Properties

Label 7406d6
Conductor $7406$
Discriminant $3.714\times 10^{12}$
j-invariant \( \frac{2251439055699625}{25088} \)
CM no
Rank $2$
Torsion structure \(\Z/{2}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, -1444446, 668069456])
 
gp: E = ellinit([1, 0, 1, -1444446, 668069456])
 
magma: E := EllipticCurve([1, 0, 1, -1444446, 668069456]);
 

\(y^2+xy+y=x^3-1444446x+668069456\)  Toggle raw display

Mordell-Weil group structure

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

Infinite order Mordell-Weil generators and heights

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

$P$ =  \(\left(688, -80\right)\)  Toggle raw display\(\left(650, 1652\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $0.71724630877032985633561404245$$3.8942126870374182468728296942$

Torsion generators

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

\( \left(\frac{2775}{4}, -\frac{2779}{8}\right) \)  Toggle raw display

Integral points

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

\( \left(-1382, 5716\right) \), \( \left(-1382, -4335\right) \), \( \left(-232, 31591\right) \), \( \left(-232, -31360\right) \), \( \left(650, 1652\right) \), \( \left(650, -2303\right) \), \( \left(688, -80\right) \), \( \left(688, -609\right) \), \( \left(694, -339\right) \), \( \left(694, -356\right) \), \( \left(734, 1484\right) \), \( \left(734, -2219\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 7406 \)  =  $2 \cdot 7 \cdot 23^{2}$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $3713924383232 $  =  $2^{9} \cdot 7^{2} \cdot 23^{6} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{2251439055699625}{25088} \)  =  $2^{-9} \cdot 5^{3} \cdot 7^{-2} \cdot 11^{3} \cdot 2383^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $1.9808480883876593029841587707\dots$
Stable Faltings height: $0.41310098042308445758078235480\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $2$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $2.7901719696349524028944679173\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.55276805836035872247836995442\dots$
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: $ 8 $  = $ 1\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: $1$ (rounded)
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^{(2)}(E,1)/2! $ ≈ $ 3.0846358842932208302860346038588449362 $

Modular invariants

Modular form   7406.2.a.a

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);
 

\( q - q^{2} - 2q^{3} + q^{4} + 2q^{6} - q^{7} - q^{8} + q^{9} - 2q^{12} - 4q^{13} + q^{14} + q^{16} - 6q^{17} - q^{18} - 2q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 76032
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

Local data

This elliptic curve is not semistable. There are 3 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$(j)_{-}$
$2$ $1$ $I_{9}$ Non-split multiplicative 1 1 9 9
$7$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$23$ $4$ $I_0^{*}$ Additive -1 2 6 0

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 8.6.0.6
$3$ 3B 9.12.0.1

$p$-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.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type nonsplit ordinary ss nonsplit ss ordinary ordinary ordinary add ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) 4 2 6,4 2 2,2 2 2 2 - 2 2 2 2 2 2
$\mu$-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, 6, 9 and 18.
Its isogeny class 7406d consists of 6 curves linked by isogenies of degrees dividing 18.

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
$2$ \(\Q(\sqrt{2}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
$2$ \(\Q(\sqrt{69}) \) \(\Z/6\Z\) Not in database
$4$ 4.0.829472.3 \(\Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{2}, \sqrt{69})\) \(\Z/2\Z \times \Z/6\Z\) Not in database
$6$ 6.0.21296252943.2 \(\Z/6\Z\) Not in database
$6$ 6.6.574998829461.2 \(\Z/18\Z\) Not in database
$8$ 8.0.44033523122176.31 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ 8.4.57513173057536.30 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ 8.0.55729927701504.11 \(\Z/12\Z\) Not in database
$12$ Deg 12 \(\Z/3\Z \times \Z/6\Z\) Not in database
$12$ Deg 12 \(\Z/18\Z\) Not in database
$12$ Deg 12 \(\Z/2\Z \times \Z/6\Z\) Not in database
$12$ Deg 12 \(\Z/2\Z \times \Z/18\Z\) 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

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.