Minimal Weierstrass equation
\(y^2+xy=x^3+x^2-7516845250x+250655549321500\)
Mordell-Weil group structure
$\Z/{2}\Z$
Torsion generators
\( \left(\frac{204635}{4}, -\frac{204635}{8}\right) \)
Integral points
None
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 308550 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 11^{2} \cdot 17$ |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | $39875031982702097652796875000 $ | = | $2^{3} \cdot 3^{3} \cdot 5^{9} \cdot 11^{18} \cdot 17 $ |
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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: | $4.4151478971942426924481478121\dots$ | ||
| 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: | $0.036074346213706753174798604758\dots$ | ||
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
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| Tamagawa product: | $ 8 $ = $ 1\cdot1\cdot2\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);
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| Special value: | $ L(E,1) $ ≈ $ 0.28859476970965402539838883806 $ | ||
Modular invariants
Modular form 308550.2.a.h
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: | 637009920 | ||
| $ \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)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{3}$ | Non-split multiplicative | 1 | 1 | 3 | 3 |
| $3$ | $1$ | $I_{3}$ | Non-split multiplicative | 1 | 1 | 3 | 3 |
| $5$ | $2$ | $I_{3}^{*}$ | Additive | 1 | 2 | 9 | 3 |
| $11$ | $4$ | $I_{12}^{*}$ | Additive | -1 | 2 | 18 | 12 |
| $17$ | $1$ | $I_{1}$ | Non-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 | 4.6.0.1 |
| $3$ | 3B | 3.4.0.1 |
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.
No Iwasawa invariant data is available for this curve.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 4, 6 and 12.
Its isogeny class 308550h
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 |
| $2$ | \(\Q(\sqrt{330}) \) | \(\Z/4\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{187}) \) | \(\Z/4\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{-55}) \) | \(\Z/6\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{187}, \sqrt{330})\) | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{-55}, \sqrt{510})\) | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{-6}, \sqrt{-55})\) | \(\Z/12\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{-55}, \sqrt{-85})\) | \(\Z/12\Z\) | Not in database |
| $6$ | 6.2.375186772125.3 | \(\Z/6\Z\) | 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 | \(\Z/8\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/2\Z \times \Z/12\Z\) | 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 | \(\Z/2\Z \times \Z/8\Z\) | 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 | \(\Z/24\Z\) | Not in database |
| $18$ | 18.0.183091994234716348674222198885513377504241943359375.1 | \(\Z/18\Z\) | 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.