# Properties

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

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## 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 form49.2.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)

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

 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

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

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.