Properties

Label 210e2
Conductor 210
Discriminant 51438240000
j-invariant \( \frac{135487869158881}{51438240000} \)
CM no
Rank 0
Torsion Structure \(\Z/{2}\Z \times \Z/{8}\Z\)

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This is the minimal-conductor curve with torsion $(\Z/2\Z) \times (\Z/8\Z)$.

Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 0, -1070, 7812]); // or
 
magma: E := EllipticCurve("210e2");
 
sage: E = EllipticCurve([1, 0, 0, -1070, 7812]) # or
 
sage: E = EllipticCurve("210e2")
 
gp: E = ellinit([1, 0, 0, -1070, 7812]) \\ or
 
gp: E = ellinit("210e2")
 

\( y^2 + x y = x^{3} - 1070 x + 7812 \)

Mordell-Weil group structure

\(\Z/{2}\Z \times \Z/{8}\Z\)

Torsion generators

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

\( \left(-36, 18\right) \), \( \left(34, 88\right) \)

Integral points

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

\( \left(-36, 18\right) \), \( \left(-26, 148\right) \), \( \left(-8, 130\right) \), \( \left(4, 58\right) \), \( \left(28, -14\right) \), \( \left(34, 88\right) \), \( \left(64, 418\right) \), \( \left(244, 3658\right) \)

Note: only one of each pair $\pm P$ is listed.

Invariants

magma: Conductor(E);
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
Conductor: \( 210 \)  =  \(2 \cdot 3 \cdot 5 \cdot 7\)
magma: Discriminant(E);
 
sage: E.discriminant().factor()
 
gp: E.disc
 
Discriminant: \(51438240000 \)  =  \(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{2} \)
magma: jInvariant(E);
 
sage: E.j_invariant().factor()
 
gp: E.j
 
j-invariant: \( \frac{135487869158881}{51438240000} \)  =  \(2^{-8} \cdot 3^{-8} \cdot 5^{-4} \cdot 7^{-2} \cdot 51361^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
 
sage: E.rank()
 
Rank: \(0\)
magma: Regulator(E);
 
sage: E.regulator()
 
Regulator: \(1\)
magma: RealPeriod(E);
 
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
Real period: \(1.02593301002\)
magma: TamagawaNumbers(E);
 
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
Tamagawa product: \( 512 \)  = \( 2^{3}\cdot2^{3}\cdot2^{2}\cdot2 \)
magma: Order(TorsionSubgroup(E));
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
Torsion order: \(16\)
magma: MordellWeilShaInformation(E);
 
sage: E.sha().an_numerical()
 
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form 210.2.a.e

magma: ModularForm(E);
 
sage: E.q_eigenform(20)
 
gp: xy = elltaniyama(E);
 
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
 

\( q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} - q^{7} + q^{8} + q^{9} + q^{10} - 4q^{11} + q^{12} - 2q^{13} - q^{14} + q^{15} + q^{16} + 2q^{17} + q^{18} + 4q^{19} + O(q^{20}) \)

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

magma: ModularDegree(E);
 
sage: E.modular_degree()
 
Modular degree: 256
\( \Gamma_0(N) \)-optimal: no
Manin constant: 1

Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
sage: r = E.rank();
 
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: ar = ellanalyticrank(E);
 
gp: ar[2]/factorial(ar[1])
 

\( L(E,1) \) ≈ \( 2.05186602004 \)

Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
prime Tamagawa number Kodaira symbol Reduction type Root number ord(\(N\)) ord(\(\Delta\)) ord\((j)_{-}\)
\(2\) \(8\) \( I_{8} \) Split multiplicative -1 1 8 8
\(3\) \(8\) \( I_{8} \) Split multiplicative -1 1 8 8
\(5\) \(4\) \( I_{4} \) Split multiplicative -1 1 4 4
\(7\) \(2\) \( I_{2} \) Non-split multiplicative 1 1 2 2

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 X193n.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 0 & 1 \end{array}\right)$ and has index 96.

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 
sage: rho = E.galois_representation();
 
sage: [rho.image_type(p) for p in rho.non_surjective()]
 

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

prime Image of Galois representation
\(2\) Cs

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]
 

All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).

Iwasawa invariants

$p$ 2 3 5 7
Reduction type split split split nonsplit
$\lambda$-invariant(s) 2 5 1 0
$\mu$-invariant(s) 0 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4 and 8.
Its isogeny class 210e consists of 8 curves linked by isogenies of degrees dividing 16.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{8}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
4 \(\Q(\sqrt{6}, \sqrt{70})\) \(\Z/2\Z \times \Z/16\Z\) Not in database
\(\Q(i, \sqrt{7})\) \(\Z/4\Z \times \Z/8\Z\) Not in database
\(\Q(\sqrt{-6}, \sqrt{10})\) \(\Z/2\Z \times \Z/16\Z\) 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.

Additional information

This is the curve of minimal conductor and torsion $(\Z/2\Z) \times (\Z/8\Z)$. Every elliptic curve $E/\Q$ with this torsion group must have conductor divisible by $210 = 2 \cdot 3 \cdot 5 \cdot 7$ (for instance, if $E$ had good reduction at $7$ then the reduction mod $7$ would have at least $16$ points, which exceeds the Weil bound $(\sqrt7+1)^2 < 14$.