Properties

Label 1470d7
Conductor 1470
Discriminant 689349609375000
j-invariant \( \frac{10316097499609}{5859375000} \)
CM no
Rank 1
Torsion Structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 0, -22222, 164284]) # or
 
sage: E = EllipticCurve("1470d7")
 
gp: E = ellinit([1, 1, 0, -22222, 164284]) \\ or
 
gp: E = ellinit("1470d7")
 
magma: E := EllipticCurve([1, 1, 0, -22222, 164284]); // or
 
magma: E := EllipticCurve("1470d7");
 

\( y^2 + x y = x^{3} + x^{2} - 22222 x + 164284 \)

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

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

\(P\) =  \( \left(3, 311\right) \)
\(\hat{h}(P)\) ≈  0.72281562331401

Torsion generators

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

\( \left(-\frac{613}{4}, \frac{613}{8}\right) \)

Integral points

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

\( \left(3, 311\right) \), \( \left(3, -314\right) \), \( \left(153, 536\right) \), \( \left(153, -689\right) \), \( \left(1253, 43436\right) \), \( \left(1253, -44689\right) \)

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 1470 \)  =  \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(689349609375000 \)  =  \(2^{3} \cdot 3 \cdot 5^{12} \cdot 7^{6} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{10316097499609}{5859375000} \)  =  \(2^{-3} \cdot 3^{-1} \cdot 5^{-12} \cdot 11^{3} \cdot 1979^{3}\)
Endomorphism Ring: \(\Z\)
Geometric Endomorphism Ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Rank: \(1\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(0.722815623314\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.437831063891\)
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: \( 24 \)  = \( 1\cdot1\cdot( 2^{2} \cdot 3 )\cdot2 \)
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)

Modular invariants

Modular form 1470.2.a.d

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);
 

\( q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{8} + q^{9} - q^{10} - q^{12} - 2q^{13} - q^{15} + q^{16} - 6q^{17} - q^{18} + 4q^{19} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 6912
\( \Gamma_0(N) \)-optimal: no
Manin constant: 1

Special L-value

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);
 

\( L'(E,1) \) ≈ \( 1.89882680012 \)

Local data

This elliptic curve is not semistable.

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\((j)_{-}\)
\(2\) \(1\) \( I_{3} \) Non-split multiplicative 1 1 3 3
\(3\) \(1\) \( I_{1} \) Non-split multiplicative 1 1 1 1
\(5\) \(12\) \( I_{12} \) Split multiplicative -1 1 12 12
\(7\) \(2\) \( I_0^{*} \) Additive -1 2 6 0

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X13.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right)$ and has index 6.

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 mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.

prime Image of Galois representation
\(2\) B
\(3\) B

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) 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 nonsplit split add ss ordinary ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ss
$\lambda$-invariant(s) 4 1 2 - 1,1 1 1 1 1,1 1 1 1 1 1 1,1
$\mu$-invariant(s) 1 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, 4, 6 and 12.
Its isogeny class 1470d consists of 8 curves linked by isogenies of degrees dividing 12.

Growth of torsion in number fields

The number fields $K$ of degree up to 7 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 \(\Q(\sqrt{-14}) \) \(\Z/4\Z\) Not in database
\(\Q(\sqrt{-21}) \) \(\Z/4\Z\) Not in database
\(\Q(\sqrt{21}) \) \(\Z/6\Z\) 2.2.21.1-300.1-j3
\(\Q(\sqrt{6}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
4 \(\Q(i, \sqrt{21})\) \(\Z/12\Z\) Not in database
\(\Q(\sqrt{-6}, \sqrt{-14})\) \(\Z/12\Z\) Not in database
\(\Q(\sqrt{6}, \sqrt{14})\) \(\Z/2\Z \times \Z/6\Z\) Not in database
\(\Q(\sqrt{6}, \sqrt{-14})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
6 6.0.20253807.1 \(\Z/6\Z\) Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.