Properties

Label 574.g2
Conductor $574$
Discriminant $7.081\times 10^{13}$
j-invariant \( \frac{801581275315909089}{70810888830976} \)
CM no
Rank $1$
Torsion structure \(\Z/{7}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, -19353, 958713])
 
gp: E = ellinit([1, -1, 1, -19353, 958713])
 
magma: E := EllipticCurve([1, -1, 1, -19353, 958713]);
 

\(y^2+xy+y=x^3-x^2-19353x+958713\)  Toggle raw display

Mordell-Weil group structure

$\Z\times \Z/{7}\Z$

Infinite order Mordell-Weil generator and height

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

$P$ =  \(\left(61, 18\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $1.0673824816878454875840596007$

Torsion generators

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

\( \left(103, 172\right) \)  Toggle raw display

Integral points

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

\( \left(-121, 1292\right) \), \( \left(-121, -1172\right) \), \( \left(-37, 1292\right) \), \( \left(-37, -1256\right) \), \( \left(-9, 1068\right) \), \( \left(-9, -1060\right) \), \( \left(39, 492\right) \), \( \left(39, -532\right) \), \( \left(61, 18\right) \), \( \left(61, -80\right) \), \( \left(103, 172\right) \), \( \left(103, -276\right) \), \( \left(133, 784\right) \), \( \left(133, -918\right) \), \( \left(159, 1292\right) \), \( \left(159, -1452\right) \), \( \left(551, 12268\right) \), \( \left(551, -12820\right) \), \( \left(7019, 584392\right) \), \( \left(7019, -591412\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 574 \)  =  $2 \cdot 7 \cdot 41$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $70810888830976 $  =  $2^{21} \cdot 7^{7} \cdot 41 $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{801581275315909089}{70810888830976} \)  =  $2^{-21} \cdot 3^{3} \cdot 7^{-7} \cdot 19^{3} \cdot 41^{-1} \cdot 43^{3} \cdot 379^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $1.3977856380968835507713534157\dots$
Stable Faltings height: $1.3977856380968835507713534157\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $1.0673824816878454875840596007\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.60020819121169191031840301132\dots$
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: $ 147 $  = $ ( 3 \cdot 7 )\cdot7\cdot1 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $7$
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) $ ≈ $ 1.9219551259947258101377176144850767847 $

Modular invariants

Modular form   574.2.a.g

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 3528
$ \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$(j)_{-}$
$2$ $21$ $I_{21}$ Split multiplicative -1 1 21 21
$7$ $7$ $I_{7}$ Split multiplicative -1 1 7 7
$41$ $1$ $I_{1}$ 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
$7$ 7B.1.1 7.48.0.1

$p$-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 split ss ordinary split ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary nonsplit ordinary ordinary
$\lambda$-invariant(s) 2 1,1 1 2 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 $d=$ 7.
Its isogeny class 574.g consists of 2 curves linked by isogenies of degree 7.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{7}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$3$ 3.3.2296.1 \(\Z/14\Z\) Not in database
$6$ 6.6.12103630336.1 \(\Z/2\Z \times \Z/14\Z\) Not in database
$8$ 8.2.14838034276107.1 \(\Z/21\Z\) Not in database
$12$ Deg 12 \(\Z/28\Z\) 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.