Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-2723763x+1730223026\)
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(homogenize, simplify) |
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\(y^2z=x^3-2723763xz^2+1730223026z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2723763x+1730223026\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(953, 8\right) \) | $1.2649732611785834700985275286$ | $\infty$ |
| \( \left(\frac{8575}{9}, \frac{98}{27}\right) \) | $1.4347519354102544938004204892$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([953:8:1]\) | $1.2649732611785834700985275286$ | $\infty$ |
| \([25725:98:27]\) | $1.4347519354102544938004204892$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(953, 8\right) \) | $1.2649732611785834700985275286$ | $\infty$ |
| \( \left(\frac{8575}{9}, \frac{98}{27}\right) \) | $1.4347519354102544938004204892$ | $\infty$ |
Integral points
\((-1519,\pm 48608)\), \((833,\pm 6272)\), \((953,\pm 8)\), \((1010,\pm 3086)\)
\([-1519:\pm 48608:1]\), \([833:\pm 6272:1]\), \([953:\pm 8:1]\), \([1010:\pm 3086:1]\)
\((-1519,\pm 48608)\), \((833,\pm 6272)\), \((953,\pm 8)\), \((1010,\pm 3086)\)
Invariants
| Conductor: | $N$ | = | \( 63504 \) | = | $2^{4} \cdot 3^{4} \cdot 7^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $1912622616576$ | = | $2^{12} \cdot 3^{4} \cdot 7^{8} $ |
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| j-invariant: | $j$ | = | \( 1168429123449 \) | = | $3^{2} \cdot 7 \cdot 2647^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $2.0909645454184311014796529912$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.26566016406775964180622937153$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0481648553214244$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.069798071335814$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.4822302159195387148837301257$ |
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| Real period: | $\Omega$ | ≈ | $0.51971354744197303882832744572$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 2^{2}\cdot1\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $9.2440214836947014986700584115 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 9.244021484 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.519714 \cdot 1.482230 \cdot 12}{1^2} \\ & \approx 9.244021484\end{aligned}$$
Modular invariants
Modular form 63504.2.a.v
For more coefficients, see the Downloads section to the right.
| Modular degree: | 564480 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{4}^{*}$ | additive | -1 | 4 | 12 | 0 |
| $3$ | $1$ | $II$ | additive | 1 | 4 | 4 | 0 |
| $7$ | $3$ | $IV^{*}$ | additive | 1 | 2 | 8 | 0 |
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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2Cn | 4.4.0.2 | $4$ |
| $7$ | 7B | 7.8.0.1 | $8$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \), index $576$, genus $16$, and generators
$\left(\begin{array}{rr} 225 & 28 \\ 224 & 29 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 14 & 29 \end{array}\right),\left(\begin{array}{rr} 15 & 14 \\ 28 & 43 \end{array}\right),\left(\begin{array}{rr} 125 & 224 \\ 0 & 251 \end{array}\right),\left(\begin{array}{rr} 1 & 28 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 116 & 1 \\ 119 & 23 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 28 & 1 \end{array}\right),\left(\begin{array}{rr} 118 & 243 \\ 63 & 109 \end{array}\right)$.
The torsion field $K:=\Q(E[252])$ is a degree-$1306368$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/252\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | additive | $2$ | \( 3969 = 3^{4} \cdot 7^{2} \) |
| $3$ | additive | $8$ | \( 784 = 2^{4} \cdot 7^{2} \) |
| $7$ | additive | $26$ | \( 1296 = 2^{4} \cdot 3^{4} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 63504y
consists of 2 curves linked by isogenies of
degree 7.
Twists
The minimal quadratic twist of this elliptic curve is 3969f2, its twist by $-84$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.3.3969.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.6.144027072.1 | \(\Z/7\Z\) | not in database |
| $8$ | 8.2.108884466432.7 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $18$ | 18.18.351496200956998572502045949952.1 | \(\Z/2\Z \oplus \Z/14\Z\) | 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 | add | add | ord | add | ord | ord | ord | ord | ord | ord | ss | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | - | 2 | - | 2 | 2 | 2 | 2 | 4 | 2 | 2,2 | 2 | 2 | 2 | 2,2 |
| $\mu$-invariant(s) | - | - | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.