Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-326209x-25271231\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-326209xz^2-25271231z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-26422956x-18501996240\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-79, 0\right) \) | $0$ | $2$ |
| \( \left(607, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-79:0:1]\) | $0$ | $2$ |
| \([607:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-714, 0\right) \) | $0$ | $2$ |
| \( \left(5460, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-527, 0\right) \), \( \left(-79, 0\right) \), \( \left(607, 0\right) \)
\([-527:0:1]\), \([-79:0:1]\), \([607:0:1]\)
\( \left(-527, 0\right) \), \( \left(-79, 0\right) \), \( \left(607, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 9408 \) | = | $2^{6} \cdot 3 \cdot 7^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $1943346986288676864$ | = | $2^{20} \cdot 3^{8} \cdot 7^{10} $ |
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| j-invariant: | $j$ | = | \( \frac{124475734657}{63011844} \) | = | $2^{-2} \cdot 3^{-8} \cdot 7^{-4} \cdot 4993^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2017770387852951255199956084$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.18910119341772050884147105449$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0649865394830864$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.432044162398092$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.21073372147730124178527317859$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{2}\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.6858697718184099342821854288 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 1.685869772 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.210734 \cdot 1.000000 \cdot 32}{4^2} \\ & \approx 1.685869772\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 147456 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{10}^{*}$ | additive | 1 | 6 | 20 | 2 |
| $3$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $7$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
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 | 8.96.0.133 | $96$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 56.192.1-56.bp.2.8, level \( 56 = 2^{3} \cdot 7 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 53 & 38 \\ 46 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 46 \\ 0 & 15 \end{array}\right),\left(\begin{array}{rr} 49 & 8 \\ 48 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 52 & 53 \end{array}\right),\left(\begin{array}{rr} 39 & 52 \\ 0 & 55 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[56])$ is a degree-$16128$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/56\Z)$.
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 | $4$ | \( 3136 = 2^{6} \cdot 7^{2} \) |
| $7$ | additive | $32$ | \( 192 = 2^{6} \cdot 3 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 9408p
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 42a3, its twist by $-56$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{14}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{-7})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{7})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{7})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.157351936.1 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.10070523904.9 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.2.341461686730752.26 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | 16.0.1622647227216566419456.13 | \(\Z/8\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | 7 |
|---|---|---|---|
| Reduction type | add | nonsplit | add |
| $\lambda$-invariant(s) | - | 0 | - |
| $\mu$-invariant(s) | - | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
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 $0$.