Properties

Label 770.f2
Conductor $770$
Discriminant $51765560000$
j-invariant \( \frac{5057359576472449}{51765560000} \)
CM no
Rank $1$
Torsion structure \(\Z/{6}\Z\)

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -3576, 81280])
 
gp: E = ellinit([1, 0, 0, -3576, 81280])
 
magma: E := EllipticCurve([1, 0, 0, -3576, 81280]);
 

\(y^2+xy=x^3-3576x+81280\)  Toggle raw display

Mordell-Weil group structure

$\Z\times \Z/{6}\Z$

Infinite order Mordell-Weil generator and height

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

$P$ =  \(\left(38, 6\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $1.0413547584596312511758409414$

Torsion generators

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

\( \left(-12, 356\right) \)  Toggle raw display

Integral points

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

\( \left(-54, 370\right) \), \( \left(-54, -316\right) \), \( \left(-12, 356\right) \), \( \left(-12, -344\right) \), \( \left(16, 160\right) \), \( \left(16, -176\right) \), \( \left(30, 20\right) \), \( \left(30, -50\right) \), \( \left(38, 6\right) \), \( \left(38, -44\right) \), \( \left(44, 76\right) \), \( \left(44, -120\right) \), \( \left(164, 1896\right) \), \( \left(164, -2060\right) \), \( \left(338, 5956\right) \), \( \left(338, -6294\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 770 \)  =  $2 \cdot 5 \cdot 7 \cdot 11$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $51765560000 $  =  $2^{6} \cdot 5^{4} \cdot 7^{6} \cdot 11 $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{5057359576472449}{51765560000} \)  =  $2^{-6} \cdot 5^{-4} \cdot 7^{-6} \cdot 11^{-1} \cdot 17^{3} \cdot 23^{3} \cdot 439^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $0.87335629813342157135760236606\dots$
Stable Faltings height: $0.87335629813342157135760236606\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $1.0413547584596312511758409414\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $1.1288970582858647608722881731\dots$
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: $ 72 $  = $ ( 2 \cdot 3 )\cdot2\cdot( 2 \cdot 3 )\cdot1 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $6$
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) $ ≈ $ 2.3511646469141299204550955530930479260 $

Modular invariants

Modular form   770.2.a.f

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 1152
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

Local data

This elliptic curve is 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$(j)_{-}$
$2$ $6$ $I_{6}$ Split multiplicative -1 1 6 6
$5$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$7$ $6$ $I_{6}$ Split multiplicative -1 1 6 6
$11$ $1$ $I_{1}$ 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 2.3.0.1
$3$ 3B.1.1 3.8.0.1

$p$-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 split ordinary nonsplit split nonsplit ordinary ss ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) 2 3 1 2 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 0

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3 and 6.
Its isogeny class 770.f consists of 4 curves linked by isogenies of degrees dividing 6.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{11}) \) \(\Z/2\Z \times \Z/6\Z\) Not in database
$4$ 4.0.2156.1 \(\Z/12\Z\) Not in database
$6$ 6.0.247066875.1 \(\Z/3\Z \times \Z/6\Z\) Not in database
$8$ 8.4.355559378944.4 \(\Z/2\Z \times \Z/12\Z\) Not in database
$8$ 8.0.8999178496.2 \(\Z/2\Z \times \Z/12\Z\) Not in database
$9$ 9.3.1356157919765880000.1 \(\Z/18\Z\) Not in database
$12$ Deg 12 \(\Z/6\Z \times \Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/24\Z\) Not in database
$18$ 18.6.1213232128892146856675046419248309862400000000.1 \(\Z/2\Z \times \Z/18\Z\) 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.