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
\(y^2+xy=x^3-x^2-2x-1\)
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(homogenize, simplify) |
\(y^2z+xyz=x^3-x^2z-2xz^2-z^3\)
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(dehomogenize, simplify) |
\(y^2=x^3-35x-98\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
\( \left(2, -1\right) \)
Integral points
\( \left(2, -1\right) \)
Invariants
Conductor: | \( 49 \) | = | $7^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-343 $ | = | $-1 \cdot 7^{3} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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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)
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Endomorphism ring: | $\Z$ | |||
Geometric endomorphism ring: | \(\Z[(1+\sqrt{-7})/2]\) | (potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $N(\mathrm{U}(1))$ | |||
Faltings height: | $-0.79913838905562241712431170973\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-1.2856159263194507434006498956\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9802957926219806\dots$ | |||
Szpiro ratio: | $3.6937251021607387\dots$ |
BSD invariants
Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $1.9333117056168115467330768390\dots$ | comment: Real Period
sage: E.period_lattice().omega()
gp: if(E.disc>0,2,1)*E.omega[1]
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
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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)
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Torsion order: | $2$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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Analytic order of Ш: | $1$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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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);
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BSD formula
$\displaystyle 0.966655853 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.933312 \cdot 1.000000 \cdot 2}{2^2} \approx 0.966655853$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 1 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There is only one prime $p$ of bad reduction:
$p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $v_p(N)$ | $v_p(\Delta)$ | $v_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$7$ | $2$ | $III$ | additive | -1 | 2 | 3 | 0 |
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.1.5 | 7.48.0.6 |
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
$\ell$ | Reduction type | Serre weight | Serre conductor |
---|---|---|---|
$7$ | additive | $20$ | \( 1 \) |
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.
Twists
This elliptic curve is its own minimal quadratic twist.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{-7}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | 2.0.7.1-49.1-CMa1 |
$4$ | 4.2.5488.1 | \(\Z/4\Z\) | not in database |
$4$ | 4.0.1372.1 | \(\Z/2\Z \oplus \Z/4\Z\) | 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.
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 $0$.