Properties

Label 1520.b2
Conductor $1520$
Discriminant $-902500000000$
j-invariant \( \frac{298091207216}{3525390625} \)
CM no
Rank $1$
Torsion structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2=x^3+x^2+884x-44280\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z=x^3+x^2z+884xz^2-44280z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3+71577x-32494878\) Copy content Toggle raw display (homogenize, minimize)

sage: E = EllipticCurve([0, 1, 0, 884, -44280])
 
gp: E = ellinit([0, 1, 0, 884, -44280])
 
magma: E := EllipticCurve([0, 1, 0, 884, -44280]);
 
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);
 

$P$ =  \(\left(\frac{229}{4}, \frac{3553}{8}\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $4.3377539088000136188400598202$

Torsion generators

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

\( \left(27, 0\right) \) Copy content Toggle raw display

Integral points

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

\( \left(27, 0\right) \) Copy content Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 1520 \)  =  $2^{4} \cdot 5 \cdot 19$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $-902500000000 $  =  $-1 \cdot 2^{8} \cdot 5^{10} \cdot 19^{2} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{298091207216}{3525390625} \)  =  $2^{4} \cdot 5^{-10} \cdot 11^{3} \cdot 19^{-2} \cdot 241^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $0.97494703747940096805237210437\dots$
Stable Faltings height: $0.51284891710610409510755069006\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $4.3377539088000136188400598202\dots$
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: $0.43534086709705489898971150582\dots$
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 $  = $ 1\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) $ ≈ $ 1.8884015479106371258984507536 $

Modular invariants

Modular form   1520.2.a.b

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);
 

\( q - 2 q^{3} - q^{5} - 2 q^{7} + q^{9} + 6 q^{13} + 2 q^{15} + 2 q^{17} + q^{19} + O(q^{20}) \) Copy content Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 1920
$ \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$(j)_{-}$
$2$ $1$ $I_0^{*}$ Additive -1 4 8 0
$5$ $2$ $I_{10}$ Non-split multiplicative 1 1 10 10
$19$ $2$ $I_{2}$ 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 4.6.0.5
sage: gens = [[381, 8, 2, 17], [401, 8, 42, 17], [753, 8, 752, 9], [1, 0, 8, 1], [5, 8, 48, 77], [189, 756, 95, 759], [457, 6, 0, 1], [3, 8, 10, 27], [1, 8, 0, 1]]
 
sage: GL(2,Integers(760)).subgroup(gens)
 
magma: Gens := [[381, 8, 2, 17], [401, 8, 42, 17], [753, 8, 752, 9], [1, 0, 8, 1], [5, 8, 48, 77], [189, 756, 95, 759], [457, 6, 0, 1], [3, 8, 10, 27], [1, 8, 0, 1]];
 
magma: sub<GL(2,Integers(760))|Gens>;
 

The image of the adelic Galois representation has level $760$, index $48$, genus $0$, and generators

$\left(\begin{array}{rr} 381 & 8 \\ 2 & 17 \end{array}\right),\left(\begin{array}{rr} 401 & 8 \\ 42 & 17 \end{array}\right),\left(\begin{array}{rr} 753 & 8 \\ 752 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 48 & 77 \end{array}\right),\left(\begin{array}{rr} 189 & 756 \\ 95 & 759 \end{array}\right),\left(\begin{array}{rr} 457 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 8 \\ 10 & 27 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right)$

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

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 1520.b 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
$2$ \(\Q(\sqrt{-1}) \) \(\Z/2\Z \oplus \Z/2\Z\) 2.0.4.1-36100.2-b1
$4$ 4.2.400.1 \(\Z/4\Z\) Not in database
$8$ 8.0.2560000.1 \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$8$ 8.0.854071705600.17 \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$8$ deg 8 \(\Z/6\Z\) Not in database
$16$ deg 16 \(\Z/4\Z \oplus \Z/4\Z\) 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.