Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-5504x+64464\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-5504xz^2+64464z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-445851x+45656730\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-40, 468\right) \) | $1.7124631758690493029962599641$ | $\infty$ |
| \( \left(12, 0\right) \) | $0$ | $2$ |
| \( \left(68, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-40:468:1]\) | $1.7124631758690493029962599641$ | $\infty$ |
| \([12:0:1]\) | $0$ | $2$ |
| \([68:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-363, 12636\right) \) | $1.7124631758690493029962599641$ | $\infty$ |
| \( \left(105, 0\right) \) | $0$ | $2$ |
| \( \left(609, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-79, 0\right) \), \((-40,\pm 468)\), \( \left(12, 0\right) \), \( \left(68, 0\right) \), \((110,\pm 882)\), \((264,\pm 4116)\)
\([-79:0:1]\), \([-40:\pm 468:1]\), \([12:0:1]\), \([68:0:1]\), \([110:\pm 882:1]\), \([264:\pm 4116:1]\)
\( \left(-79, 0\right) \), \((-40,\pm 468)\), \( \left(12, 0\right) \), \( \left(68, 0\right) \), \((110,\pm 882)\), \((264,\pm 4116)\)
Invariants
| Conductor: | $N$ | = | \( 30576 \) | = | $2^{4} \cdot 3 \cdot 7^{2} \cdot 13$ |
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| Minimal Discriminant: | $\Delta$ | = | $8978700616704$ | = | $2^{10} \cdot 3^{2} \cdot 7^{8} \cdot 13^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{153091012}{74529} \) | = | $2^{2} \cdot 3^{-2} \cdot 7^{-2} \cdot 13^{-2} \cdot 337^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.1789316444799729807344559606$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.37164608051430476299924717900$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8759622866765943$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.6264119120385807$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.7124631758690493029962599641$ |
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| Real period: | $\Omega$ | ≈ | $0.65016133758252873040352673118$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2^{2}\cdot2\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.4535093959353849258270203545 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.453509396 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.650161 \cdot 1.712463 \cdot 64}{4^2} \\ & \approx 4.453509396\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 61440 |
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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 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 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $7$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $13$ | $2$ | $I_{2}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2Cs | 2.6.0.1 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2184 = 2^{3} \cdot 3 \cdot 7 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 935 & 2182 \\ 0 & 2183 \end{array}\right),\left(\begin{array}{rr} 1683 & 2 \\ 334 & 2183 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2181 & 4 \\ 2180 & 5 \end{array}\right),\left(\begin{array}{rr} 1635 & 2180 \\ 1090 & 2181 \end{array}\right),\left(\begin{array}{rr} 1457 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1093 & 4 \\ 2 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[2184])$ is a degree-$81155063808$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2184\Z)$.
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 | $2$ | \( 49 = 7^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 10192 = 2^{4} \cdot 7^{2} \cdot 13 \) |
| $7$ | additive | $32$ | \( 624 = 2^{4} \cdot 3 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 30576k
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 2184b2, its twist by $28$.
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 |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{-14}, \sqrt{-26})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{14})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{-91})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | 16.0.132513778481397717569346994176.4 | \(\Z/4\Z \oplus \Z/4\Z\) | 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.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | nonsplit | ord | add | ord | split | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 1 | 3 | - | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | 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.