Minimal Weierstrass equation
\(y^2+xy=x^3-300673810x+2005244394572\)
Mordell-Weil group structure
$\Z/{2}\Z$
Torsion generators
\( \left(\frac{40927}{4}, -\frac{40927}{8}\right) \)
Integral points
None
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 61710 \) | = | $2 \cdot 3 \cdot 5 \cdot 11^{2} \cdot 17$ |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | $2552002046892934249779000 $ | = | $2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 11^{18} \cdot 17 $ |
|
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
|
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| j-invariant: | \( \frac{1696892787277117093383481}{1440538624914939000} \) | = | $2^{-3} \cdot 3^{-3} \cdot 5^{-3} \cdot 11^{-12} \cdot 17^{-1} \cdot 181^{3} \cdot 227^{3} \cdot 2903^{3}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $3.6104289409771925051477681455\dots$ | ||
| Stable Faltings height: | $2.4114813045780072331167963565\dots$ | ||
BSD invariants
sage: E.rank()
magma: Rank(E);
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| Analytic rank: | $0$ | ||
sage: E.regulator()
magma: Regulator(E);
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| Regulator: | $1$ | ||
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | $0.080664690377710455785118165313\dots$ | ||
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
| |||
| Tamagawa product: | $ 108 $ = $ 3\cdot3\cdot3\cdot2^{2}\cdot1 $ | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | $2$ | ||
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Analytic order of Ш: | $4$ = $2^2$ (exact) | ||
sage: r = E.rank();
gp: ar = ellanalyticrank(E);
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
| |||
| Special value: | $ L(E,1) $ ≈ $ 8.7117865607927292247927618538 $ | ||
Modular invariants
Modular form 61710.2.a.da
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: | 26542080 | ||
| $ \Gamma_0(N) $-optimal: | no | ||
| Manin constant: | 1 | ||
Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
| $3$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
| $5$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
| $11$ | $4$ | $I_{12}^{*}$ | Additive | -1 | 2 | 18 | 12 |
| $17$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
| prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
|---|---|---|
| $2$ | 2B | 8.12.0.6 |
| $3$ | 3B | 3.4.0.1 |
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 11 | 17 |
|---|---|---|---|---|---|
| Reduction type | split | split | split | add | split |
| $\lambda$-invariant(s) | 6 | 7 | 1 | - | 1 |
| $\mu$-invariant(s) | 0 | 0 | 0 | - | 0 |
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, 4, 6 and 12.
Its isogeny class 61710cz
consists of 8 curves linked by isogenies of
degrees dividing 12.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{510}) \) | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{66}) \) | \(\Z/4\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{935}) \) | \(\Z/4\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{-11}) \) | \(\Z/6\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{66}, \sqrt{510})\) | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $4$ | 4.4.568719360.6 | \(\Z/8\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{-11}, \sqrt{510})\) | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{-6}, \sqrt{-11})\) | \(\Z/12\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{-11}, \sqrt{-85})\) | \(\Z/12\Z\) | Not in database |
| $6$ | 6.2.3001494177.2 | \(\Z/6\Z\) | 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 |
| $8$ | Deg 8 | \(\Z/2\Z \times \Z/8\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/2\Z \times \Z/12\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/24\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/3\Z \times \Z/6\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/12\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/12\Z\) | 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 | \(\Z/2\Z \times \Z/12\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/24\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/24\Z\) | Not in database |
| $18$ | 18.0.1464735953877730789393777591084107020033935546875.3 | \(\Z/18\Z\) | 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.