Minimal Weierstrass equation
\(y^2=x^3-x^2-2033x+125937\)
Mordell-Weil group structure
\(\Z^2 \times \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
| \(P\) | = | \( \left(-23, 400\right) \) | \( \left(27, 300\right) \) |
| \(\hat{h}(P)\) | ≈ | $1.4564795120956974054498000352$ | $2.0379310876681977269466850058$ |
Torsion generators
\( \left(-63, 0\right) \)
Integral points
\( \left(-63, 0\right) \), \((-23,\pm 400)\), \((1,\pm 352)\), \((27,\pm 300)\), \((187,\pm 2500)\), \((193,\pm 2624)\), \((728937,\pm 622350000)\)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 62400 \) | = | \(2^{6} \cdot 3 \cdot 5^{2} \cdot 13\) |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(-6240000000000 \) | = | \(-1 \cdot 2^{14} \cdot 3 \cdot 5^{10} \cdot 13 \) |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( -\frac{3631696}{24375} \) | = | \(-1 \cdot 2^{4} \cdot 3^{-1} \cdot 5^{-4} \cdot 13^{-1} \cdot 61^{3}\) |
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
BSD invariants
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sage: E.rank()
magma: Rank(E);
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| Analytic rank: | \(2\) | ||
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sage: E.regulator()
magma: Regulator(E);
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| Regulator: | \(2.9018522720478558373967042193\) | ||
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sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | \(0.64897213893208033119450411885\) | ||
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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: | \( 16 \) = \( 2^{2}\cdot1\cdot2^{2}\cdot1 \) | ||
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sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | \(2\) | ||
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sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Analytic order of Ш: | \(1\) (rounded) | ||
Modular invariants
Modular form 62400.2.a.f
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: | 147456 | ||
| \( \Gamma_0(N) \)-optimal: | yes | ||
| Manin constant: | 1 | ||
Special L-value
\( L^{(2)}(E,1)/2! \) ≈ \( 7.5328851034232562716760369091759045185 \)
Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(4\) | \(I_4^{*}\) | Additive | 1 | 6 | 14 | 0 |
| \(3\) | \(1\) | \(I_{1}\) | Non-split multiplicative | 1 | 1 | 1 | 1 |
| \(5\) | \(4\) | \(I_4^{*}\) | Additive | 1 | 2 | 10 | 4 |
| \(13\) | \(1\) | \(I_{1}\) | 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 X13d.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 4 & 1 \end{array}\right)$ and has index 12.
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
Note: \(p\)-adic regulator data only exists for primes \(p\ge 5\) of good ordinary reduction.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | nonsplit | add | ordinary | ordinary | split | ordinary | ss | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary |
| $\lambda$-invariant(s) | - | 2 | - | 2 | 2 | 3 | 2 | 2,2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| $\mu$-invariant(s) | - | 0 | - | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
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 and 4.
Its isogeny class 62400bi
consists of 4 curves linked by isogenies of
degrees dividing 4.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-39}) \) | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{10}) \) | \(\Z/4\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{-390}) \) | \(\Z/4\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{10}, \sqrt{-39})\) | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $4$ | 4.2.249600.3 | \(\Z/8\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/8\Z\) | Not in database |
| $8$ | 8.0.94758543360000.4 | \(\Z/2\Z \times \Z/8\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/6\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/4\Z \times \Z/4\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/8\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/16\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/12\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/12\Z\) | 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.