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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

sage: E = EllipticCurve([1, -1, 0, -2, -1])

gp: E = ellinit([1, -1, 0, -2, -1])

magma: E := EllipticCurve([1, -1, 0, -2, -1]);

$$y^2+xy=x^3-x^2-2x-1$$ ## Mordell-Weil group structure

$\Z/{2}\Z$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(2, -1\right)$$ ## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(2, -1\right)$$ ## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  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: E.omega  magma: RealPeriod(E); Real period: $1.9333117056168115467330768390\dots$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $2$  = $2$ sage: E.torsion_order()  gp: elltors(E)  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/factorial(ar)  magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12); Special value: $L(E,1)$ ≈ $0.96665585280840577336653841951483552682$

## Modular invariants

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

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

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 7.48.0.6

## $p$-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

$p$ 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 49a 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:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-7})$$ $$\Z/2\Z \times \Z/2\Z$$ 2.0.7.1-49.1-CMa1 $4$ 4.2.5488.1 $$\Z/4\Z$$ Not in database $4$ 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 $8$ 8.0.30118144.2 $$\Z/4\Z \times \Z/4\Z$$ Not in database $8$ 8.0.30118144.1 $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ 8.2.257298363.1 $$\Z/6\Z$$ Not in database $12$ 12.0.126548911552.1 $$\Z/2\Z \times \Z/28\Z$$ Not in database $16$ 16.4.59447875862838378496.1 $$\Z/8\Z$$ Not in database $16$ 16.0.232218265089212416.1 $$\Z/4\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/16\Z$$ Not in database $16$ 16.0.66202447602479769.1 $$\Z/6\Z \times \Z/6\Z$$ Not in database $20$ 20.0.11194501700250570391613.1 $$\Z/2\Z \times \Z/22\Z$$ 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.