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Properties

Label 8280p2
Conductor $8280$
Discriminant $533110118400$
j-invariant \( \frac{28697814}{13225} \)
CM no
Rank $1$
Torsion structure \(\Z/{2}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2=x^3-2187x+17766\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z=x^3-2187xz^2+17766z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-2187x+17766\) Copy content Toggle raw display (homogenize, minimize)

sage: E = EllipticCurve([0, 0, 0, -2187, 17766])
 
gp: E = ellinit([0, 0, 0, -2187, 17766])
 
magma: E := EllipticCurve([0, 0, 0, -2187, 17766]);
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

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

$P$ =  \(\left(-50, 46\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $2.7298485579550391527016207490$

Torsion generators

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

\( \left(42, 0\right) \) Copy content Toggle raw display

Integral points

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

\((-50,\pm 46)\), \( \left(42, 0\right) \) Copy content Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 8280 \)  =  $2^{3} \cdot 3^{2} \cdot 5 \cdot 23$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $533110118400 $  =  $2^{11} \cdot 3^{9} \cdot 5^{2} \cdot 23^{2} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{28697814}{13225} \)  =  $2 \cdot 3^{15} \cdot 5^{-2} \cdot 23^{-2}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $0.94489886062954333837668380351\dots$
Stable Faltings height: $-0.51444527138482213046887956885\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $2.7298485579550391527016207490\dots$
sage: E.period_lattice().omega()
 
gp: if(E.disc>0,2,1)*E.omega[1]
 
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
 
Real period: $0.82845558675985639309958052028\dots$
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\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)
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) $ ≈ $ 4.5231165776923796037203438943 $

Modular invariants

Modular form   8280.2.a.q

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);
 

\( q + q^{5} - 4 q^{11} + 4 q^{13} - 4 q^{17} + 4 q^{19} + O(q^{20}) \) Copy content Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 7680
$ \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$(j)_{-}$
$2$ $1$ $II^{*}$ Additive -1 3 11 0
$3$ $2$ $III^{*}$ Additive 1 2 9 0
$5$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$23$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 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
sage: gens = [[1, 2, 2, 5], [1844, 1, 919, 0], [1, 4, 0, 1], [2757, 4, 2756, 5], [1657, 4, 554, 9], [2, 1, 1379, 0], [1, 0, 4, 1], [1036, 1729, 345, 2416], [3, 4, 8, 11], [1201, 4, 2402, 9]]
 
sage: GL(2,Integers(2760)).subgroup(gens)
 
magma: Gens := [[1, 2, 2, 5], [1844, 1, 919, 0], [1, 4, 0, 1], [2757, 4, 2756, 5], [1657, 4, 554, 9], [2, 1, 1379, 0], [1, 0, 4, 1], [1036, 1729, 345, 2416], [3, 4, 8, 11], [1201, 4, 2402, 9]];
 
magma: sub<GL(2,Integers(2760))|Gens>;
 

The image of the adelic Galois representation has level $2760$, index $12$, genus $0$, and generators

$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1844 & 1 \\ 919 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2757 & 4 \\ 2756 & 5 \end{array}\right),\left(\begin{array}{rr} 1657 & 4 \\ 554 & 9 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 1379 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 1036 & 1729 \\ 345 & 2416 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1201 & 4 \\ 2402 & 9 \end{array}\right)$

$p$-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 add add split ss ord ord ord ord nonsplit ord ord ord ss ord ss
$\lambda$-invariant(s) - - 2 1,1 1 1 1 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

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.
Its isogeny class 8280p 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
$2$ \(\Q(\sqrt{6}) \) \(\Z/2\Z \oplus \Z/2\Z\) Not in database
$4$ 4.4.11426400.1 \(\Z/4\Z\) Not in database
$8$ 8.8.8356007485440000.8 \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$8$ deg 8 \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$8$ 8.2.391687850880000.6 \(\Z/6\Z\) 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.