Minimal Weierstrass equation
\( y^2 + x y = x^{3} + x^{2} - 20930167323 x - 1883681914046823 \)
Mordell-Weil group structure
Infinite order Mordell-Weil generator and height
| \(P\) | = | \( \left(\frac{15290300560336145784416667491093032534255737588366974525873}{23430821155086049467940641448306837303371717776015089}, \frac{1835905916689334865577826566575814517941349914356412103523088227129921753252822229004288}{3586586121976703577676656544441443407675996708806265945894928043949132439802313}\right) \) |
| \(\hat{h}(P)\) | ≈ | 134.106779816 |
Torsion generators
\( \left(\frac{710347}{4}, -\frac{710347}{8}\right) \)
Integral points
Invariants
|
magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
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| Conductor: | \( 46410 \) | = | \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 17\) | ||
|
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
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| Discriminant: | \(-946022301497270062664245652323500 \) | = | \(-1 \cdot 2^{2} \cdot 3^{4} \cdot 5^{3} \cdot 7^{5} \cdot 13^{4} \cdot 17^{16} \) | ||
|
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
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| j-invariant: | \( -\frac{1014009007595272988562623757184248889}{946022301497270062664245652323500} \) | = | \(-1 \cdot 2^{-2} \cdot 3^{-4} \cdot 5^{-3} \cdot 7^{-5} \cdot 13^{-4} \cdot 17^{-16} \cdot 1004648031529^{3}\) | ||
| Endomorphism ring: | \(\Z\) | (no Complex Multiplication) | |||
| Sato-Tate Group: | $\mathrm{SU}(2)$ | ||||
BSD invariants
|
magma: Rank(E);
sage: E.rank()
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| Rank: | \(1\) | ||
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magma: Regulator(E);
sage: E.regulator()
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| Regulator: | \(134.106779816\) | ||
|
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
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| Real period: | \(0.00604403498818\) | ||
|
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: | \( 16 \) = \( 2\cdot2\cdot1\cdot1\cdot2\cdot2 \) | ||
|
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
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| Torsion order: | \(2\) | ||
|
magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
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| Analytic order of Ш: | \(1\) (exact) | ||
Modular invariants
Modular form 46410.2.a.d
|
magma: ModularDegree(E);
sage: E.modular_degree()
|
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| Modular degree: | 253624320 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 1 | ||
Special L-value
\( L'(E,1) \) ≈ \( 3.24218427744 \)
Local data
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(2\) | \( I_{2} \) | Non-split multiplicative | 1 | 1 | 2 | 2 |
| \(3\) | \(2\) | \( I_{4} \) | Non-split multiplicative | 1 | 1 | 4 | 4 |
| \(5\) | \(1\) | \( I_{3} \) | Non-split multiplicative | 1 | 1 | 3 | 3 |
| \(7\) | \(1\) | \( I_{5} \) | Non-split multiplicative | 1 | 1 | 5 | 5 |
| \(13\) | \(2\) | \( I_{4} \) | Non-split multiplicative | 1 | 1 | 4 | 4 |
| \(17\) | \(2\) | \( I_{16} \) | Non-split multiplicative | 1 | 1 | 16 | 16 |
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 X13.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right)$ and has index 6.
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
\(p\)-adic regulators are not yet computed for curves that are not \(\Gamma_0\)-optimal.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | nonsplit | nonsplit | nonsplit | ordinary | nonsplit | nonsplit | ss | ordinary | ordinary | ordinary | ordinary | ordinary | ss | ordinary |
| $\lambda$-invariant(s) | 9 | 3 | 1 | 1 | 1 | 1 | 1 | 3,1 | 1 | 1 | 3 | 1 | 1 | 1,1 | 1 |
| $\mu$-invariant(s) | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 |
Isogenies
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class 46410a
consists of 4 curves linked by isogenies of
degrees dividing 4.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base-change curve |
|---|---|---|---|
| 2 | \(\Q(\sqrt{-35}) \) | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
| \(\Q(\sqrt{7}) \) | \(\Z/4\Z\) | Not in database | |
| \(\Q(\sqrt{-5}) \) | \(\Z/4\Z\) | Not in database | |
| 4 | \(\Q(\sqrt{-5}, \sqrt{7})\) | \(\Z/2\Z \times \Z/4\Z\) | 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.