Properties

Label 714.f6
Conductor $714$
Discriminant $-2.929\times 10^{18}$
j-invariant \( \frac{4738217997934888496063}{2928751705237796928} \)
CM no
Rank $0$
Torsion structure \(\Z/{4}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 1, 349916, 20936165])
 
gp: E = ellinit([1, 1, 1, 349916, 20936165])
 
magma: E := EllipticCurve([1, 1, 1, 349916, 20936165]);
 

\(y^2+xy+y=x^3+x^2+349916x+20936165\)  Toggle raw display

Mordell-Weil group structure

$\Z/{4}\Z$

Torsion generators

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

\( \left(541, 18937\right) \)  Toggle raw display

Integral points

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

\( \left(541, 18937\right) \), \( \left(541, -19479\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 714 \)  =  $2 \cdot 3 \cdot 7 \cdot 17$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $-2928751705237796928 $  =  $-1 \cdot 2^{6} \cdot 3^{4} \cdot 7^{16} \cdot 17 $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{4738217997934888496063}{2928751705237796928} \)  =  $2^{-6} \cdot 3^{-4} \cdot 7^{-16} \cdot 17^{-1} \cdot 47^{3} \cdot 191^{3} \cdot 1871^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $2.2325553790430451078261910604\dots$
Stable Faltings height: $2.2325553790430451078261910604\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $0$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $1$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.15689245972130971411926011680\dots$
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: $ 192 $  = $ ( 2 \cdot 3 )\cdot2\cdot2^{4}\cdot1 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $4$
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) $ ≈ $ 1.8827095166557165694311214016361321632 $

Modular invariants

Modular form   714.2.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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 15360
$ \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
$3$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$7$ $16$ $I_{16}$ Split multiplicative -1 1 16 16
$17$ $1$ $I_{1}$ 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 16.96.0.103

$p$-adic regulators

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

All $p$-adic regulators are identically $1$ since the rank is $0$.

Iwasawa invariants

$p$ 2 3 7 17
Reduction type split nonsplit split split
$\lambda$-invariant(s) 3 4 3 1
$\mu$-invariant(s) 2 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ 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 714.f consists of 4 curves linked by isogenies of degrees dividing 8.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{-17}) \) \(\Z/2\Z \times \Z/4\Z\) Not in database
$2$ \(\Q(\sqrt{2}) \) \(\Z/8\Z\) Not in database
$2$ \(\Q(\sqrt{-34}) \) \(\Z/8\Z\) Not in database
$4$ \(\Q(\sqrt{2}, \sqrt{-17})\) \(\Z/2\Z \times \Z/8\Z\) Not in database
$4$ 4.2.34816.2 \(\Z/16\Z\) Not in database
$8$ 8.0.6179217664.3 \(\Z/4\Z \times \Z/4\Z\) Not in database
$8$ 8.0.1581879721984.17 \(\Z/2\Z \times \Z/8\Z\) Not in database
$8$ 8.0.101240302206976.26 \(\Z/16\Z\) Not in database
$8$ 8.0.1401249857536.7 \(\Z/2\Z \times \Z/16\Z\) Not in database
$8$ 8.2.35523982503387.4 \(\Z/12\Z\) Not in database
$16$ Deg 16 \(\Z/4\Z \times \Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/4\Z \times \Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/16\Z\) Not in database
$16$ Deg 16 \(\Z/32\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/12\Z\) Not in database
$16$ Deg 16 \(\Z/24\Z\) Not in database
$16$ Deg 16 \(\Z/24\Z\) 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.