Z\times \Z/{2}\Z$

Infinite order Mordell-Weil generator and height

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

$P$ =  \(\left(49635, 142888\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $3.2992227161195872435314184732$

Torsion generators

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

\( \left(\frac{197451}{4}, -\frac{197455}{8}\right) \)  Toggle raw display

Integral points

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

\( \left(49635, 142888\right) \), \( \left(49635, -192524\right) \), \( \left(280459, 142215060\right) \), \( \left(280459, -142495520\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 381150 \)  =  $2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 11^{2}$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $1372984558732935073183104000 $  =  $2^{10} \cdot 3^{11} \cdot 5^{3} \cdot 7^{10} \cdot 11^{8} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{260744057755293612689909}{8504954620259328} \)  =  $2^{-10} \cdot 3^{-5} \cdot 7^{-10} \cdot 11^{-2} \cdot 29^{3} \cdot 2202961^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $4.3012559055839139153437888729\dots$
Stable Faltings height: $2.1506426467421487039650046321\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $3.2992227161195872435314184732\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.044892784119774158231961095136\dots$
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: $ 320 $  = $ ( 2 \cdot 5 )\cdot2\cdot2\cdot2\cdot2^{2} $
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) $ ≈ $ 11.848903452624925754879941636787605804 $

Modular invariants

Modular form 381150.2.a.je

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);
 

\( q + q^{2} + q^{4} - q^{7} + q^{8} - 4 q^{13} - q^{14} + q^{16} - 2 q^{17} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 368640000
$ \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$(j)_{-}$
$2$ $10$ $I_{10}$ Split multiplicative -1 1 10 10
$3$ $2$ $I_{5}^{*}$ Additive -1 2 11 5
$5$ $2$ $III$ Additive -1 2 3 0
$7$ $2$ $I_{10}$ Non-split multiplicative 1 1 10 10
$11$ $4$ $I_{2}^{*}$ Additive -1 2 8 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
$5$ 5B.4.1 5.12.0.1

$p$-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, 5 and 10.
Its isogeny class 381150je consists of 4 curves linked by isogenies of degrees dividing 10.

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
$2$ \(\Q(\sqrt{15}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
$2$ \(\Q(\sqrt{33}) \) \(\Z/10\Z\) Not in database
$4$ 4.4.8893500.4 \(\Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{15}, \sqrt{33})\) \(\Z/2\Z \times \Z/10\Z\) Not in database
$8$ Deg 8 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ 8.8.11389585284000000.2 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ Deg 8 \(\Z/6\Z\) Not in database
$8$ 8.8.711849080250000.1 \(\Z/20\Z\) Not in database
$16$ Deg 16 \(\Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/20\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/20\Z\) Not in database
$16$ Deg 16 \(\Z/30\Z\) Not in database
$20$ 20.0.10019151533337487082567413330078125.2 \(\Z/10\Z\) 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.