Label 49a3
Conductor $49$
Discriminant $-40353607$
j-invariant \( -3375 \)
CM yes (\(D=-7\))
Rank $0$
Torsion structure \(\Z/{2}\Z\)

Related objects


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Show commands: Magma / Oscar / PariGP / SageMath

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy=x^3-x^2-107x+552\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz=x^3-x^2z-107xz^2+552z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-1715x+33614\) Copy content Toggle raw display (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


magma: MordellWeilGroup(E);

Torsion generators

\( \left(-12, 6\right) \) Copy content Toggle raw display

comment: Torsion subgroup
sage: E.torsion_subgroup().gens()
gp: elltors(E)
magma: TorsionSubgroup(E);
oscar: torsion_structure(E)

Integral points

\( \left(-12, 6\right) \) Copy content Toggle raw display

comment: Integral points
sage: E.integral_points()
magma: IntegralPoints(E);


Conductor: \( 49 \)  =  $7^{2}$
comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
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
magma: Regulator(E);
Real period: $1.9333117056168115467330768390\dots$
comment: Real Period
sage: E.period_lattice().omega()
gp: if(E.disc>0,2,1)*[1]
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[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
oscar: tamagawa_numbers(E)
Torsion order: $2$
comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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();
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

Modular form   49.2.a.a

\( q + q^{2} - q^{4} - 3 q^{8} - 3 q^{9} + 4 q^{11} - q^{16} - 3 q^{18} + O(q^{20}) \) Copy content Toggle raw display

comment: q-expansion of modular form
sage: E.q_eigenform(20)
\\ actual modular form, use for small N
[mf,F] = mffromell(E)
\\ or just the series
magma: ModularForm(E);

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

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)[5]
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

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)];


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.


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:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{-7}) \) \(\Z/2\Z \oplus \Z/2\Z\)
$3$ \(\Q(\zeta_{7})^+\) \(\Z/14\Z\)
$4$ 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

$p$ 2 3 5 7
Reduction type ord 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.

$p$-adic regulators

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

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 []. By a result of Gross and Rohrlich published in Inventiones Math. 1978 [], 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.