Minimal Weierstrass equation
magma: E := EllipticCurve("21a4");
sage: E = EllipticCurve("21a4")
gp: E = ellinit("21a4")
\( y^2 + x y = x^{3} + x \)
Mordell-Weil group structure
Torsion generators
\( \left(1, 1\right) \)
Integral points
\( \left(0, 0\right) \), \( \left(1, 1\right) \)
Invariants
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magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
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| Conductor: | \( 21 \) | = | \(3 \cdot 7\) | ||
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magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
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| Discriminant: | \(-63 \) | = | \(-1 \cdot 3^{2} \cdot 7 \) | ||
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magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
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| j-invariant: | \( \frac{103823}{63} \) | = | \(3^{-2} \cdot 7^{-1} \cdot 47^{3}\) | ||
| Endomorphism ring: | \(\Z\) | (no Complex Multiplication) | |||
| Sato-Tate Group: | $\mathrm{SU}(2)$ | ||||
BSD invariants
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magma: Rank(E);
sage: E.rank()
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| Rank: | \(0\) | ||
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magma: Regulator(E);
sage: E.regulator()
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| Regulator: | \(1\) | ||
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magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
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| Real period: | \(3.60892324311\) | ||
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magma: TamagawaNumbers(E);
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
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| Tamagawa product: | \( 2 \) = \( 2\cdot1 \) | ||
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magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
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| Torsion order: | \(4\) | ||
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magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
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| Analytic order of Ш: | \(1\) (exact) | ||
Modular invariants
Modular form 21.2.a.a
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
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magma: ModularDegree(E);
sage: E.modular_degree()
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| Modular degree: | 2 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 2 | ||
Special L-value
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
gp: ar[2]/factorial(ar[1])
\( L(E,1) \) ≈ \( 0.451115405388 \)
Local data
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(3\) | \(2\) | \( I_{2} \) | Split multiplicative | -1 | 1 | 2 | 2 |
| \(7\) | \(1\) | \( I_{1} \) | Non-split multiplicative | 1 | 1 | 1 | 1 |
Galois representations
The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X120d.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^4\Z_2)$ generated by $\left(\begin{array}{rr} 7 & 7 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 8 & 5 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 0 & 1 \end{array}\right)$ and has index 48.
sage: [rho.image_type(p) for p in rho.non_surjective()]
The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.
| prime | Image of Galois representation |
|---|---|
| \(2\) | B |
$p$-adic data
$p$-adic regulators
All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).
Iwasawa invariants
| $p$ | 2 | 3 | 7 |
|---|---|---|---|
| Reduction type | ordinary | split | nonsplit |
| $\lambda$-invariant(s) | 1 | 1 | 0 |
| $\mu$-invariant(s) | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
Isogenies
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class 21.a
consists of 6 curves linked by isogenies of
degrees dividing 8.
Growth of torsion in number fields
The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base-change curve |
|---|---|---|---|
| 2 | \(\Q(\sqrt{-7}) \) | \(\Z/2\Z \times \Z/4\Z\) | 2.0.7.1-63.1-a2 |
| \(\Q(\sqrt{21}) \) | \(\Z/8\Z\) | 2.2.21.1-21.1-b2 | |
| \(\Q(\sqrt{-3}) \) | \(\Z/8\Z\) | 2.0.3.1-147.2-a4 | |
| 4 | 4.0.189.1 | \(\Z/16\Z\) | Not in database |
| \(\Q(\sqrt{-3}, \sqrt{-7})\) | \(\Z/2\Z \times \Z/8\Z\) | Not in database |
We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.