Label 49.a4
Conductor $49$
Discriminant $-343$
j-invariant \( -3375 \)
CM yes (\(D=-7\))
Rank $0$
Torsion structure \(\Z/{2}\Z\)

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This is a model for the modular curve $X_0(49)$. This is the largest level $N \in \mathbb{N}$ such that $X_0(N)$ is of genus $1$, so this elliptic curve is the one of largest conductor to have modular degree $1$.

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy=x^3-x^2-2x-1\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz=x^3-x^2z-2xz^2-z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-35x-98\) Copy content Toggle raw display (homogenize, minimize)

sage: E = EllipticCurve([1, -1, 0, -2, -1])
gp: E = ellinit([1, -1, 0, -2, -1])
magma: E := EllipticCurve([1, -1, 0, -2, -1]);
sage: E.short_weierstrass_model()
magma: WeierstrassModel(E);

Mordell-Weil group structure


Torsion generators

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

\( \left(2, -1\right) \) Copy content Toggle raw display

Integral points

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

\( \left(2, -1\right) \) Copy content Toggle raw display


sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
Conductor: \( 49 \)  =  $7^{2}$
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
Discriminant: $-343 $  =  $-1 \cdot 7^{3} $
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
j-invariant: \( -3375 \)  =  $-1 \cdot 3^{3} \cdot 5^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z[(1+\sqrt{-7})/2]\) (potential complex multiplication)
Sato-Tate group: $N(\mathrm{U}(1))$
Faltings height: $-0.79913838905562241712431170973\dots$
Stable Faltings height: $-1.2856159263194507434006498956\dots$

BSD invariants

sage: E.rank()
magma: Rank(E);
Analytic rank: $0$
sage: E.regulator()
magma: Regulator(E);
Regulator: $1$
sage: E.period_lattice().omega()
gp: if(E.disc>0,2,1)*[1]
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
Real period: $1.9333117056168115467330768390\dots$
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: $ 2 $  = $ 2 $
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
Torsion order: $2$
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) $ ≈ $ 0.96665585280840577336653841951 $

Modular invariants

Modular form   49.2.a.a

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);

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

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

sage: E.modular_degree()
magma: ModularDegree(E);
Modular degree: 1
$ \Gamma_0(N) $-optimal: yes
Manin constant: 1

Local data

This elliptic curve is not semistable. There is only one prime 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$(j)_{-}$
$7$ $2$ $III$ Additive -1 2 3 0

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.1.5

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) 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 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.


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 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
$2$ \(\Q(\sqrt{-7}) \) \(\Z/2\Z \oplus \Z/2\Z\)
$4$ 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
$12$ 12.0.126548911552.1 \(\Z/2\Z \oplus \Z/28\Z\) 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
$21$ 21.3.3219905755813179726837607.1 \(\Z/14\Z\) 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.