Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-16743266x-11929605948\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-16743266xz^2-11929605948z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-21699272763x-556522597291626\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{7}\Z\)
Torsion generators
\( \left(4744, 121774\right) \)
Integral points
\( \left(4744, 121774\right) \), \( \left(4744, -126518\right) \), \( \left(8506, 674788\right) \), \( \left(8506, -683294\right) \), \( \left(32332, 5749726\right) \), \( \left(32332, -5782058\right) \)
Invariants
Conductor: | \( 61446 \) | = | $2 \cdot 3 \cdot 7^{2} \cdot 11 \cdot 19$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $238934485136491906962816 $ | = | $2^{7} \cdot 3^{7} \cdot 7^{2} \cdot 11^{7} \cdot 19^{7} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{10593712059133697959507441}{4876213982377385856384} \) | = | $2^{-7} \cdot 3^{-7} \cdot 7 \cdot 11^{-7} \cdot 19^{-7} \cdot 61^{3} \cdot 1882147^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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Endomorphism ring: | $\Z$ | |||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
| |
Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $3.1806336398583684034460518405\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $2.8563152816824828525951597166\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
BSD invariants
Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.077986963295661760447658101830\dots$ | comment: Real Period
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);
|
Tamagawa product: | $ 2401 $ = $ 7\cdot7\cdot1\cdot7\cdot7 $ | comment: Tamagawa numbers
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
oscar: tamagawa_numbers(E)
|
Torsion order: | $7$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
|
Analytic order of Ш: | $1$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 3.8213612014874262619352469897 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
|
BSD formula
$\displaystyle 3.821361201 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.077987 \cdot 1.000000 \cdot 2401}{7^2} \approx 3.821361201$
Modular invariants
Modular form 61446.2.a.cv
For more coefficients, see the Downloads section to the right.
Modular degree: | 6750240 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $7$ | $I_{7}$ | Split multiplicative | -1 | 1 | 7 | 7 |
$3$ | $7$ | $I_{7}$ | Split multiplicative | -1 | 1 | 7 | 7 |
$7$ | $1$ | $II$ | Additive | -1 | 2 | 2 | 0 |
$11$ | $7$ | $I_{7}$ | Split multiplicative | -1 | 1 | 7 | 7 |
$19$ | $7$ | $I_{7}$ | Split multiplicative | -1 | 1 | 7 | 7 |
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 |
---|---|---|
$7$ | 7B.1.1 | 7.48.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 35112 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \cdot 19 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 7393 & 14 \\ 16639 & 99 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 8779 & 17570 \\ 0 & 28843 \end{array}\right),\left(\begin{array}{rr} 35099 & 14 \\ 35098 & 15 \end{array}\right),\left(\begin{array}{rr} 26335 & 14 \\ 8785 & 99 \end{array}\right),\left(\begin{array}{rr} 11705 & 14 \\ 11711 & 99 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 17557 & 14 \\ 17563 & 99 \end{array}\right),\left(\begin{array}{rr} 19153 & 14 \\ 28735 & 99 \end{array}\right)$.
The torsion field $K:=\Q(E[35112])$ is a degree-$2516252884992000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/35112\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 61446.cv
consists of 2 curves linked by isogenies of
degree 7.
Twists
This elliptic curve is its own minimal quadratic twist.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{7}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$3$ | 3.3.245784.1 | \(\Z/14\Z\) | Not in database |
$6$ | 6.6.303015429674496.1 | \(\Z/2\Z \oplus \Z/14\Z\) | Not in database |
$8$ | deg 8 | \(\Z/21\Z\) | Not in database |
$12$ | deg 12 | \(\Z/28\Z\) | 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.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | split | split | ord | add | split | ss | ord | split | ord | ord | ord | ord | ss | ord | ord |
$\lambda$-invariant(s) | 1 | 1 | 0 | - | 1 | 0,0 | 0 | 1 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 |
$\mu$-invariant(s) | 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.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.