Properties

Label 102a1
Conductor $102$
Discriminant $612$
j-invariant \( \frac{1771561}{612} \)
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, 0, -2, 0])
 
gp: E = ellinit([1, 1, 0, -2, 0])
 
magma: E := EllipticCurve([1, 1, 0, -2, 0]);
 

Minimal equation

Minimal equation

Simplified equation

\(y^2+xy=x^3+x^2-2x\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz=x^3+x^2z-2xz^2\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-3267x+45630\) Copy content Toggle raw display (homogenize, minimize)

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

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

$P$ =  \(\left(2, 2\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $0.14325389294088007147627441303$

Torsion generators

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

\( \left(0, 0\right) \) Copy content Toggle raw display

Integral points

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

\( \left(-2, 2\right) \), \( \left(-2, 0\right) \), \( \left(-1, 2\right) \), \( \left(-1, -1\right) \), \( \left(0, 0\right) \), \( \left(1, 0\right) \), \( \left(1, -1\right) \), \( \left(2, 2\right) \), \( \left(2, -4\right) \), \( \left(8, 20\right) \), \( \left(8, -28\right) \), \( \left(9, 24\right) \), \( \left(9, -33\right) \), \( \left(2738, 141932\right) \), \( \left(2738, -144670\right) \) Copy content Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 102 \)  =  $2 \cdot 3 \cdot 17$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $612 $  =  $2^{2} \cdot 3^{2} \cdot 17 $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{1771561}{612} \)  =  $2^{-2} \cdot 3^{-2} \cdot 11^{6} \cdot 17^{-1}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $-0.76352819056830652556218343290\dots$
Stable Faltings height: $-0.76352819056830652556218343290\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $0.14325389294088007147627441303\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $4.7278638235414655119977559586\dots$
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: $ 4 $  = $ 2\cdot2\cdot1 $
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) $ ≈ $ 0.67728489801666901020123734615 $

Modular invariants

Modular form   102.2.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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 8
$ \Gamma_0(N) $-optimal: yes
Manin constant: 1

Local data

This elliptic curve is 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$(j)_{-}$
$2$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$3$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$17$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1

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 8.6.0.4

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]
 

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type nonsplit nonsplit ord ord ss ord nonsplit ord ord ord ord ord ord ord ord
$\lambda$-invariant(s) 1 1 1 1 1,1 1 1 1 3 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

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 102a consists of 2 curves linked by isogenies of degree 2.

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
$2$ \(\Q(\sqrt{17}) \) \(\Z/2\Z \oplus \Z/2\Z\) 2.2.17.1-612.1-a2
$4$ 4.0.1088.1 \(\Z/4\Z\) Not in database
$8$ 8.4.2002066523136.1 \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$8$ 8.0.342102016.4 \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$8$ 8.2.236727913392.1 \(\Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \oplus \Z/6\Z\) 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.

Additional information

This is the elliptic curve $E$ associated to the [Somos-5 sequence ] $\{a(n)\}$. Let $T$ be the $2$-torsion point $(0,0)$, and $P$ the point $(2,2)$ such that $E(\Q) = \Z P \oplus \{0, T\}$. Then the $x$- and $y$-coordinates of $nP+T$ have denominators $d_n^2$ and $d_n^3$ where $$d_n = 1, 1, 2, 3, 5, 11, 37, 83, 274, 1217$$ for $1 \leq n \leq 10$, and $d_n = a(n+2)$ in general, satisfying the Somos-5 recurrence $$ d_n d_{n+5} = d_{n+1} d_{n+4} + d_{n+2} d_{n+3}. $$ Thus the regulator of $E$, which is the canonical height $\hat h(P) = 0.143\ldots$, controls the growth of the $a(n)$: asymptotically $\log a_n \sim \frac12 \hat h(P) n^2$.