Minimal Weierstrass equation
\(y^2+xy+y=x^3+x^2-2160x-39540\)
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
\( \left(-\frac{109}{4}, \frac{105}{8}\right) \)
Integral points
None
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 15 \) | = | \(3 \cdot 5\) |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(405 \) | = | \(3^{4} \cdot 5 \) |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( \frac{1114544804970241}{405} \) | = | \(3^{-4} \cdot 5^{-1} \cdot 103681^{3}\) |
| 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: | \(0\) | ||
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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: | \(0.70030152116630101159009041840\) | ||
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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: | \( 2 \) = \( 2\cdot1 \) | ||
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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: | \(1\) (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: | 4 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 1 | ||
Special L-value
\( L(E,1) \) ≈ \( 0.35015076058315050579504520920239987269 \)
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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(3\) | \(2\) | \(I_{4}\) | Non-split multiplicative | 1 | 1 | 4 | 4 |
| \(5\) | \(1\) | \(I_{1}\) | 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 X225g.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^4\Z_2)$ generated by $\left(\begin{array}{rr} 5 & 5 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 5 & 5 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 5 \\ 0 & 3 \end{array}\right)$ and has index 96.
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 | 5 |
|---|---|---|---|
| Reduction type | ordinary | nonsplit | split |
| $\lambda$-invariant(s) | 0 | 0 | 1 |
| $\mu$-invariant(s) | 3 | 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, 8 and 16.
Its isogeny class 15.a
consists of 5 curves linked by isogenies of
degrees dividing 16.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{5}) \) | \(\Z/2\Z \times \Z/2\Z\) | 2.2.5.1-45.1-a9 |
| $2$ | \(\Q(\sqrt{-5}) \) | \(\Z/4\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/4\Z\) | 2.0.4.1-225.2-a10 |
| $4$ | \(\Q(i, \sqrt{5})\) | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $4$ | 4.2.2000.1 | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $4$ | 4.0.32000.1 | \(\Z/8\Z\) | Not in database |
| $4$ | \(\Q(\zeta_{8})\) | \(\Z/8\Z\) | Not in database |
| $4$ | \(\Q(i, \sqrt{10})\) | \(\Z/8\Z\) | Not in database |
| $8$ | 8.0.64000000.3 | \(\Z/4\Z \times \Z/4\Z\) | Not in database |
| $8$ | 8.0.1024000000.6 | \(\Z/2\Z \times \Z/8\Z\) | Not in database |
| $8$ | 8.0.40960000.1 | \(\Z/2\Z \times \Z/8\Z\) | Not in database |
| $8$ | 8.2.414720000000.4 | \(\Z/2\Z \times \Z/8\Z\) | Not in database |
| $8$ | 8.0.33973862400.10 | \(\Z/16\Z\) | Not in database |
| $8$ | 8.0.21233664000000.7 | \(\Z/16\Z\) | Not in database |
| $8$ | 8.0.21233664000000.8 | \(\Z/16\Z\) | Not in database |
| $8$ | 8.2.110716875.2 | \(\Z/6\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/4\Z \times \Z/8\Z\) | Not in database |
| $16$ | 16.0.16777216000000000000.3 | \(\Z/4\Z \times \Z/8\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/16\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/16\Z\) | Not in database |
| $16$ | 16.0.450868486864896000000000000.3 | \(\Z/2\Z \times \Z/16\Z\) | 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.