Properties

Label 49.a2
Conductor 49
Discriminant -40353607
j-invariant \( -3375 \)
CM yes (\(D=-7\))
Rank 0
Torsion Structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 0, -107, 552]); // or
 
magma: E := EllipticCurve("49a3");
 
sage: E = EllipticCurve([1, -1, 0, -107, 552]) # or
 
sage: E = EllipticCurve("49a3")
 
gp: E = ellinit([1, -1, 0, -107, 552]) \\ or
 
gp: E = ellinit("49a3")
 

\( y^2 + x y = x^{3} - x^{2} - 107 x + 552 \)

Mordell-Weil group structure

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

Torsion generators

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

\( \left(-12, 6\right) \)

Integral points

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

\( \left(-12, 6\right) \)

Invariants

magma: Conductor(E);
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
Conductor: \( 49 \)  =  \(7^{2}\)
magma: Discriminant(E);
 
sage: E.discriminant().factor()
 
gp: E.disc
 
Discriminant: \(-40353607 \)  =  \(-1 \cdot 7^{9} \)
magma: jInvariant(E);
 
sage: E.j_invariant().factor()
 
gp: E.j
 
j-invariant: \( -3375 \)  =  \(-1 \cdot 3^{3} \cdot 5^{3}\)
Endomorphism ring: \(\Z[(1+\sqrt{-7})/2]\)   ( Complex Multiplication)
Sato-Tate Group: $N(\mathrm{U}(1))$

BSD invariants

magma: Rank(E);
 
sage: E.rank()
 
Rank: \(0\)
magma: Regulator(E);
 
sage: E.regulator()
 
Regulator: \(1\)
magma: RealPeriod(E);
 
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
Real period: \(1.93331170562\)
magma: TamagawaNumbers(E);
 
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
Tamagawa product: \( 2 \)  = \( 2 \)
magma: Order(TorsionSubgroup(E));
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
Torsion order: \(2\)
magma: MordellWeilShaInformation(E);
 
sage: E.sha().an_numerical()
 
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form 49.2.a.a

magma: ModularForm(E);
 
sage: E.q_eigenform(20)
 
gp: xy = elltaniyama(E);
 
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
 

\( q + q^{2} - q^{4} - 3q^{8} - 3q^{9} + 4q^{11} - q^{16} - 3q^{18} + O(q^{20}) \)

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

magma: ModularDegree(E);
 
sage: E.modular_degree()
 
Modular degree: 7
\( \Gamma_0(N) \)-optimal: no
Manin constant: 1

Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
sage: r = E.rank();
 
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: ar = ellanalyticrank(E);
 
gp: ar[2]/factorial(ar[1])
 

\( L(E,1) \) ≈ \( 0.966655852808 \)

Local data

This elliptic curve is not semistable.

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
prime Tamagawa number Kodaira symbol Reduction type Root number ord(\(N\)) ord(\(\Delta\)) ord\((j)_{-}\)
\(7\) \(2\) \( III^{*} \) Additive -1 2 9 0

Galois representations

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 
sage: rho = E.galois_representation();
 
sage: [rho.image_type(p) for p in rho.non_surjective()]
 

The mod \( p \) Galois representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois representation
\(7\) B.1.2

For all other primes \(p\), the image is the normalizer of a split Cartan subgroup if \(\left(\frac{ -7 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -7 }{p}\right)=-1\).

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]
 

All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).

Iwasawa invariants

$p$ 2 3 5 7
Reduction type ordinary ss ss add
$\lambda$-invariant(s) ? 0,0 0,0 -
$\mu$-invariant(s) ? 0,0 0,0 -

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 11$ of good reduction are zero.

An entry ? indicates that the invariants have not yet been computed.

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, 7 and 14.
Its isogeny class 49.a consists of 4 curves linked by isogenies of degrees dividing 14.

Growth of torsion in number fields

The number fields $K$ of degree up to 7 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 \(\Q(\sqrt{-7}) \) \(\Z/2\Z \times \Z/2\Z\) 2.0.7.1-49.1-CMa1
3 \(\Q(\zeta_{7})^+\) \(\Z/14\Z\) 3.3.49.1-49.1-a2
4 4.2.5488.1 \(\Z/4\Z\) Not in database
4.0.1372.1 \(\Z/2\Z \times \Z/4\Z\) Not in database
6 \(\Q(\zeta_{7})\) \(\Z/2\Z \times \Z/14\Z\) Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.

Additional information

This curve $E$ is the quotient of the Fermat curve $F_7$ of degree $7$ by the action of $S_3$ that permutes the variables of the symmetrical form $X^7 + Y^7 + Z^7 = 0$ of $F_7$. Since $E$ has rank zero (and that fact can be shown by descent using the rational $2$-isogeny), this yields a proof of the exponent-$7$ case of Fermat's last theorem that is almost as elementary as Fermat's for $n=4$ (and certainly easier than the known proofs for $n=5$). This proof was given by Genocchi in 1855: he wrote, towards the end of his paper "Intorno all'equazione $x^7+y^7+z^7 = 0$", Annali di Mat. Pura ed Applicata 6 (1864), 287-288), that he announced these results in "Cimento di Torino, vol. VI, fasc. VIII, 1855"; see pages 75-76 of [http://library.msri.org/books/Book35/files/elkies.pdf]. By a result of Gross and Rohrlich published in Inventiones Math. 1978 [https://link.springer.com/article/10.1007/BF01403161], the Jacobian of $F_p$ has infinite order for all primes $p>7$, suggesting that Genocchi's elementary proof for $p=7$ is the last one of its kind.