Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-424x+3140\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-424xz^2+3140z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-34371x+2392146\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(14, 12\right) \) | $0.61683986429579455126410563298$ | $\infty$ |
| \( \left(8, 18\right) \) | $0.72352496198785813676582755115$ | $\infty$ |
| \( \left(10, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([14:12:1]\) | $0.61683986429579455126410563298$ | $\infty$ |
| \([8:18:1]\) | $0.72352496198785813676582755115$ | $\infty$ |
| \([10:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(129, 324\right) \) | $0.61683986429579455126410563298$ | $\infty$ |
| \( \left(75, 486\right) \) | $0.72352496198785813676582755115$ | $\infty$ |
| \( \left(93, 0\right) \) | $0$ | $2$ |
Integral points
\((-22,\pm 48)\), \((-16,\pm 78)\), \((-8,\pm 78)\), \((2,\pm 48)\), \((8,\pm 18)\), \( \left(10, 0\right) \), \((14,\pm 12)\), \((19,\pm 48)\), \((23,\pm 78)\), \((35,\pm 180)\), \((62,\pm 468)\), \((110,\pm 1140)\), \((478,\pm 10452)\)
\([-22:\pm 48:1]\), \([-16:\pm 78:1]\), \([-8:\pm 78:1]\), \([2:\pm 48:1]\), \([8:\pm 18:1]\), \([10:0:1]\), \([14:\pm 12:1]\), \([19:\pm 48:1]\), \([23:\pm 78:1]\), \([35:\pm 180:1]\), \([62:\pm 468:1]\), \([110:\pm 1140:1]\), \([478:\pm 10452:1]\)
\((-22,\pm 48)\), \((-16,\pm 78)\), \((-8,\pm 78)\), \((2,\pm 48)\), \((8,\pm 18)\), \( \left(10, 0\right) \), \((14,\pm 12)\), \((19,\pm 48)\), \((23,\pm 78)\), \((35,\pm 180)\), \((62,\pm 468)\), \((110,\pm 1140)\), \((478,\pm 10452)\)
Invariants
| Conductor: | $N$ | = | \( 10608 \) | = | $2^{4} \cdot 3 \cdot 13 \cdot 17$ |
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| Minimal Discriminant: | $\Delta$ | = | $238298112$ | = | $2^{10} \cdot 3^{4} \cdot 13^{2} \cdot 17 $ |
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| j-invariant: | $j$ | = | \( \frac{8251733668}{232713} \) | = | $2^{2} \cdot 3^{-4} \cdot 13^{-2} \cdot 17^{-1} \cdot 19^{3} \cdot 67^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.38623977422520258818906994263$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.19138287624141850299195682525$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8566015986690725$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.2111331355453414$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.41902423558957649829601988062$ |
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| Real period: | $\Omega$ | ≈ | $1.7532960946835002159244561253$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{2}\cdot2^{2}\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $5.8773884466955473366373201762 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.877388447 \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 1.753296 \cdot 0.419024 \cdot 32}{2^2} \\ & \approx 5.877388447\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 7168 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{2}^{*}$ | additive | 1 | 4 | 10 | 0 |
| $3$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $13$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $17$ | $1$ | $I_{1}$ | nonsplit 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 2.3.0.1 | $3$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1768 = 2^{3} \cdot 13 \cdot 17 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 210 & 1 \\ 1663 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1765 & 4 \\ 1764 & 5 \end{array}\right),\left(\begin{array}{rr} 225 & 1548 \\ 1104 & 663 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 1497 & 4 \\ 1226 & 9 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 885 & 4 \\ 2 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[1768])$ is a degree-$262787825664$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1768\Z)$.
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 | $2$ | \( 17 \) |
| $3$ | split multiplicative | $4$ | \( 3536 = 2^{4} \cdot 13 \cdot 17 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 816 = 2^{4} \cdot 3 \cdot 17 \) |
| $17$ | nonsplit multiplicative | $18$ | \( 624 = 2^{4} \cdot 3 \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 10608f
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 5304g1, its twist by $-4$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{17}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.735488.3 | \(\Z/4\Z\) | not in database |
| $8$ | 8.4.1044287785216.12 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.156332410863616.16 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | split | ord | ord | ord | nonsplit | nonsplit | ord | ord | ord | ss | ord | ss | ord | ord |
| $\lambda$-invariant(s) | - | 3 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.