Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-66x-216\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-66xz^2-216z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-5373x-141372\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(21, 90\right)\)
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| $\hat{h}(P)$ | ≈ | $2.7077321322842255603795325494$ |
Torsion generators
\( \left(-4, 0\right) \), \( \left(9, 0\right) \)
Integral points
\( \left(-6, 0\right) \), \( \left(-4, 0\right) \), \( \left(9, 0\right) \), \((21,\pm 90)\)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 6240 \) | = | $2^{5} \cdot 3 \cdot 5 \cdot 13$ |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | $2433600 $ | = | $2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2} $ |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( \frac{504358336}{38025} \) | = | $2^{6} \cdot 3^{-2} \cdot 5^{-2} \cdot 13^{-2} \cdot 199^{3}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $-0.029371143300508013491975127215\dots$ | ||
| Stable Faltings height: | $-0.37594473358048066820059118794\dots$ | ||
BSD invariants
sage: E.rank()
magma: Rank(E);
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| Analytic rank: | $1$ | ||
sage: E.regulator()
magma: Regulator(E);
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| Regulator: | $2.7077321322842255603795325494\dots$ | ||
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: | $1.6808628335731643632398454780\dots$ | ||
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: | $ 16 $ = $ 2\cdot2\cdot2\cdot2 $ | ||
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) $ ≈ $ 4.5513263044283697000553108219 $ | ||
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: | 768 | ||
| $ \Gamma_0(N) $-optimal: | yes | ||
| Manin constant: | 1 | ||
Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III$ | Additive | -1 | 5 | 6 | 0 |
| $3$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
| $5$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
| $13$ | $2$ | $I_{2}$ | 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 |
The image of the adelic Galois representation has level $1560$, index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 937 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 521 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1081 & 4 \\ 602 & 9 \end{array}\right),\left(\begin{array}{rr} 783 & 2 \\ 1558 & 1559 \end{array}\right),\left(\begin{array}{rr} 1173 & 2 \\ 778 & 1559 \end{array}\right),\left(\begin{array}{rr} 1557 & 4 \\ 1556 & 5 \end{array}\right)$
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | split | nonsplit | ss | ss | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | 2 | 1 | 1,1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 3 | 1 | 3 | 1 | 1,1 |
| $\mu$-invariant(s) | - | 0 | 0 | 0,0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 |
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.
Its isogeny class 6240ba
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{-2}, \sqrt{-15})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{13}, \sqrt{15})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{-13})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
| $16$ | 16.0.8979181539709000089600000000.18 | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | 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.