Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-4385x+94815\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-4385xz^2+94815z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-355212x+70185744\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{4}\Z\)
Torsion generators
\( \left(103, 864\right) \)
Integral points
\( \left(49, 0\right) \), \((103,\pm 864)\)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 960 \) | = | $2^{6} \cdot 3 \cdot 5$ |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | $1393140695040 $ | = | $2^{19} \cdot 3^{12} \cdot 5 $ |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( \frac{35578826569}{5314410} \) | = | $2^{-1} \cdot 3^{-12} \cdot 5^{-1} \cdot 11^{3} \cdot 13^{3} \cdot 23^{3}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $1.0545806117222147987174678853\dots$ | ||
| Stable Faltings height: | $0.014859840882296834591619703113\dots$ | ||
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: | $0.81910691718318947792851914759\dots$ | ||
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: | $ 48 $ = $ 2^{2}\cdot( 2^{2} \cdot 3 )\cdot1 $ | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | $4$ | ||
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) $ ≈ $ 2.4573207515495684337855574428 $ | ||
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: | 1536 | ||
| $ \Gamma_0(N) $-optimal: | no | ||
| Manin constant: | 1 | ||
Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{9}^{*}$ | Additive | -1 | 6 | 19 | 1 |
| $3$ | $12$ | $I_{12}$ | Split multiplicative | -1 | 1 | 12 | 12 |
| $5$ | $1$ | $I_{1}$ | 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 | 4.12.0.7 |
| $3$ | 3B | 3.4.0.1 |
The image of the adelic Galois representation has level $120$, index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 89 & 96 \\ 90 & 119 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 25 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 97 & 24 \\ 96 & 25 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 14 & 11 \end{array}\right),\left(\begin{array}{rr} 61 & 24 \\ 92 & 29 \end{array}\right),\left(\begin{array}{rr} 78 & 41 \\ 43 & 28 \end{array}\right),\left(\begin{array}{rr} 112 & 3 \\ 117 & 34 \end{array}\right)$
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.
Iwasawa invariants
| $p$ | 2 | 3 | 5 |
|---|---|---|---|
| Reduction type | add | split | split |
| $\lambda$-invariant(s) | - | 3 | 1 |
| $\mu$-invariant(s) | - | 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 960o
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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{10}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/12\Z\) | 2.0.8.1-450.2-a4 |
| $4$ | 4.4.92160.2 | \(\Z/8\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{-5})\) | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
| $6$ | 6.2.34560000.1 | \(\Z/12\Z\) | Not in database |
| $8$ | 8.0.262144000000.9 | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
| $8$ | 8.8.849346560000.5 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
| $8$ | 8.0.5184000000.12 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
| $8$ | 8.0.8493465600.20 | \(\Z/24\Z\) | Not in database |
| $12$ | 12.0.1194393600000000.1 | \(\Z/3\Z \oplus \Z/12\Z\) | Not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
| $12$ | deg 12 | \(\Z/24\Z\) | Not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | Not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/12\Z\) | 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 |
| $18$ | 18.0.172713999781822935859200000000.1 | \(\Z/36\Z\) | 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.