This is the minimal-conductor elliptic curve over $\Q$ with torsion subgroup $\Z/2\Z \times \Z/6\Z$.
Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-19x+26\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-19xz^2+26z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-24003x+1296702\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{6}\Z\)
Torsion generators
\( \left(3, -2\right) \), \( \left(1, 2\right) \)
Integral points
\( \left(-5, 2\right) \), \( \left(-2, 8\right) \), \( \left(-2, -7\right) \), \( \left(1, 2\right) \), \( \left(1, -4\right) \), \( \left(3, -2\right) \), \( \left(4, 2\right) \), \( \left(4, -7\right) \), \( \left(13, 38\right) \), \( \left(13, -52\right) \)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 30 \) | = | $2 \cdot 3 \cdot 5$ |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | $72900 $ | = | $2^{2} \cdot 3^{6} \cdot 5^{2} $ |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( \frac{702595369}{72900} \) | = | $2^{-2} \cdot 3^{-6} \cdot 5^{-2} \cdot 7^{3} \cdot 127^{3}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $-0.33171374939767582011699635763\dots$ | ||
| Stable Faltings height: | $-0.33171374939767582011699635763\dots$ | ||
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: if(E.disc>0,2,1)*E.omega[1]
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
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| Real period: | $3.3519482592414964494482281339\dots$ | ||
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: | $ 24 $ = $ 2\cdot( 2 \cdot 3 )\cdot2 $ | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | $12$ | ||
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Analytic order of Ш: | $1$ (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) $ ≈ $ 0.55865804320691607490803802232 $ | ||
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) $-optimal: | no | ||
| Manin constant: | 1 | ||
Local data
This elliptic curve is semistable. There are 3 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
| $3$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
| $5$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
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 |
|---|---|---|
| $2$ | 2Cs | 8.12.0.1 |
| $3$ | 3B.1.1 | 3.8.0.1 |
The image of the adelic Galois representation has level $120$, index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 31 & 12 \\ 66 & 73 \end{array}\right),\left(\begin{array}{rr} 67 & 6 \\ 114 & 115 \end{array}\right),\left(\begin{array}{rr} 109 & 12 \\ 108 & 13 \end{array}\right),\left(\begin{array}{rr} 97 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 9 & 4 \\ 104 & 113 \end{array}\right),\left(\begin{array}{rr} 81 & 4 \\ 22 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right)$
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.
Iwasawa invariants
| $p$ | 2 | 3 | 5 |
|---|---|---|---|
| Reduction type | nonsplit | split | nonsplit |
| $\lambda$-invariant(s) | 0 | 1 | 0 |
| $\mu$-invariant(s) | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 30a
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{6}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{3}, \sqrt{-5})\) | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{-3})\) | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{5})\) | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
| $6$ | 6.0.270000.1 | \(\Z/6\Z \oplus \Z/6\Z\) | Not in database |
| $9$ | 9.3.143489070000.1 | \(\Z/2\Z \oplus \Z/18\Z\) | Not in database |
| $12$ | 12.0.1194393600000000.1 | \(\Z/6\Z \oplus \Z/12\Z\) | Not in database |
| $16$ | 16.0.11007531417600000000.1 | \(\Z/4\Z \oplus \Z/12\Z\) | Not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | Not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | Not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | 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.
Additional information
This is the curve of minimal conductor and torsion $(\Z/2\Z) \times (\Z/6\Z)$. Every elliptic curve $E/\Q$ with this torsion group must have conductor divisible by $30$ (for instance, if $E$ had good reduction at $5$ then the reduction mod $5$ would have at least $12$ points, which exceeds the Weil bound $(\sqrt5+1)^2 < 11$.