This is a model for the modular curve $X_0(36)$.
Minimal Weierstrass equation
\(y^2=x^3+1\)
Mordell-Weil group structure
\(\Z/{6}\Z\)
Torsion generators
\( \left(2, 3\right) \)
Integral points
\( \left(-1, 0\right) \), \((0,\pm 1)\), \((2,\pm 3)\)
Invariants
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor: | \( 36 \) | = | \(2^{2} \cdot 3^{2}\) |
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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Discriminant: | \(-432 \) | = | \(-1 \cdot 2^{4} \cdot 3^{3} \) |
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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j-invariant: | \( 0 \) | = | \(0\) |
Endomorphism ring: | \(\Z\) | ||
Geometric endomorphism ring: | \(\Z[(1+\sqrt{-3})/2]\) | (potential complex multiplication) | |
Sato-Tate group: | $N(\mathrm{U}(1))$ |
BSD invariants
sage: E.rank()
magma: Rank(E);
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Analytic rank: | \(0\) | ||
sage: E.regulator()
magma: Regulator(E);
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Regulator: | \(1\) | ||
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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Real period: | \(4.2065463159763627835250572371\) | ||
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
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Tamagawa product: | \( 6 \) = \( 3\cdot2 \) | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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Torsion order: | \(6\) | ||
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Analytic order of Ш: | \(1\) (exact) |
Modular invariants
For more coefficients, see the Downloads section to the right.
sage: E.modular_degree()
magma: ModularDegree(E);
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Modular degree: | 1 | ||
\( \Gamma_0(N) \)-optimal: | yes | ||
Manin constant: | 1 |
Special L-value
\( L(E,1) \) ≈ \( 0.70109105266272713058750953952426833642 \)
Local data
This elliptic curve is not semistable. There are 2 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
---|---|---|---|---|---|---|---|
\(2\) | \(3\) | \(IV\) | Additive | -1 | 2 | 4 | 0 |
\(3\) | \(2\) | \(III\) | Additive | 1 | 2 | 3 | 0 |
Galois representations
The mod \( p \) Galois representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois representation |
---|---|
\(2\) | B |
\(3\) | B.1.1 |
For all other primes \(p\), the image is the normalizer of a split Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=-1\).
$p$-adic data
$p$-adic regulators
All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).
Iwasawa invariants
$p$ | 2 | 3 |
---|---|---|
Reduction type | add | add |
$\lambda$-invariant(s) | - | - |
$\mu$-invariant(s) | - | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
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, 3 and 6.
Its isogeny class 36.a
consists of 4 curves linked by isogenies of
degrees dividing 6.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{-3}) \) | \(\Z/2\Z \times \Z/6\Z\) | 2.0.3.1-144.1-CMa1 |
$4$ | 4.2.1728.1 | \(\Z/12\Z\) | Not in database |
$6$ | 6.0.34992.1 | \(\Z/6\Z \times \Z/6\Z\) | Not in database |
$8$ | 8.0.2985984.1 | \(\Z/4\Z \times \Z/12\Z\) | Not in database |
$9$ | 9.3.918330048.1 | \(\Z/18\Z\) | Not in database |
$12$ | 12.0.84687918336.1 | \(\Z/2\Z \times \Z/42\Z\) | Not in database |
$16$ | 16.4.2337302235907620864.1 | \(\Z/24\Z\) | Not in database |
$18$ | 18.0.2529990231179046912.1 | \(\Z/6\Z \times \Z/18\Z\) | Not in database |
We only show fields where the torsion growth is primitive.