Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-19x+685\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-19xz^2+685z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-25299x+32335470\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
\( \left(-10, 5\right) \)
Integral points
\( \left(-10, 5\right) \)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 78 \) | = | $2 \cdot 3 \cdot 13$ |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | $-207028224 $ | = | $-1 \cdot 2^{16} \cdot 3^{5} \cdot 13 $ |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( -\frac{822656953}{207028224} \) | = | $-1 \cdot 2^{-16} \cdot 3^{-5} \cdot 13^{-1} \cdot 937^{3}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $0.27493047175759567185621147284\dots$ | ||
| Stable Faltings height: | $0.27493047175759567185621147284\dots$ | ||
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: | $1.4504359261765928469744084519\dots$ | ||
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: | $ 2 $ = $ 2\cdot1\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: | $1$ (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) $ ≈ $ 0.72521796308829642348720422595 $ | ||
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: | 40 | ||
| $ \Gamma_0(N) $-optimal: | yes | ||
| Manin constant: | 1 | ||
Local data
This elliptic curve is semistable. There are 3 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{16}$ | Non-split multiplicative | 1 | 1 | 16 | 16 |
| $3$ | $1$ | $I_{5}$ | Non-split multiplicative | 1 | 1 | 5 | 5 |
| $13$ | $1$ | $I_{1}$ | 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 |
The image of the adelic Galois representation has level $312$, index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 7 & 6 \\ 306 & 307 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 121 & 120 \\ 286 & 127 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 43 & 42 \\ 130 & 283 \end{array}\right),\left(\begin{array}{rr} 112 & 3 \\ 109 & 2 \end{array}\right),\left(\begin{array}{rr} 305 & 8 \\ 304 & 9 \end{array}\right),\left(\begin{array}{rr} 268 & 1 \\ 215 & 6 \end{array}\right)$
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.
Iwasawa invariants
| $p$ | 2 | 3 | 13 |
|---|---|---|---|
| Reduction type | nonsplit | nonsplit | split |
| $\lambda$-invariant(s) | 0 | 0 | 1 |
| $\mu$-invariant(s) | 0 | 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 and 4.
Its isogeny class 78.a
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-39}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{13}) \) | \(\Z/4\Z\) | 2.2.13.1-468.1-e1 |
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/4\Z\) | 2.0.3.1-2028.2-a1 |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{13})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
| $4$ | 4.2.6591.1 | \(\Z/8\Z\) | Not in database |
| $8$ | 8.0.900798402816.14 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
| $8$ | 8.0.31539456.4 | \(\Z/8\Z\) | Not in database |
| $8$ | 8.0.390971529.2 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
| $8$ | 8.2.80951927472.1 | \(\Z/6\Z\) | Not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
| $16$ | 16.0.28410691590485508096.3 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | Not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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.