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
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 36 \) | = | \(2^{2} \cdot 3^{2}\) |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(-432 \) | = | \(-1 \cdot 2^{4} \cdot 3^{3} \) |
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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
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sage: E.rank()
magma: Rank(E);
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| Analytic rank: | \(0\) | ||
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sage: E.regulator()
magma: Regulator(E);
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| Regulator: | \(1\) | ||
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sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | \(4.2065463159763627835250572371\) | ||
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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: | \( 6 \) = \( 3\cdot2 \) | ||
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sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | \(6\) | ||
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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.
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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 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.