Properties

Label 5070.t3
Conductor $5070$
Discriminant $1.264\times 10^{21}$
j-invariant \( \frac{453198971846635561}{261896250564000} \)
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, 0, 0, -2704426, 64216580])
 
gp: E = ellinit([1, 0, 0, -2704426, 64216580])
 
magma: E := EllipticCurve([1, 0, 0, -2704426, 64216580]);
 

\(y^2+xy=x^3-2704426x+64216580\)  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);
 

$P$ =  \(\left(-766, 41450\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $0.51007216446073725318235397291$

Torsion generators

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

\( \left(\frac{95}{4}, -\frac{95}{8}\right) \)  Toggle raw display

Integral points

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

\( \left(-1624, 13994\right) \), \( \left(-1624, -12370\right) \), \( \left(-766, 41450\right) \), \( \left(-766, -40684\right) \), \( \left(1664, 12290\right) \), \( \left(1664, -13954\right) \), \( \left(3446, 176234\right) \), \( \left(3446, -179680\right) \), \( \left(42254, 8657960\right) \), \( \left(42254, -8700214\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 5070 \)  =  $2 \cdot 3 \cdot 5 \cdot 13^{2}$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $1264123179288570276000 $  =  $2^{5} \cdot 3^{18} \cdot 5^{3} \cdot 13^{8} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{453198971846635561}{261896250564000} \)  =  $2^{-5} \cdot 3^{-18} \cdot 5^{-3} \cdot 13^{-2} \cdot 47^{3} \cdot 59^{3} \cdot 277^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $2.7385545879776548270423400201\dots$
Stable Faltings height: $1.4560799092468864590155962993\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $0.51007216446073725318235397291\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.12997156027386942001329828371\dots$
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: $ 360 $  = $ 5\cdot( 2 \cdot 3^{2} )\cdot1\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) $ ≈ $ 5.9665387560508572652813787088734685062 $

Modular invariants

Modular form   5070.2.a.t

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);
 

\( q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} - 2 q^{7} + q^{8} + q^{9} - q^{10} + q^{12} - 2 q^{14} - q^{15} + q^{16} + q^{18} - 2 q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 241920
$ \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$(j)_{-}$
$2$ $5$ $I_{5}$ Split multiplicative -1 1 5 5
$3$ $18$ $I_{18}$ Split multiplicative -1 1 18 18
$5$ $1$ $I_{3}$ Non-split multiplicative 1 1 3 3
$13$ $4$ $I_{2}^{*}$ 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
$3$ 3B 3.4.0.1

$p$-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 split split nonsplit ordinary ss add ss ordinary ordinary ss ordinary ordinary ordinary ordinary ss
$\lambda$-invariant(s) 2 2 1 1 1,1 - 1,3 1 1 1,1 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 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 5070.t 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
$2$ \(\Q(\sqrt{10}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
$2$ \(\Q(\sqrt{13}) \) \(\Z/6\Z\) Not in database
$4$ 4.0.243360.3 \(\Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{10}, \sqrt{13})\) \(\Z/2\Z \times \Z/6\Z\) Not in database
$6$ 6.0.160398576.1 \(\Z/6\Z\) Not in database
$8$ 8.0.94758543360000.6 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ Deg 8 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ 8.0.59224089600.7 \(\Z/12\Z\) Not in database
$12$ Deg 12 \(\Z/3\Z \times \Z/6\Z\) Not in database
$12$ Deg 12 \(\Z/2\Z \times \Z/6\Z\) Not in database
$12$ Deg 12 \(\Z/12\Z\) Not in database
$16$ Deg 16 \(\Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/12\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/12\Z\) Not in database
$18$ 18.6.7333751368868611304482312500000000.1 \(\Z/18\Z\) 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.