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## Simplified equation

 $$y^2+xy=x^3-x^2-107x+552$$ y^2+xy=x^3-x^2-107x+552 (homogenize, simplify) $$y^2z+xyz=x^3-x^2z-107xz^2+552z^3$$ y^2z+xyz=x^3-x^2z-107xz^2+552z^3 (dehomogenize, simplify) $$y^2=x^3-1715x+33614$$ y^2=x^3-1715x+33614 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, -1, 0, -107, 552])

gp: E = ellinit([1, -1, 0, -107, 552])

magma: E := EllipticCurve([1, -1, 0, -107, 552]);

oscar: E = EllipticCurve([1, -1, 0, -107, 552])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

$$\Z/{2}\Z$$

magma: MordellWeilGroup(E);

## Torsion generators

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

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

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

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$49$$ = $7^{2}$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E);  oscar: conductor(E) Discriminant: $-40353607$ = $-1 \cdot 7^{9}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$-3375$$ = $-1 \cdot 3^{3} \cdot 5^{3}$ comment: j-invariant  sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E);  oscar: j_invariant(E) Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z[(1+\sqrt{-7})/2]$$ (potential complex multiplication) sage: E.has_cm()  magma: HasComplexMultiplication(E); Sato-Tate group: $N(\mathrm{U}(1))$ Faltings height: $0.17381668547203423542836466200\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $-1.2856159263194507434006498956\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E)

## BSD invariants

 Analytic rank: $0$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $1$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $1.9333117056168115467330768390\dots$ comment: Real Period  sage: E.period_lattice().omega()  gp: if(E.disc>0,2,1)*E.omega  magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E); Tamagawa product: $2$  = $2$ comment: Tamagawa numbers  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E);  oscar: tamagawa_numbers(E) Torsion order: $2$ comment: Torsion order  sage: E.torsion_order()  gp: elltors(E)  magma: Order(TorsionSubgroup(E));  oscar: prod(torsion_structure(E)) Analytic order of Ш: $1$ (exact) comment: Order of Sha  sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Special value: $L(E,1)$ ≈ $0.96665585280840577336653841951$ comment: Special L-value  r = E.rank(); E.lseries().dokchitser().derivative(1,r)/r.factorial()  gp: [r,L1r] = ellanalyticrank(E); L1r/r!  magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

## BSD formula

$\displaystyle 0.966655853 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.933312 \cdot 1.000000 \cdot 2}{2^2} \approx 0.966655853$

# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)

E = EllipticCurve(%s); r = E.rank(); ar = E.analytic_rank(); assert r == ar;

Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical();

omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order();

assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))

/* self-contained Magma code snippet for the BSD formula (checks rank, computes analyiic sha) */

E := EllipticCurve(%s); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar;

sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1);

reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E);

assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);

## Modular invariants

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

comment: q-expansion of modular form

sage: E.q_eigenform(20)

\\ actual modular form, use for small N

[mf,F] = mffromell(E)

Ser(mfcoefs(mf,20),q)

\\ or just the series

Ser(ellan(E,20),q)*q

magma: ModularForm(E);

Modular degree: 7
comment: Modular degree

sage: E.modular_degree()

gp: ellmoddegree(E)

magma: ModularDegree(E);

$\Gamma_0(N)$-optimal: no
Manin constant: 1
comment: Manin constant

magma: ManinConstant(E);

## Local data

This elliptic curve is not semistable. There is only one prime of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$7$ $2$ $III^{*}$ Additive -1 2 9 0

comment: Local data

sage: E.local_data()

gp: ellglobalred(E)

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

oscar: [(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]

## Galois representations

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.1.2 7.48.0.3

comment: mod p Galois image

sage: rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

## Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 7 and 14.
Its isogeny class 49a consists of 4 curves linked by isogenies of degrees dividing 14.

## Twists

The minimal quadratic twist of this elliptic curve is 49a1, its twist by $-7$.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-7})$$ $$\Z/2\Z \oplus \Z/2\Z$$ 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 $4$ 4.0.1372.1 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $6$ $$\Q(\zeta_{7})$$ $$\Z/2\Z \oplus \Z/14\Z$$ Not in database $8$ 8.0.30118144.2 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.30118144.1 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.2.257298363.1 $$\Z/6\Z$$ Not in database $12$ 12.6.8099130339328.1 $$\Z/28\Z$$ Not in database $12$ 12.0.126548911552.1 $$\Z/2\Z \oplus \Z/28\Z$$ Not in database $16$ 16.4.59447875862838378496.1 $$\Z/8\Z$$ Not in database $16$ 16.0.232218265089212416.1 $$\Z/4\Z \oplus \Z/8\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/16\Z$$ Not in database $16$ 16.0.66202447602479769.1 $$\Z/6\Z \oplus \Z/6\Z$$ Not in database $20$ 20.0.11194501700250570391613.1 $$\Z/2\Z \oplus \Z/22\Z$$ 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.

## Iwasawa invariants

$p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 ord 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.

## $p$-adic regulators

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

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.