Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-13430667x-24063029746\)
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(homogenize, simplify) |
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\(y^2z=x^3-13430667xz^2-24063029746z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-13430667x-24063029746\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(30107224622185/4899160036, 121472831365829035677/342911807559784)$ | $25.971923317950300653629151261$ | $\infty$ |
| $(10945, 1067742)$ | $0$ | $4$ |
Integral points
\( \left(4354, 0\right) \), \((10945,\pm 1067742)\)
Invariants
| Conductor: | $N$ | = | \( 18720 \) | = | $2^{5} \cdot 3^{2} \cdot 5 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-95090362794826944929280$ | = | $-1 \cdot 2^{9} \cdot 3^{13} \cdot 5 \cdot 13^{12} $ |
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| j-invariant: | $j$ | = | \( -\frac{717825640026599866952}{254764560814329735} \) | = | $-1 \cdot 2^{3} \cdot 3^{-7} \cdot 5^{-1} \cdot 13^{-12} \cdot 37^{3} \cdot 120997^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.1200190038081716510224545043$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.0508524740541578232619077947$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0433838995113014$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.2345087338964635$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $25.971923317950300653629151261$ |
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| Real period: | $\Omega$ | ≈ | $0.038740872204537583613938048941$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 96 $ = $ 2\cdot2^{2}\cdot1\cdot( 2^{2} \cdot 3 ) $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.0370497730005740234628613401 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.037049773 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.038741 \cdot 25.971923 \cdot 96}{4^2} \\ & \approx 6.037049773\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1720320 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_0^{*}$ | additive | -1 | 5 | 9 | 0 |
| $3$ | $4$ | $I_{7}^{*}$ | additive | -1 | 2 | 13 | 7 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $13$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
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 |
|---|---|---|
| $2$ | 2B | 4.12.0.7 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 512 & 1557 \\ 515 & 1558 \end{array}\right),\left(\begin{array}{rr} 1081 & 8 \\ 1204 & 33 \end{array}\right),\left(\begin{array}{rr} 628 & 1 \\ 1271 & 6 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 1554 & 1555 \end{array}\right),\left(\begin{array}{rr} 1553 & 8 \\ 1552 & 9 \end{array}\right),\left(\begin{array}{rr} 1369 & 1368 \\ 598 & 1375 \end{array}\right),\left(\begin{array}{rr} 1368 & 593 \\ 187 & 174 \end{array}\right)$.
The torsion field $K:=\Q(E[1560])$ is a degree-$19322634240$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1560\Z)$.
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 | $4$ | \( 45 = 3^{2} \cdot 5 \) |
| $3$ | additive | $8$ | \( 160 = 2^{5} \cdot 5 \) |
| $5$ | split multiplicative | $6$ | \( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 18720.bk
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 6240.c4, its twist by $-3$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-30}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.2.11681280.5 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.191102976000000.11 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.54580920975360000.132 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.2.7255941120000.36 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | 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 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | add | split | ss | ord | split | ord | ord | ord | ord | ss | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | - | 4 | 1,1 | 1 | 2 | 1 | 1 | 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 | 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.